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% THE FIRST HALF OF ``Fall, Flow and Heat'', the first volume
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% The golden file check list is in part 6a
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% versione 9.130 full OED spellcheck Jul 2014
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% change ``allow(s/ed/ing) to'' to ``allow ..ing'' Jul 2014
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% changed all \lived to \livedplace not yet
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% every \emph (yes) and par (no) in index May 2004
% subject index: completed Dec 2016
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% triple exclamation marks: total 24, !.!1: (2), !.!2: (3) Sep 2016
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% professional editor (L. Osb.) Apr 2005
% professional editor II (Penny Sucharov) Apr 2006
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%  Add all solutions by the reader who wanted solution money in 2016
%  Add ``R. Müller, Kl. Mechanik: vom Weitsprung zum Marsflug''
% www.tubraunschweig.de/ifdn/physik/veröffentlichungen/buch
%  Add more magic tricks
%  Formelsammlungen einarbeiten
%  Add in the Earth segment image: convection in mantle and liquid core!
%  Add more machine and experiment pictures
%
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% Per la versione 9.90 (finale)
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%  Shorten text  too many words
%  Add professional index
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% (OK) For every concept in physics, explain (to interest biol, med, women,
% exper, etc.)
%  anecdote
%  appearance (table)
%  properties (other table)
%  effects (on humans, daily life)
%  limits (in appearance table)
%  biological scaling (graph)
%  sensors (photos, mix biological and technical)
%  measurement
%  records
%  technical applications
% (OK) Altlasten der Physik eingearbeitet (F. Herrmann)
% (OK) inverted pendulum and quantph/0102065
% (OK) Scaling of gait
% (OK) add citations by Agustinus, says Saverio.
% (OK) check volume references in all \seepaget (done for all volumes)
% (OK) add tensegrity image (never put in AA message)
% (OK) finish vignettes as in vol VI
% (OK) add a figure of wallpaper symmetries
% (OK) add a figure of crystal symmetries
% (OK) More lively! more daring! more fluid!
% (OK) More weird and queer stuff!
% (NO) clearer motivational boxed asides (NO, says Zedler)
% (NO) add: epikie, constructiver ungehorsam (or elsewhere?)
% (NO) Friedrich Christoph Oetinger \lived(17021782)
% (OK) force: the lie from Quito in Peru
% (OK) 4D thinking: how? (p.47)
% (NO) calorimetry
% (OK) add Indian rubbish movie site http://www.arvindguptatoys.com/
% (HALF) escape velocity table
%
% Dec 2016 9.13 many figures added, many corrections throughout
% Mar 2012 9.06 many topics and figures added
% Feb 2012 9.05 extensive correction of everything
% Sep 2011 9.00 rewritten atom section in 1b
% Aug 2011 9.00 start of final version
% Nov 2009 8.93 preparation for first lulu edition: 23.8
% Nov 2009 8.92 many small details, separate volume
% Nov 2008 8.90 summaries, new pictures, eliminated many triple excl. points
% Jun 2007 8.85 many new pictures, started eliminating triple excl. points
% Apr 2006 8.82 corrections by Sucharov, many details
% Jul 2005 8.77 corrections by Osborn, many details
% Jul 2003 8.72 reduced many triple esclamation marks; numerous details;
% checked all `blankblank' (changed to `blankblank')
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% May 2002 started adding Carole's corrections
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% Dec 2001 sent to USA
% introduced British spelling
% Oct 2001 prepared exercise structure
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% Summer 2001 improved motor table
% put indices in \rm
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\typeout{1a}
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% {La tavola delle materie specifica}
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\index{astronautsee{cosmonaut}}%
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\index{digits!Arabicsee{number system, Indian}}%
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%
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\index{gravitysee{gravitation}}%
\index{gravitation!universalsee{universal gravitation}}%
\index{gravitationseealso{universal gravitation}}%
\index{wave!watersee{water wave}}%
%
%
%
\partfigure[scale=0.4]{fblitzrotviolett2}%
%
% {scale=0.3}{An example of motion observed in
% nature ({\textcopyright}~\iinn{Harald Edens})}
% NO EMAIL  ALLOWED FORM THE WEBSITE OF NASA
% EMAILED FEB 2008  edens@weatherphotography.com
%
% Impr. Feb 2012
%
\partsubtitle{%
In our quest to learn how things move,\phantom{Wpygqj}\\%
the experience of hiking and other motion\phantom{Wpygqj}\\%
leads us to introduce the concepts of\phantom{Wpygqj}\\%
velocity, time, length, mass and temperature.\phantom{Wpygqj}\\%
We learn to use them to \emph{measure change}\phantom{Wpygqj}\\% % corrects a bug
and find that nature minimizes it.\phantom{Wpygqj}\\% % corrects a bug
We discover how to float in free space,\phantom{Wpygqj}\\% % vomit comet,
% % astronaut
why we have legs instead of wheels,\phantom{Wpygqj}\\%
why disorder can never be eliminated,\phantom{Wpygqj}\\% % third princ of thermo
and why one of the most difficult open issues\phantom{Wpygqj}\\%
in science is the flow of water through a tube.\phantom{Wpygqj}\\% % turbulence
%
% allowing us to discover limits to\\%
% speed, entropy, force and charge,\\%
% and thus to understand  among other things \\%
%
% why we have legs instead of wheels,\\%
% how empty space can bend, wobble and move,\\%
% what love has to do with magnets and amber,\\%
% and why we can see the stars.\\%
}%
%
\part[Fall, Flow and Heat]{Grawitacja, Płyny, Ciepło}
%
%
%
%
%\setcounter{chapter}{1} % cosi diventa il capitolo numero due
\pagestyle{nomovieheadings} % da non togliere
% Impr. Jan 2011, June 2014, reread Sep 2016
\chapter{Po co zajmować się ruchem?}
% Index ok
\markboth{\thesmallchapter\ po co zajmować się
ruchem?}{\thesmallchapter\ po co zajmować się ruchem?}
\begin{quote}
Wszelki ruch jest iluzją.\\
%Following
Zenon z Elei\footnote{Zenon z Elei (ok. 450r. p.n.e.),\indname{Zenon z Elei}
jeden z głównych przedstawicieli elejskiej szkoły filozoficznej.}
\end{quote}
\csini{T}{rach!} Piorun uderzający w pobliskie drzewo gwałtownie zakłóca naszą
% The word 'trach' can be replaced with 'bum' or 'łup', but I think
% that 'trach' is the most appropriate
cichą leśną wędrówkę, powodując że nasze serca zaczynają bić szybciej. W
górnych częściach drzew widzimy naprzemian pojawiający się i znikający ogień.
Delikatny wiatr poruszając liśćmi, pomaga nam przywrócić spokój tego
miejsca.\label{zenoill} W pobliżu płynie woda w małej rzece. Podąrzając
skomplikowaną ścieżką w dół doliny,odbijaja na swej powierzchni nieustannie
zmieniające się kształty chmur.
% \footnotetext{Since the final chapter is also the first, this is the second
% chapter.}
% Somebody falling
%
% or: the Thais falling from the little bus
%
% a puma jumping on a prey
%
% a cat jumping onto a bird
%
% a moving tree
%
% a tornado in the back and a tree in the foreground
%
% a stormy shore in the back and a tree in the foreground
%
% a colibri
%
% a whale jumping
Ruch jest wszędzie:\index{motion!is everywhere} groźny i przyjazny, piękny i
okropny. Jest fundamentem ludzkiej egzystencji. Ruch towarzyszy nam przy
rozwoju, nauce, myśleniu, pomaga utrzymać zdrowie % Jun 2017
i cieszyć się życiem. Korzystamy z ruchu podczas przechadzek po
lesie, % with our legs,
nasłuchiwania dochodzących z niego ogłosów % with our eardrums
i mówienia o tym wszystkim. Jak wszystkie zwierzęta, polegamy na ruchu by
zdobyć jedzenie i uniknąć niebezpieczeństw.
%
Podobnie jak wszystkie żywe istoty, potrzebujemy ruchu, aby się rozmnażać,\index{reproduction}
oddychać i trawić. Tak jak we wszystkich przedmiotach, ruch utrzymuje w nas ciepło.
Ruch jest najbardziej podstawowym obiektem\index{motion!is fundamental} obserwacji możliwej
możliwej do przeprowadzenia. Okazuje się, że \textit{wszystko} co się dzieje
na świecie to jakiś rodzaj ruchu. Nie ma wyjątków. Ruch jest tak
podstawową częścią naszych obserwacji, że nawet pochodzenie tego słowa jest zagubione w
ciemnościach indoeuropejskiej historii językowej. Fascynacja ruchem zawsze była
lubionym obiektem ciekawości.
%Already at the beginning of written thought, during
Do % sixth
% Jun 2005: changed to
piątego wieku p.n.e. in ancient Greece, w starożytnej Grecji, jego studium otrzymało
nazwę:\cite{russo} \ii{fizyka}.
%
% introduced by whom? sci.classics does not know
% age of word in OED?: not found first use
%
% by Democritus
% \lived(460370 {\bce}) or 356
% Aristotle \lived(384322 {\bce}) or by Epicuros
% \lived(341270 {\bce}), among the oldest conservative and humanist
% thinkers.
%
% EARLIER NAMING OF
% PHYSICS? OTHER PEOPLE? Democritus? SERGE AND GEORGES DO NOT KNOW.
Ruch jest także ważny\index{motion!is important} dla człowieka.
o możemy wiedzieć? Skąd pochodzi \iin{świat}? Kim jesteśmy? Skąd
pochodzimy?\index{origin!human} Co będziemy robić? Co powinniśmy robić?
Co przyniesie przyszłość? %Where do people come from? Where do they go?
Czym jest \iin[death!origin]{śmierć}? Dokąd życie prowadzi? Wszystkie te pytania
dotyczą ruchu. Badanie ruchu dostarcza odpowiedzi, które są równie głębokie, jak i
zaskakujące.
Ruch jest tajemniczy.\index{mystery!of motion}\index{motion!is mysterious}
Pomimo, że można go spotkać wszędzie  w gwiazdach, w pływach,\cite{hewex} w naszych
powiekach  ani starożytni myśliciele, ani miriady innych w ciągu 25
stuleci od tego czasu nie byli w stanie rzucić światła na centralną zagadkę:
\emph{Czym jest ruch?} Przekonamy się, że standardowa odpowiedź `ruch jest
zmianą miejsca w czasie', jest poprawna, ale niewystarczająca. Niedawno
znaleziono wreszcie pełną odpowiedź. To jest historia drogi do jej odnalezienia.
% Not only is the subject of motion fascinating and important; above all, it
% is vast. Its study resembles the exploration of a large unknown tropical
% island. Destiny, the waves of the sea, has carried us to its shore. But
% the size of the unknown jungle on the island is overwhelming. We wonder
% where to start, knowing that a whole lifetime does not suffice to explore it
% all.
\csepsfnb{imiland9}{scale=1.0}{Experience Island, with \protect\iin{Motion
Mountain} and the trail to be followed.}
Ruch jest częścią ludzkiego doświadczenia.\index{motion!is part of being human}
Jeśli wyobrazimy sobie ludzkie doświadczenie jako wyspę, to przeznaczenie, symbolizowane
przez fale morza, doprowadziło nas do brzegu. W pobliżu centrum
wyspy\indexs{Island, Experience}\indexs{Experience Island} wyróżnia się wysoka góra.
{}Ze jej szczytu widzimy cały krajobraz i dostrzegamy relację między
wszystkimi ludzkimi doświadczeniami, zwłaszcza pomiędzy różnymi
przykładami ruchu.\index{motion!manifestations}
%
To jest przewodnik na szczyt, który nazwałem \iin{Mt. Mountain  Góra Ruchu} (zobacz
\figureref{imiland9};
% Oct 2008, Jan 2015
mniej symboliczna i dokładniejsza wersja jest przedstwawiona na
\figureref{iphysicsstructure}).\seepageone{iphysicsstructure}
%
Trasa jest jedną z najpiękniejszych przygód ludzkiego umysłu.
Pierwsze pytanie\index{motion!manifestations}, które należy zadać:
%
% Impr. Jan 2011, Jul 2016
\subsection{Czy ruch istnieje?}
% Index OK
\begin{quote}\selectlanguage{german}\emph{Das Rätsel} gibt es nicht.
Wenn sich %\\
eine Frage überhaupt stellen läßt, %\\
so \emph{kann} sie beantwortet
werden.\selectlanguage{polish}\footnote{`\emph{Zagadka} nie istnieje.
Jeżeli w ogóle {można} zadać pytanie, to \emph{można} na nie również odpowiedzieć.'}\\
% Odgen translation
\iinn{Ludwig Wittgenstein}, \bt Tractatus/ 6.5\index{riddle}
\end{quote}
% \begin{quote}
% Any fool can ask more questions\\
% than seven sages can answer.\\
% %(Popular saying)
% \end{quote}
% %
% % Double
\np To sharpen the mind for the issue of motion's
existence,\index{motion!existence of} spójrz na\figureref{ifakerotationleft}
lub \figureref{irotsnake} i postępuj zgodnie z instrukcjami.
\cite{motillus} We wszystkich przypadkach obrazy zdają się
obracać. \index{motion!illusions, figures showing}\index{illusions!of motion}
% Apr 2006, impr. Jan 2011
YPodobnych efektów można doświadczyć przechadzając się po brukowanej nawierzchni,
która jest ułożona w łukowate wzory lub patrząc na liczne iluzje ruchu
zebrane przez \iname{Kitaoka Akiyoshi} na stronie
\url{www.ritsumei.ac.jp/~akitaoka}.\cite{kitaoka}
%
Jak możemy się upewnić, że prawdziwy ruch różni się od tych lub
podobnych\challengenor{motill} złudzeń?
Wielu uczonych twierdziło, że ruch nie istnieje w ogóle.\index{motion!does not exist}
Ich argumenty miały silny wpływ na badania
ruchu przez wiele stuleci.\cite{asoc}
% in the past and they still continue do so.
%
% The issues
% raised by them will accompany us throughout our adventure.
%
%
Przykładowo, grecki filozof Parmenides \iname[Parmenides of Elea]{Parmenides}
(urodzony około 515r. p.n.e. w Elei, małym mieście niedaleko
Neapolu) %, in southern Italy)
argumentował, że skoro nic nie pochodzi z niczego, zmiana nie może mieć miejsca.
Podkreślał \emph{stałość} natury i konsekwentnie utrzymywał, \index{permanence!of nature}
że wszelkie zmiany, a tym samym ruch, są \iin[motion!as an illusion]{iluzją}.\cite{a5} %
\cstepsfnb{ifakerotationleft}{scale=1}%
{ifakerotationright}{scale=0.9}{Złudzenia ruchu: patrz na grafikę po lewej,
delikatnie poruszając kartką lub patrz na białą kropkę na środku obrazka po prawej,
poruszając głową do tyłu i do przodu.}
\inames{Heraklit} \livedca(ok. 540r.ok. 480r. p.n.e.) utrzymywał przeciwny
pogląd. Wyraził to w swoim słynnym stwierdzeniu % {%\tau\acute\upsilon\
%\pi\acute\alpha\nu\tau\alpha\ \varrho\epsilon\acute\iota}
%
\csgreekok{p'anta {\PBS\raggedright}p{64mm}%
@{\extracolsep{\fill}}>{\PBS\raggedright}p{70mm}@{}}
%
\toprule
%
\tabheadf{Motion topics} & \tabhead{Motion topics} \\[0.5mm]
%
\midrule
%
%
motion pictures and digital effects & motion as therapy for cancer, diabetes,
acne and depression\\
motion perception \cite{percres} & motion sickness\\ motion for fitness and
wellness& motion for meditation\\
%
motion control and training in sport and
singing &
motion ability as health check\\ perpetual motion & motion in dance, music
and other performing arts\\ motion as proof of various gods \cite{a6} & motion
of planets, stars and \iin{angels} \cite{suaq}\\
economic efficiency of motion &
the connection between motional and emotional habits\\
% Jun 2005
motion as help to overcome trauma & motion in psychotherapy \cite{ilikeit}\\
%
locomotion of insects, horses, animals and robots & motion of cells and
plants\\
collisions of atoms, cars, stars and galaxies & growth of multicellular
beings, mountains, sunspots and galaxies\\
motion of springs, joints, mechanisms, liquids and gases & motion of
continents, bird flocks, shadows and empty space \\
commotion and violence& motion in martial arts\\ motions in
parliament & movements in art, sciences and politics\\
%
movements in watches & movements in the stock market\\
movement teaching and learning & movement development in children
\cite{pikler}\\
musical movements & troop movements \cite{chandlernap}\\
religious movements & bowel movements \\ moves
in chess & cheating moves in casinos \cite{marcus}\\
\multicolumn{2}{@{}l}{connection between gross national product and citizen
mobility}\\
%
%
\bottomrule
\end{tabular*}
\end{table}
}
\np Dobrym miejscem do zrobienia ogólnego przeglądu rodzajów ruchu jest wielka
biblioteka.\label{laglibll} \tableref{molitab} pokazuje jego rezultaty.
Dziedziny, w któych ruch gra główną rolę, są rzeczywiście zróżnicowane.
% One can find
% books on motion pictures and on motion therapy, on motion perception and on
% motion sickness, on motion for fitness and for meditation, on perpetual
% motion, on motion in dance, in music and in the other arts, on motion as
% proof
% of existence of various gods,\cite{a6} on motion of insects, horses, robots,
% stars and \iin{angels}, on the economic efficiency of motion, on emotion,
% locomotion and commotion, on motions in parliament, on movements in art, in
% the sciences and in politics, on movements in watches and in the stock
% market,
% on movement teaching and learning, on movement development in children, on
% musical and on troop movements, on religious and on bowel movements, on
% moves
% in chess, and many more.
%
Już antyczni myśliciele z antycznej Grecji  zebrani na
\figureref{itimeline}  podejrzewali, że wszelkie rodzaje ruchu, tak jak
inne rodzaje zmiany są powiązane.\index{motion!types
of}\index{motion!and change} Powszechnie ruch\index{change!types of}
\label{greekchanzz} dzieli się na trzy kategorie:\label{threemotiontypes}
\cssmallepsfnb{fvolcano2}{scale=0.9}{An example of transport, at Mount Etna
({\textcopyright}~\protect\iinn{Marco Fulle}).} % (OK) not referenced
\begin{Strich} % (NO) old style figures in items?
\item[{1.}] \emph{Transport}.\index{transport} Rodzaj zmiany, który na codzień
nazywamy ruchem to {transport} matrialny, np. podczas chodzenia, liść spadający
z drzewa, czy grający instrument myzyczny. Transport jest zmianą pozycji lub
orientacji obiektów, włacznie z płynami. Ta kategoria ma szeroki zakres, obejmujący
również \iin{zachowanie} ludzi.
\smallskip
\item[{2.}] \emph{Transformacja}.\index{transformation!of matter and bodies}
Inną rodzają zmiany jest na przykład wytącanie soli z wody, formowanie się lodu
podczas zamarzania, gnicie drewna,
% not putrefaction of wood, that is for protein only
gotowanie potraw, krzepnięcie krwi, topnienie i
stapianie metali. Te zmiany koloru, jasności, twardości, temperatury i innych właściwości
są nazywane {transformacjami}. Do tej kategorii, kilku antycznych myślicieli dodało
emisję i absorbcję światła. W dwudziestym wieku, te zjawiska zostały potwiedzone
jako specjalne rodzaje transformacji, tak samo jak nowo odkryte pojawianie się i
znikanie materii, obserwowanej w zjawiskach na Słońcu oraz w radioaktywności.
\ii[mind!change]{Zmiana myślenia}, taka jak zmiany humoru, zdrowia, uczenie się,
czy zmiana charakteru jest także (zazwyczaj) odmianą transformacji.\cite{a7}
\smallskip
\item[{3.}] \emph{Wzrost}.\index{growth} Ostatnia i szczególnie ważna kategoria zmiany\cite{a7before},
zauważalna podczas obserwacji zwierząt, roślin, kryształów, gór, planet, gwiazd, a nawet
galaktyk. W XIX wieku, zmiany populacji w systemach\ii[evolution!biological]{ewolucja biologiczna},
i w XX wieku, zmiana rozmiarów wszechświata\ii[evolution!cosmic]{cosmic
evolution}, zostały dodane do tej kategorii. Tradycyjnie, ten fenomen był analizowany
przez różnych naukowców. Wszyscy doszli niezależnie do wniosku, że wzrost jest połączeniem
transportu i transformacji. Różnicą jest jego złożoność oraz skala czasowa.
\end{Strich}
\np W początkach nauki, podczas renesansu, wyłącznie transport był zagadnieniem fizycznym.
Ruch jednoznacznie oznaczał transport. Pozostałe kategorie były zaniedbywane przez fizyków.
Pomimo tego ograniczenia, ten obszar pokrywa dużą część \iin{Wyspy doświadczenia}.
\seepageone{imiland9} Pierwsi uczeni rozróżniali transport ze względu na jego pochodzenie.
Ruchy takie jak, porszanie nogami podczas chodzenia zostały zakwalifikowane jako
\ii[motion!volitional]{wolicjonalny}, ponieważ są kontrolowane przez czyjąś wolę,
podczas gdy ruchy zewnętrznych ciał, jak opady śniegu, na które czyjaś wola nie ma wpływu,
zostały zakwalifikowane jako \ii[motion!passive]{pasywne}.
% Mar 2012
Młodzi ludzie, szczególnie płci męskiej, spędzają dużo czasu na opanowaniu
wolicjonalnych ruchów, co pokazuje \figureref{itsukanov}.
Pełne rozróżnienie pomiędzy pasywnym, a wolicjonalnym ruchem jest robione przez
dzieci przed 6 rokiem życia, i oznacza to centralny krok w rozwoju każdego
człowieka w kierunku precyzyjnego opisu środowiska naturalnego.%
%
\footnote{Niepowodzenie na tym etapie może spowodować, że osoba ma przeróżne
dziwne przekonania, takie jak wiara w zdolność do wpływania na kulki ruletki,
co można spotkać u kompulsywnych graczy,\index{play} lub w zdolność do poruszania innymi
ciałami za pomocą myśli, jak to ma miejsce u wielu innych zdrowo wyglądających ludzi.
\index{psychokinesis} Zabawne i pouczające sprawozdanie o wszystkich oszustwach i oszustwach
związanych z tworzeniem i utrzymywaniem tych wierzeń\index{belief!collection} zostało wydane przez
\asi Jamesa Randi/ \bt The Faith Healers/. Profesjonalny magik, przedstawia wiele podobnych tematów
w kilku innych swoich książkach. Więcej informacji można znaleźć na jego stronie internetowej \url{www.randi.org}.}%
%
{}Z tego rozróżnienia wynika historyczna, ale już przestarzała definicja
\index{physics!outdated definition} fizyki jako nauki o ruchu rzeczy
nieożywionych. % !!!4 add a reference one day
\cssmallepsfnb{iplissonbook}{scale=0.52}{Transport, wzrost i transformacja ({\textcopyright}~\protect\iinn{Philip Plisson}).}
% (OK) not referenced
\cssmallepsfnb{itsukanov}{scale=1}{Jeden z najtrudniejszych znanych ruchów
wolicjonalnych, wykonany przez \protect\iinn{Alexandra Tsukanova}, pierwszego
zdolnego do tego człowieka:
% Zukanov on the internet, Tsukanov in Montecarlo
skakanie z jednego \protect\iin[wheel!ultimate]{ultimate wheel} na
drugie({\textcopyright}~\protect\iname{Moscow State Circus}).}
% How many types of motion of different speed can you find on
% \figureref{iplissonbook}?\challengenor{pliss}
%
% {pliss} Wind; sea waves; growth of grass; building of light tower;
% erosion of coast; rise of continental crust.
Pojawienie się maszyn zmusiło naukowców do ponownego przemyślenia
rozróżnienia pomiędzy ruchem wolicjonalnym i biernym. Podobnie jak istoty żywe,
maszyny same się poruszają, a więc naśladują ruchy wolicjonalne. Jednak
uważna obserwacja pokazuje, że każda część maszyny jest przenoszona przez inną,
więc ich ruch jest w rzeczywistości bierny. Czy żywe istoty są również maszynami?
Czy ludzkie działania są również przykładem ruchu biernego?
Nagromadzenie obserwacji w ciągu ostatnich 100
lat dało jasno do zrozumienia, że ruch wolicjonalny%
%
\comment{\footnote{The word `{movement}' is rather modern;\indexs{movement} it was
imported into English from the old French and became popular only at the end
of the eighteenth century. It is never used by \iname[Shakespeare,
William]{Shakespeare}.}} %
%
rzeczywiście ma te same właściwości fizyczne, co bierny ruch w systemach
nieożywionych. Rozróżnienie pomiędzy dwoma rodzajami ruchu jest zatem niepotrzebne.
Oczywiście, z emocjonalnego punktu widzenia różnice są ważne; na przykład,
\ii{wdzięk} można przypisać tylko ruchom wolicjonalnym.\cite{a8}
Ponieważ ruch bierny i ruch wolicjonalny mają te same właściwości, poprzez
badanie ruchu nieożywionych obiektów możemy nauczyć się czegoś o ludzkich
warunkach. Jest to najbardziej widoczne, kiedy poruszamy tematy
determinizmu% (\cspageref{deter})
, przyczynowości%(\cspageref{causal})
, prawdopodobieństwa%(\cspageref{probab})
, nieskończoności%(\cspageref{infin} and \pageref{infin2})
, czasu%(\cspageref{timeaway})
, miłości i śmierci%
, aby wymienić tylko kilka tematów, które napotkamy podczas naszej przygody.%
W XIX i XX wieku inne klasyczne przekonania o ruchu, zeszły na manowce.
% , even more of the historical restrictions on the study of motion
% were put into question.
Rozległe obserwacje wykazały, że wszystkie transformacje i wszystkie
zjawiska wzrostu, w tym zmiana zachowań i ewolucja, są również przykładami
transportu. Innymi słowy, ponad 2\,000 badań wykazało, że starożytna
klasyfikacja obserwacji była bezużyteczna::
%
\begin{quotation}
\noindent \csrhd {Każda zmiana jest transportem.}\index{change! and
transport}\index{transport!and change}
\end{quotation}
%
\np Oraz
%
\begin{quotation}
\noindent \csrhd {Transport i ruch są tym samym.}\index{motion!is
transport}\index{transport!is motion}
\end{quotation}
%
\np W połowie\index{motion!is due to particles} XX wieku badania nad ruchem
zakończyły się eksperymentalnym potwierdzeniem jeszcze bardziej konkretnej
idei, uprzednio artykułowanej w starożytnej Grecji:
\begin{quotation}
\noindent \csrhd {Każdy rodzaj zmiany wynika z ruchu cząstek.}
\end{quotation}
\np Potrzeba czasu i pracy, aby dojść do tego wniosku, który pojawia się
tylko wtedy, gdy bezustannie dążymy do coraz większej precyzji w opisie
natury. Pierwsze pięć części tej przygody odtwarza ścieżkę do tego celu.
(Zgadzasz się z tym?)\challengenor{motvac}
Ostatnia dekada XX wieku ponownie całkowicie zmieniła opis ruchu:
pomysł cząstek okazuje się ograniczony i błędny.
Ten niedawny rezultat, osiągnięty dzięki połączeniu uważnej obserwacji i dedukcji,
%, already suggested by advanced quantum theory,
zostanie zbadany w ostatniej części naszej przygody. Przed nami jednak jeszcze
długa droga do osiągnięcia tego rezultatu, tuż przed szczytem naszej podróży.
Podsumowując, historia pokazała, że klasyfikacja różnych rodzajów ruchu nie jest
produktywna. Tylko dążąc do osiągnięcia maksymalnej precyzji możemy mieć nadzieję,
że dojdziemy do podstawowych właściwości ruchu. \emph{Precyzja, nie zaś
klasyfikacja, jest drogą, którą należy podążać.} Jak żartobliwie powiedział
\iinnq{Ernest Rutherford}: `Cała nauka to albo fizyka, albo zbieranie znaczków.'
Aby osiągnąć precyzję w naszym opisie ruchu, musimy wybrać konkretne przykłady
ruchu i dokładnie je przestudiować. Jest to intuicyjnie oczywiste, że najdokładniejszy
opis można uzyskać w\emph{najprostszych} możliwych przykładach.\index{motion!simplest}
W życiu codziennym dotyczy to ruchu każdego nieożywionego, solidnego i sztywnego
ciała w naszym otoczeniu, takiego jak \iin[stones]{kamień} wyrzucany w powietrze.
Rzeczywiście, jak wszyscy ludzie, nauczyliśmy się rzucać\index{throw!and motion}
przedmiotami na długo zanim nauczyliśmy się chodzić.\cite{childev} Rzucanie jest
jednym z pierwszych eksperymentów fizycznych, które sami przeprowadziliśmy. %
%
%{\footnote{Read: Agostino, Il maestro e la parola, Rusconi, ed.
%Maria Bettetini. Spiega le conseguenze dell'atto del passeggiare. Eco
% dice: una delizia.}} % non è il mio parere
%
% Footnote out in August 2013:
%
Znaczenie \iin[throwing!importance of]{rzucania} jest również widoczne na
podstawie pochodzących od niego określeń: w łacinie \emph{temat} pochodzi od 'rzucony w dół',
\emph{obiekt} od 'rzucony z przodu', wtrącenie od \emph{wrzucony pomiędzy}; w grece akt rzucania
jest widoczny w słowach takich jak \emph{symbol} ('wrzucony razem'), \emph{problem} ('rzucony od przodu'),
\emph{emblemat} ('wrzucony do'), a nawet \emph{diabeł} ('wyrzucony').
%
%
I rzeczywiście, we wczesnym dzieciństwie, rzucając kamieniami, zabawkami i
innymi przedmiotami, aż nasi rodzice obawiali się o każdy kawałek domu, badaliśmy
percepcję i właściwości ruchu. To samo robimy tutaj.
% During our early childhood, by throwing stones and similar objects until our
% parents feared for every piece of the household, we explored the properties
% of motion; first of all we learned that in order to describe and to
% understand motion we needed to distinguish \ii{permanent} aspects, such as
% objects and images, and \ii{variable} aspects, such as dimensions, position
% and instants.
\begin{quoteunder}\selectlanguage{german}Die Welt ist unabhängig von
meinem Willen.\selectlanguage{polish}%
\footnote{`Świat jest niezależny od mojej woli.'}\\ % Odgen translation
Ludwig Wittgenstein, \bt Tractatus/ 6.373\indname{Wittgenstein, Ludwig}
\end{quoteunder}
%
% Impr. July 2016
\subsection{Percepcja, trwałość i zmiana}
% Index OK
\begin{quote}
Tylko mięczaki badają jedynie ogólny przypadek; prawdziwi naukowcy poszukują przykładów.\\
\iinn{Beresford Parlett}
\end{quote}
\np Ludzie\label{pppiiittt} czerpią przyjemność z postrzegania. Percepcja zaczyna
się przed narodzinami i nadal cieszymy się nią tak długo, jak możemy. To dlatego
\iin{telewizja}, nawet pozbawiona treści devoid of content, odnosi taki sukces.
Podczas naszej wędrówki po lesie u podnóża Góry Ruchu nie obędzie się bez percepcji.
Percepcja to przede wszystkim umiejętność \ii{rozróżniania}. Korzystamy z tej podstawowej
zdolności w niemal każdej chwili życia;
na przykład w dzieciństwie nauczyliśmy się najpierw rozróżniać {znane} rzeczy
od {nieznanych}.\index{familiarity} Jest to możliwe w połączeniu z inną
podstawową umiejętnością, a mianowicie umiejętnością \ii[memory]{zapamiętywania}
doświadczeń. Pamięć pozwala nam doświadczać, rozmawiać, a tym samym badać
przyrodę. Postrzeganie, klasyfikowanie i zapamiętywanie razem tworzą\ii{naukę}.
Bez żadnej z tych trzech umiejętności nie moglibyśmy studiować ruchu.
%But let us continue. Using the senses together with one's memory,
Dzieci szybko uczą się odróżniać \ii{trwałość} od \ii{zmienności}.
% aspects.
Uczą się \emph{rozpoznawać} ludzkie\index{recognition} twarze, mimo że nigdy nie
wyglądają dokładnie tak samo za każdym razem, gdy są widoczne. Od rozpoznawania
twarzy, dzieci rozszerzają rozpoznawanie na wszystkie inne obserwacje. Rozpoznanie
działa całkiem dobrze w życiu codziennym; dobrze jest rozpoznawać przyjaciół
nawet w nocy, i nawet po wielu piwach (to nie jest wyzwanie). Akt
\iin{rozpoznawania} {zawsze} używa więc formy \emph{uogólnienia}. Kiedy
obserwujemy, zawsze mamy jakieś ogólne wyobrażenie w naszym umyśle. Określmy główne z nich.
% Later
% on in our walk we will encounter situations where this generalization cannot
% be applied, and where thus recognition is not possible at all.
% {}From the beginning of their life, humans and most other animals apply
% these abilities first of all to motion itself. Motion being of central
% importance to animal life, evolution took care that all animals developed
% effective means to perceive movements, both of themselves and of the bodies
% around them. For example, when we take a stone and throw it against a tree,
% we can follow its path with several \iin{senses}. We can use the \ii{eyes}
% to watch it, the \ii{ears} to listen to the sounds produced when it flies
% through the air; during the time the stone is still in our hand, we can
% follow it with our \ii{kinesthetic sense}, i.e.,{} the sense that provides
% information about the position of the arm and the hand relative to the body;
% it is this sense which gives us the possibility to \ii[aim, ability to]{aim}
% for a certain spot. Additionally, we can use the \ii{sense of equilibrium},
% which allows us to follow the position of our head relative to the surface
% of the Earth, in order to avoid to fall over after the launch. The
% \ii{sense of touch} allows us to tell when the stone\index{stones} leaves
% the hand. Other senses, like the \iin{sense of coldness}, the \iin{sense of
% warmness}, \iin{smell}, or \iin{taste} have perhaps played a role in the
% preparation of the throw.
% All this is common knowledge and experience. However, our walk through the
% details of motion will show that perception is severely limited; evolution
% has optimised the human perception system for survival and for reproduction,
% not for understanding nature. The study of motion will show that the
% intrinsic limitations of any perceiving system, also the human one, are the
% reason that quantum mechanics is so difficult to understand. Moreover, our
% basic prejudice that classification of observation is always possible will
% be confronted by some hard to swallow counterexamples. We will discover
% that the simple fact that we possess memory is one of the main reasons that
% motion is so difficult to describe with precision.
% transition
Siedząc na trawie, na\iin{leśnej} polanie u dołu Mt. Mountain,
w otoczeniu drzew i ciszy typowej dla takich miejsc, ogarnia nas uczucie spokoju.
Myślimy o istocie percepcji. Nagle coś porusza się w krzakach; natychmiast
nasze oczy się obracają, a nasza uwaga wytęża się.
% All these reactions are built into our body.
Komórki nerwowe, które wykrywają ruch, są częścią najstarszej części naszego
mózgu, dzielonej z ptakami i gadami: the \iin[brain!stem]{pniem mózgu}.\cite{reptibrain}
% (The brain stem also controls all our
% involuntary motions, such as the blinking of the eyes.)
Następnie \iin{kora mózgu}, przejmuje kontrolę rodzaju ruchu i określa jego pochodzenie.
%
%\comment{Space, spatiu(m), perhaps connected to pat\=ere, to be open}%
%
Obserwując ruch w naszym polu widzenia, obserwujemy dwie jednostki:
nieruchomy krajobraz i poruszające się zwierzę. Po rozpoznaniu zwierzęcia
jako \iin{jelenia}, wracamy do odpoczynku.
% May 2007
\cssmallepsfnb{ideer}{scale=1}{Jak rozróżniamy jelenia od jego otoczenia?
({\textcopyright}~\protect\iinn{Tony Rodgers}).}
Jak odróżniliśmy, w przypadku \figureref{ideer}, krajobraz od jelenia?
Percepcja obejmuje kilka procesów zachodzących w oczach i mózgu.
% Several steps in the eye
% and in the brain are involved.
%
Zasadniczą częścią tych procesów jest ruch, co najlepiej można wywnioskować
%from the most difficult cases, such as
z \iin{kineografu} pokazanego w kilku lewych dolnych rogach stron tej książki.
\cite{zank} Każdy\index{figures!in corners}\index{pattern!in corners}
obraz przedstawia tylko prostokąt wypełniony matematycznie
\iin[pattern!random]{przypadkowym wzorem}.\index{corner film!lower left} Kiedy jednak strony są
widziane w krótkim ostępie czasu, rozpoznaje się kształt  kwadrat 
poruszający się na niezmiennym tle. Na pojedynczej stronie nie można odróżnić kwadratu od tła;
nie ma widocznego obiektu w żadnej konkretnej chwili. Niemniej jednak łatwo jest dostrzec jego ruch.%
%
\footnote{Ludzkie \iin{oko} jest raczej dobre w wykrywaniu ruchu. Na przykład,
oko może wykryć ruch punktu świetlnego, nawet jeśli zmiana kąta jest mniejsza od tego,
który można odróżnić na stałym obrazie. Szczegóły tego i podobnych tematów dla innych
zmysłów są domeną badań percepcji.\cite{percres}} %
%
Eksperymenty percepcyjne, takie jak ten, były przeprowadzane na wiele sposobów.
Eksperymenty takie wykazały, że wykrywanie poruszającego się kwadratu na przypadkowym
tle nie jest niczym szczególnym dla człowieka; \iin{muchy} mają taką samą zdolność,
jak w rzeczywistości wszystkie zwierzęta, mające \iin[eye]{oczy}.
\iin{Kineograf} w \iin[corner film!lower left]{lewym dolnym rogu},
jak wiele podobnych eksperymentów, ilustruje dwa główne atrybuty ruchu.
Po pierwsze, ruch jest postrzegany tylko wtedy, gdy \ii{obiekt} można odróżnić od
\ii{tła} lub \ii{otoczenia}. Wiele złudzeń ruchu skupia się na tym punkcie.%
%
\footnote{Temat percepcji ruchu jest pełen ciekawych aspektów. Doskonałym wprowadzeniem do
tegoż teamtu jest rozdział 6 książki autorstwa \asi Donald D.
Hoffmana/ \bt Visual Intelligence  How We Create What We See/ W.W. Norton
\& Co., \yrend 1998/ Jego zbiór podstawowych złudzeń ruchu może być doświadczany i eksplorowany na powiązanych stonach: % two words added sep 2003
% \url{www.socsci.uci.edu/~ddhoff} % Impr. July 2010, also ok
\url{www.cogsci.uci.edu/~ddhoff} % Impr. July 2010
.}
%
Po drugie, ruch jest niezbędny do \emph{zdefiniowania} zarówno obiektu,
jak i otoczenia oraz do odróżnienia ich od siebie.
%
\label{enviobj}%
%
Pojęcie przestrzeni jest w istocie m.in. abstrakcją idei tła. Tło jest przedłużone;
ruchoma jednostka jest zlokalizowana. Czy to wydaje się nudne? Nie jest; po prostu czytaj dalej.
% Aug 2005, Mar 2014, June 2014
Nazywamy zlokalizowany podmiot badawczy, który można zmienić lub przenieść
\ii[system!physical]{systemem fizycznymm}  lub po prostu systemem. System jest rozpoznawalną,
a więc trwałą częścią natury. Systemy mogą być obiektami  zwanymi również "ciałami fizycznymi" 
lub promieniowaniem.\index{radiation!as physical system} Dlatego obrazy\index{image!definition},
które są zrobione z promieniowania, są aspektami systemów fizycznych, same w sobie zaś nie są systemami fizycznymi.
Powiązania te podsumowano w \tableref{bascon}. Czy więc \iin[hole!system or
not?]{dziury} są systemami fizycznymi\challengenor{holephsys}?
Innymi słowy, nazywamy zbiór zlokalizowanych aspektów, które pozostają niezmienne lub stałe podczas ruchu,
takie jak rozmiar, kształt, kolor itp., razem dając (fizyczny) \ii[object!defintion]{obiekt} lub (fizyczne)
\ii[body!definition]{ciało}. Wkrótce dopracujemy definicję, aby odróżnić obiekty od obrazów.
%
% In personal history, the
% understanding of permanence is completed only at the age of about six
% years.
% In other words, at six we finish learning that \ii{objects} are those
% permanent parts of nature which move against a permanent background.
Zauważamy, że aby określić stale ruchome obiekty, musimy je odróżnić od otoczenia.
Innymi słowy, od samego początku doświadczamy ruchu jako procesu \ii[motion!is relative]{względnego};
jest postrzegany w relacji i w opozycji do środowiska.
% The concept of an object is therefore also, at least partially, relative.
% Impr. Jan 2015
Ważne jest koncepcyjne rozróżnienie pomiędzy zlokalizowanymi, odizolowanymi obiektami i rozszerzonym środowiskiem.
Prawda, wydaje się to\challengenor{circdef} błędnym kołem. (Zgadzasz się z tym?) Rzeczywiście,
wytłumaczenie tej kwesti chwilę zajmie\seepageone{strangesumm}. Z drugiej strony, jesteśmy tak przyzwyczajeni
do naszej zdolności izolowania lokalnych systemów od środowiska, że uznajemy to za oczywiste. Jednak,
jak odkryjemy później w naszej wędrówce, to rozróżnienie okazuje się logicznie i eksperymentalnie
% in the last part of
\seepagesix{objenvi} niemożliwe!%
%
\footnote{W przeciwieństwie do tego, co często można przeczytać w literaturze popularnonaukowej, rozróżnienie to
\emph{jest} możliwe w \iin{teorii kwantowej}. Staje się to niemożliwe tylko wtedy, gdy teorię kwantową połączy się z ogólną teorią względności.} %
%
Powód tej niemożności okaże się fascynujący. Aby odkryć niemożliwość, w pierwszym kroku zauważamy,
że oprócz poruszających się podmiotów i stałego tła, musimy także opisać ich relacje.
Niezbędne koncepcje podsumowano w \tableref{bascon}.
%
% entity=object or image
%
% object = permanent, moving against background
%
% permanent = boundary in motion ...
\begin{quoteunder}
Mądrość jest jedną rzeczą: zrozumieniem myśli, która kieruje wszystkim przez wszystkie rzeczy.\\
Heraclitus of Ephesus\cite{herharm}\indname{Heraklit z Efezu}
\end{quoteunder}
%
% {Table of main physical concepts}
%
\begin{table}[t]\small\centering\dirrtabularnolines
% {\hbox{{\ \ \ \ }}}
\caption[Family tree of the basic physical concepts.]{Drzewo genealogiczne\protect{\index{tree!family, of physical concepts}}
podstawowych pojęć fizycznych.\label{bascon}}
%
% The figure is related to the text dimensions!
% \ \hbox to 20mm{\hss\vbox to
% 0pt{\vskip8mm\csepsffile{iworldtree1}\vss}\hss}\ \\
% since Aug 2001 changed to:
%
\ \hbox to 5mm{\hss\vbox to
0pt{\vskip 12.2mm\csepsffile{iworldtree1}\vss}\hss}\ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \\
% was 8.7 without the rules
%
\dirrtabularstarnolines
%
\begin{tabular*}{\textwidth}{@{\hspace{0em}}
p{21mm}@{\extracolsep{\fill}}%
p{21mm}@{\extracolsep{\fill}}%
p{21mm}@{\extracolsep{\fill}}%
p{21mm}@{\extracolsep{\fill}}%
p{21mm}@{\extracolsep{\fill}}%
p{21mm}@{\hspace{0em}}}
%
\toprule
\multicolumn{6}{@{\hspace{0em}}c}{\ii{ruch}}\\
\multicolumn{6}{@{\hspace{0em}}c}{podstawowy rodzaj zmiany}\\ %
%
\ \\
%
\multicolumn{2}{@{\hspace{0em}}c}{\ii[part!of systems]{elementy/systemy}} &
\multicolumn{2}{@{\hspace{0em}}c}{\ii[relation!among nature's parts]{relacje}} &
\multicolumn{2}{@{\hspace{0em}}c}{\ii{tło}}\cstabhlinedown \\
%
\multicolumn{2}{@{\hspace{0em}}c}{stałe} &
\multicolumn{2}{@{\hspace{0em}}c}{} &
% \multicolumn{2}{@{\hspace{0em}}c}{variable} & % !.!1 bad says a reviewer,
% % opposite also true
\multicolumn{2}{@{\hspace{0em}}c}{mierzalne} \\
%
\multicolumn{2}{@{\hspace{0em}}c}{ograniczone} &
% \multicolumn{2}{@{\hspace{0em}}c}{unbounded} & % !.!1 bad says a reviewer,
% % opposite also true
\multicolumn{2}{@{\hspace{0em}}c}{tworzą granice} &
\multicolumn{2}{@{\hspace{0em}}c}{nieograniczone} \\
% \multicolumn{2}{@{\hspace{0em}}c}{extended} \\
%
\multicolumn{2}{@{\hspace{0em}}c}{posiada kształt} &
\multicolumn{2}{@{\hspace{0em}}c}{tworzą kształty} &
\multicolumn{2}{@{\hspace{0em}}c}{obszerne} \\
%
\ \\
%
\ii[object]{obiekty} & \ii{promieniowanie} & \ii[state]{stany} &
\ii[interaction]{interakcje} &
\ii{przestrzeń fazowa} & \ii{czasoprzestrzeń}\cstabhlinedown \\
%
nieprzepuszczalny & przenikalny & globalne & lokalne & złożona &
prosta\cstabhlinedown\\
%
\multicolumn{3}{@{\hspace{0em}}l}{Powiązane aspekty:\cstabhlineup\cstabhlinedown} \\
%
masa\cstabhlineup & intensywność & niezwłoczność & źródło & wymiar & krzywizna \\
rozmiar & kolor & pozycja & domena & dystans & topologia\\
ładunek & obraz & pęd & siła & objętość & dystans\\
spin & wygląd & energia & kierunek & podprzestrzeń & obszar\\
itp. & itp.\ & itp.\ & itp.\ & itp.\ & itp.\ \\ %
%
\ \\
%
\multicolumn{6}{@{\hspace{0em}}c}{\ii{świat}  \ii{natura} 
\ii{wrzechświat}  \ii{kosmos}}\cstabhlinedown\\
%
\multicolumn{6}{@{\hspace{0em}}c}{\rm zbiór wszystkich części, relacji i tła}\\
%
\bottomrule
%
\end{tabular*}
%
%
%\hfil\break (the last three concepts are not distinguished
%in this text; see \cspageref{threeconc}) %
%
\end{table}\index{evolution}\index{variation}
%
%
% Impr. Mar 2014
\subsection{Czy świat potrzebuje stany?}
% Index OK
\begin{quote}\selectlanguage{german}%
Das Feste, das Bestehende und der Gegenstand sind Eins. Der Gegenstand ist
das Feste, Bestehende; die Konfiguration ist das Wechselnde,
Unbeständige.\selectlanguage{polish}%
%
\footnote{`Stałość, istnienie i obiekt są jednością. Obiekt jest nieruchomy, istnieje; konfiguracja jest zmienną.'} % Odgen translation
%
\\
Ludwig Wittgenstein, \bt Tractatus/ 2.027 
2.0271\indname{Wittgenstein, Ludwig}
\end{quote}
\label{statedef}%
%
\np Co odróżnia różne wzory w lewych dolnych rogach\index{corner film!lower left}
\index{film!lower left corner} tej książki? W życiu codziennym powiedzielibyśmy: {sytuacja} lub
{konfiguracja} zaangażowanych podmiotów. {Sytuacja} jakoś opisuje wszystkie te aspekty,
które mogą się różnić w zależności od przypadku. Powszechnie przywołuje się listę wszystkich
\emph{zmiennych} aspektów zestawu obiektów \ii[state!of
motion]{(fizycznego) stanu ruchu} lub po prostu ich \ii[state!physical]{stanu}. Jak stan jest scharakteryzowany?
Poszczególne klatki kineografu różnią się przede wszystkim w
\emph{czasie}. Czas sprawia, że przeciwieństwa są możliwe: dziecko jest w domu, a to samo dziecko znajduje się poza domem. % from Serge
Czas opisuje i rozwiązuje tę sprzeczność. Ale stan nie tylko rozróżnia sytuacje w czasie: stan zawiera \emph{wszystkie}
aspekty \emph{systemu}  np. grupy obiektów\index{system!definition}  które odróżniają go od wszystkich \emph{odobnych}
systemów. Dwa podobne obiekty mogą się różnić w każdej chwili pod względem
%
\begin{Strich}
\item położenia,
\item prędkości,
\item orientacji lub
\item prędkości kątowej.
\end{Strich}
%
Właściwości te determinują stan\index{state!allows to distinguish} i wskazują
\ii{indywidualność} systemu fizycznego wśród jego \emph{dokładnych kopii}.
Stan ten opisuje również relację obiektu lub systemu w stosunku do jego otoczenia. Lub równoważnie:
%
\begin{quotation}
\noindent \csrhd \iin[state!definition]{\emph{Stan} opisuje wszystkie aspekty systemu,
które zależą od obserwatora.}
\end{quotation}
%
\np Definicja stanu nie jest wcale nudna  rozważmy chociby to: Czy \emph{wrzechświat} ma stan?
\challengenor{unist} Albo: czy powyższa lista właściwości stanu jest \emph{kompletna}?
% Mar 2014
Ponadto, fizyczne systemy są opisane przez ich stałe,
\emph{wewnętrzne właściwości}, którymi przykładami są\index{property!intrinsic}
%
\begin{Strich}
\item masa,
\item kształt,
\item kolor,
\item struktura.
\end{Strich}
%
Właściwości wewnętrzne nie zależą od obserwatora i są niezależne od stanu systemu.
Są one stałe  przynajmniej przez pewien okres czasu. Te właściwości pozwalają również
odróżnić poszczególne systemy fizyczne. I tu rodzi się kolejne pytanie: Jaki jest \emph{kompletny}
wykaz swoistych właściwości w naturze? Oraz czy wszechświat ma takowe właściwości?\challengenor{unintrin}
Różne aspekty obiektów i ich stanów nazywane są (fizycznymi)
\ii[observable!definition]{obserwowalnymi}. Poniżej krok po kroku doprecyzujemy
tę wstępną, przybliżoną definicję. %step by step
% Improved Feb 2010, Mar 2014
Opisując przyrodę jako zbiór stałych jednostek oraz zmiany stanów jest początkiem
badania ruchu. Każda obserwacja ruchu wymaga rozróżnienia stałych, wewnętrznych
właściwości  opisu poruszających się obiektów  oraz zmian stanów  opisu sposobu,
w jaki dane obiekty się poruszają.\index{property!permanent}\index{property!intrinsic}\index{state}
Bez tego rozróżnienia nie ma ruchu. Bez niego nie można nawet \emph{mówić} o ruchu.
% Impr. Mar 2014
Korzystając z właśnie wprowadzonych pojęć, możemy powiedzieć
%
\begin{quotation}
\noindent \csrhd \iin[motion!as change of state of permanent objects]{Ruch jest
zmianą stanu stałych obiektów.}
\end{quotation}
%
Dokładne rozdzielenie tych aspektów, należących do obiektu, trwałych \emph{właściwości
wewnętrznych} od tych, które należą do stanu, zmieniającymi się \emph{właściwościami stanu}
zależy od precyzji obserwacji. Na przykład długość kawałka drewna nie jest stała; drewno
kurczy się i wygina z powodu procesów zachodzących na poziomie molekularnym. A dokładniej
długość kawałka drewna nie jest aspektem obiektu, ale aspektem jego stanu.
Precyzyjność obserwacji \emph{zmienia} zatem rozróżnienie między obiektem a jego stanem;
samo rozróżnienie nie znika  przynajmniej nie w pierwszych pięciu tomach naszej przygody.
%
% Only in the third part of this walk a surprising twist will appear.
%
% \comment{Only in the final part of our walk we will arrive at a point where
% one is forced to introduce a description of nature which overcomes the
% opposition between objects and their states. This surprising conclusion is
% one of the main results of modern physics.\seepage{Parmend}}
% Mar 2014 % (NO) need ref
Pod koniec XX wieku neuronauka wykazała, że rozróżnienie między stanami zmiennymi
a obiektami trwałymi nie jest dokonywane wyłącznie przez naukowców i inżynierów.
W naturze rozróżnienie również jest obecne. W rzeczywistości ten podział robiony jest nawet
w mózgu! Wykorzystując sygnały wyjściowe z kory wzrokowej, która przetwarza to, co obserwują oczy,
sąsiadujący obszar w \emph{górnej} części ludzkiego mózgu  \ii[stream!dorsal]{strumień grzbietowy} 
przetwarza \emph{stan obiektów}, które są widziane, takie jak odległość i ruch, podczas gdy sąsiedni obszar
w \emph{niżej położonej} części ludzkiego mózgu  strumień brzuszny  przetwarza wewnętrzne właściwości,
takie jak kształty, kolory i wzory.
Podsumowując,\label{statintaim} do opisu ruchu stany są potrzbne. Podobnie jak stałe,
wewnętrzne właściwości. Aby kontynuować i uzyskać \emph{pełny} opis ruchu, potrzebujemy więc
kompletnego opisu możliwych stanów i pełnego opisu wewnętrznych właściwości obiektów.
Pierwsze podejście do tego celu nazywa się \iin{fizyką galileuszową}; zaczyna się od\index{physics!Galilean} określenia naszego
\emph{codziennego} środowiska i ruchu w nim tak precyzyjnie, jak to możliwe.\index{physics!everyday}
%
% is approximation is based on the precise description
% of what all children know about motion.
% This program can indeed be realised: several
% \emph{consistent} descriptions of motion are possible, depending on the
% level
% of precision required. Each description is an approximation of another,
% finally leading to the {complete} description of motion. At each of these
% levels, the characteristic ingredient is the description of the state of
% objects, and thus in particular the description of their velocity, their
% position, and their interactions with time.
%
% New June 2008, reworked Sep, Dec 2008, Oct 2012, Mar 2014, Jul 2016
\subsection{Fizyka Galileusza w sześciu interesujących stwierdzeniach}
% Index OK
Badania\label{Galsixint} nad\index{Galilean physics!in six statements} codziennym
ruchem, fizyka galileuszowa, jest sama w sobie warta uwagi: odkryjemy wiele wyników,\index{motion!has six properties(}
które są sprzeczne z naszymi doświadczeniami. Na przykład jeśli przypomnimy sobie naszą
przeszłość, wszyscy doświadczyliśmy jak ważne, zachwycająće lub niepożądane mogą być \emph{niespodzianki}.
Niemniej jednak badanie codziennych ruchów pokazuje, że w przyrodzie \emph{nie ma} żadnych niespodzianek.
% Motion is predictable. We will see that all results that are in
% contrast to our everyday experience make sense.
Ruch, a tym samym świat, jest \emph{przewidywalny} lub \emph{deterministyczny}.
Główną niespodzianką naszych poszukiwań ruchu jest to, że w naturze nie ma niespodzianek.
Przyroda jest przewidywalna. W rzeczywistości odkryjemy sześć aspektów przewidywalności codziennego ruchu:
\begin{Strich}
% Impr. Oct 2012
\item[{1.}] \emph{Ciągłość.} Wiemy, że oczy, kamery i wszelka aparatura pomiarowa
ma skończoną \iin[resolution!of measurements is finite]{rozdzielczość}. Wszystko ma
określoną najmniejszą odległość, która może być zaobserwowana. Wiadomo, że zegary
mają określony najkrótszy czas, który mogą zmierzyć. Pomimo tych ograniczeń,
w codziennym życiu wszystkie ruchy, stany, przestrzeń, jak i czas są \emph{ciągłe}.\index{motion!is continuous}
\smallskip
% Impr. Oct 2012
\item[{2.}] \emph{Zachowanie.} Wszyscy obserwujemy, że ludzie muzyka i wiele innych
rzeczy w ruchu po pewnym czasie zatrzymują się. Badanie ruchu daje odwrotny rezultat:
ruch nigdy się nie zatrzymuje.\index{motion!is conserved} W rzeczywistości trzy aspekty
ruchu nie zmieniają się, ale są \emph{zachowane}: pęd, moment pędu i energia są zachowane
osobno we wszystkich przykładach ruchu. Nigdy nie zaobserwowano jeszcze żadnego wyjątku od
tego. (Dla kontrastu, masa jest bardzo często, ale nie zawsze zachowywana). Ponadto odkryjemy,
że zachowanie oznacza, że ruch i jego właściwości są takie same we wszystkich miejscach i przez
cały czas: ruch jest \emph{uniwersalny}.
\smallskip
% Impr. Oct 2012
\item[{3.}] \emph{Względność.} Wszyscy wiemy,\index{motion!is relative} że ruch różni
się od spoczynku. Pomimo naszych doświadczeń, rzetelne badania wykazują, że nie ma żadnej
zasadniczej różnicy między nimi. Ruch zależy od obserwatora. Ruch jest \emph{względny}.
Tak samo spoczynek. Jest to pierwszy krok prowadzący do zrozumienia teorii względności.
\smallskip
% Impr. Oct 2012
\item[{4.}] \emph{Odwracalność.} Wszyscy obserwujemy, że wiele procesów zachodzi
w tylko jednym kierunku. Na przykład rozalane mleko nigdy samoczynnie nie wróci do
kartonu. Mimo takich obserwacji, badania ruchu pokazują, że ruch jest \emph{odwracalny}.
Fizycy nazywają to\indexs{motion!reversal invariance}\index{motion!is reversible} niezmiennością
codzinnego ruchu w \emph{odwróceniu ruchu.} Potocznie, lecz niepoprawnie nazywanym 'odwróceniem czasu'.
\smallskip
% Impr. Oct 2012
\item[{5.}] \emph{Niezmienność lustrzana.} Większość z nas ma problemy z używaniem
nożyczek lewą reką, trudności z pisaniem inną ręką niż zwykle i ma serce po lewej stronie.
%
% and we know that many organic molecules in our bodies exist only in one type
% of handedness.
Pomimo tych obserwacji, nasze odkrywanie ruchu pokaże, że jest on \emph{lustrzanie niezmienny} (albo
\emph{parzyście niezmienny}).\index{parity!invariance}
Lustrzane\index{mirror!invariance}\index{motion!everyday, is mirrorinvariant}
procesy są zawsze możliwe w codziennym życiu.
\smallskip
% Impr. Nov 2010, Jan 2011, Oct 2012
\item[{6.}] \emph{Minimalizacja zmiany.} Wszyscy\index{motion!minimizes action} jesteśmy
zaskoczeni licznymi obserwacjami, które możemy przeprowadzić: kolory, kształty, dźwięki,
wzrost, katastrofy, szczęście, przyjaźń, miłość. Różnorodność, piękno i złożoność natury
są niesamowite. Potwierdzimy, że wszystkie obserwacje wynikają z ruchu. I pomimo pozorów
złożoności każdy ruch jest {prosty}. \index{motion!is lazy}\index{nature!is lazy}
Nasze badania pokażą, że wszystkie obserwacje można podsumować w prosty sposób: Natura jest leniwa.
Wszelki ruch zachodzi w taki sposób, aby \emph{zminimalizować zmianę}. Zmiana może być zmierzona za
pomocą jednostki zwanej 'akcją', a natura ograniacza ją do minimum. Sytuacje  lub stany, jak
mawiają fizycy  ewoluują, minimalizując zmiany. Natura jest leniwa.
\end{Strich}
% Impr. Jan 2011, June 2014
\np Te sześć aspektów ma zasadnicze znaczenie dla zrozumienia ruchu w sporcie,
muzyce, zwierzętach, maszynach i wśród gwiazd. Ten pierwszy tom naszej przygody
będzie eksploracją takich ruchów. W szczególności, wbrew pozorom potwierdzimy,
wspomniane sześć kluczowych właściwości we wszystkich przypadkach codziennego ruchu.\index{motion!has six properties)}
%
% Impr. Jul 2016
\subsection[Curiosities and fun challenges about motion]{Ciekawostki i wyzwania związane z
ruchem\footnote{Sekcje zatytułowane '{ciekawostki}' to zbiory tematów i problemów,
które pozwalają sprawdzić i rozszerzyć wykorzystanie już wprowadzonych koncepcji.}}
% Index OK
% !!!2 needs figures of fool's and of Spanish burton
\csepsfnb{iblockandtackle}{scale=1}{Maszyny proste\protect\index{block
and tackle}. Wielokrążek zwykły i różnicowy (po lewej), w zastosowaniu rolniczym (po prawej).}
\begin{curiosity}
\item[] W przeciwieństwie do większości zwierząt, istoty osiadłe, takie jak
rośliny czy ukwiały, nie mają nóg i nie mogą zbytnio się poruszać; dla samoobrony
wykształciły \ii{trucizny}. Przykładami takich roślin są pokrzywa zwyczajna, tytoń,
naparstnica, dzwonnica i mak; trucizny obejmują kofeinę, nikotynę i kurarę. Takie trucizny
są podstawami większości \iin{leków}. Dlatego większość leków istnieje głównie dlatego, że
\iin[plats!and legs]{rośliny} nie mają \iin{nóg}.\index{legs!and plants}
\item Człowiek wspina się na górę od 9.00 do 13.00. Śpi na szczycie i schodzi następnego dnia od 9.00 do 13.00.
Czy istnieje takie miejsce na szlaku, w którym znalazł się w ciągu tych dwóch dni o tej samej godzinie?\challengenor{yes}
% March 2007
\item Za każdym razem, gdy \iin[soap!bubble bursting]{bańka mydlana} pęka,
ruch powierzchni podczas pęknięcia jest taki sam, chociaż jest on zbyt szybki,
aby mógł być dostrzeżony gołym okiem. Czy potrafisz wyobrazić sobie szczegóły?\challengenor{burstingbubble}
\item Czy ruch \iin[ghosts]{ducha} jest przykładem ruchu?\challengenor{ghost1}
\item Czy coś może przestać się poruszać?\challengenor{stopmo} Jak byś to pokazał(a)?
% Impr. 2010
\item Czy ciało poruszające się zawsze po linii prosteje\index{motion!possibly
infinite?} pokazuje, że natura lub przestrzeń są nieskończone?\challengenor{evermomo}
\item Jaką długość liny należy ciągnąć, aby przenieść masę o odległość $h$ za
pomocą \iin{wielokrążka} z czterema krążkami, jak pokazano po lewej stronie
\figureref{iblockandtackle}?\challengenor{flaschenzug} Czy rolnik po prawej
stronie robi coś rozsądnego?
% October 2017
W przeszłości wielokrążki były bardzo ważnymi maszynami. Dwie szczególnie użyteczne
wersje to wielokrążek \emph{różnicowy} oraz \ii{wielokrążek potęgowy}, który jest
najefektywniejszy przy małej ilości krążków.\cite{tacklesima} \challengn
% October 2017
Wszystkie te urządzenia są przykładami \ii{złotej zasady mechaniki}: tyle ile
zyskujesz na sile, tracisz na przemieszczeniu. Albo, równoważnie: iloczyn
siły i przemieszczenia  w fizyce nazwyany \\ii(pracą)  pozostaje niezmieniony,
niezależnie od urządzenia jakiego używasz. Jest to szczególny przypadek.
\item Czy wrzechświat ?\challengenor{unimove}
\item Aby mówić precyzyjnie o \iin[precision!measuring it]{precyzji}, musimy zmierzyć
precyzję. Jak to zrobić?\challengenor{presme}
\item Czy można obserwować ruch nie mając \iin{pamięci}?\challengenor{memmo}
\item Jaka jest \iin[speed!lowest]{najmniejsza prędkość}, jaką możesz zaobserwować?\challengenor{zerosp}
Czy w naturze istnieje minimalna prędkość?
\item Według legendy, \iname{Sissa ben Dahir}, indyjski twórca gry \ii{czaturanga} lub
\iin{szachów}, zażądał od \iinn{króla Shirhama} następującej nagrody za swą grę:
chciał jednego ziarna przenicy na pierwszym polu, dwóch na drugim, czterech na trzecim,
ośmiu na czwratym i tak dalej. Jak długo zajełoby wyprodukowanie tylu zboża, mając do
dyspozycji wszystkie pola przenicy na świecie?\challengenor{wheat}
\item Gdy zapalona \iin[candle!motion]{świeca} jest poruszona, {płomień} pozostaje
za świecą. Jak zachowuje się płomień, gdy płonąca świeca znajduje się wewnątrz szklanki,
a szklanka jest przyspieszana?\challengenor{candl}
% Apr 2005
\item Dobrym sposobem na zarobienie pieniędzy jest budowa \iin[motion!detector]{czujników ruchu}.
Czujnik ruchu to małe pudełko z kilkoma przewodami. Skrzynka wytwarza sygnał elektryczny podczas
ruchu skrzynki. Jakie typy czujników ruchu możesz sobie wyobrazić? Jak tanio można zrobić takie pudełko?
Jak precyzyjne?\challengedif{motdetct}
% Fig added May 2007
\cstftlepsf{itableball}{scale=1}{What happens?}%
% Fig added May 2007
[15mm]{iblockrolling}{scale=1}{What is the speed of the rollers? Are
other roller shapes possible?}
\item Doskonale beztarciowa kula leży w pobliżu krawędzi idealnie płaskiego i poziomego stołu
jak pokazano na \figureref{itableball}. Co się stanie? W jakiej skali czasowej?\challengedif{tough}
\item Wchodzisz do zamkniętego pudła bez okien. Skrzynka jest przesuwana przez nieznane Ci siły zewnętrzne.
Czy potrafisz określić sposób poruszania się pudła, bez wychodzenia z niego?\challengenor{inbox}
\item Kiedy blok jest przesuwany po cylindrach, jak pokazano na \figureref{iblockrolling},\index{roll!speed}
jaka jest zależność między prędkością bloku a prędkością cylindrów?\challengenor{blkrolling}
\item Nie lubisz wzorów?\index{formulae!liking them} Jeśli\index{fear of formulae}
tak, wypróbuj ten trzyminutowy sposób\cite{a7}, aby to zmienić.\cite{a7} %pioneered by \iin{Richard Bandler} and \iin{John Grinder}.
Warto spróbować, bo to sprawi, że będziesz cieszyć się tą książką o wiele bardziej\challengenor{nlp}
Życie jest krótkie; jak najwięcej czynności, które wykonujesz tak jak czytanie tego tekstu, powinno być przyjemnością.
Pionierami tej metody byli \iin{Richard Bandler} i \iin{John Grinder}.
\begin{Strich}
\item[{1.}] Zamknij oczy i przypomnij sobie doświadczenie, które było \emph{absolutnie
wspaniałe}, sytuację, w której czułeś(aś) się podekscytowany(a), zainteresowany(a) i pozytywny(a).
\item[{2.}] Otwórz oczy na sekundę lub dwie i spójrz na
\cspageref{formulapage} % this vol I
 lub inną stronę zawierającą wiele wzorów.
\item[{3.}] Następnie zamknij oczy i wróć do swojego cudownego doświadczenia.
\item[{4.}] Powtórz kroki 2 i 3 jeszcze trzykrotnie.
\end{Strich}
\np Następnie pozostaw to wspomnienie, rozejrzyj się dookoła siebie, aby wrócić tu i
teraz i sprawdź się. Spójrz ponownie na \cspageref{formulapage}. % this vol I
Co teraz myślisz o wzorach?
\item W XVI wieku \iinns{Niccolò Tartaglia}\footnote{Niccolò
Fontana Tartaglia \lived(14991557), istotny renesansowy matematyk.}
zaproponował następujący problem. Trzy młode pary chcą przeprawić się przez rzekę.
Dostępna jest tylko mała łódka, do której zmieszczą się tylko dwie osoby. Mężczyźni są niezwykle
zazdrośni i nigdy nie puściliby żon z innym mężczyzną. Ile przepraw przez rzekę jest koniecznych?
\challengenor{tarta}
\item Walce mogą być\index{rolling!puzzle} używane do toczenia płaskiego przedmiotu nad podłogą,
jak pokazano na \figureref{iblockrolling}. Cylindry utrzymują płaszczyznę obiektu zawsze w tej samej odległości od
podłogi. Jakie przekroje inne niż okrągłe, tzw. figury o stałej szerokości, zadziałają tak samo jak walec?
Ile przykładów można znaleźć?\challengenor{roll} Czy są możliwe inne przedmioty niż cylindry?
% August 2013, Nov 2016, margin overlap corrected in Jun 2017
\item Wieszanie obrazów na ścianie nie jest łatwe. Pierwszy problem:
jaki jest najlepszy sposób\index{painting puzzle}\index{nail puzzle} powieszenia obrazu na jednym gwoździu?index{puzzle!painting}\index{puzzle!nail}
Metoda ta musi umożliwiać przesunięcie obrazu w pozycji poziomej po umieszczeniu
gwoździa w ścianie, w przypadku gdy ciężar nie będzie równomiernie rozłożony.\challengenor{firstpictnail}
Drugi problem: Czy obraz można powiesić na ścianie  tym razem z pomocą długiej linki\cite{phpuzz} 
na dwóch gwoździach w taki sposób, że naciągnięcie jednego z nich spowoduje upadek obrazu? A z trzema gwoździami? A $n$ gwoździami?
\challengenor{pictnail}
\end{curiosity}
%
% Oct 2008, Feb 2012, Jul 2016
\subsection{Podsumowanie ruchu}
% Index OK
Ruch\index{motion!is fundamental}\index{motion!has six properties} jest najbardziej
fundamentalnym zjawiskiem jaki możemy zaoobserwować w naturze. Codzienny ruch jest
przewidywalny i deterministyczny. Przewidywalność jest odzwierciedlona w sześciu aspektach
ruchu: ciągłości, zachowalnośći, odwracalności, lustrzanej niezmienności, względności i mnimalizacji.
Niektóre z tych aspektów są ważne dla \emph{wszystkich} ruchów, a inne tylko dla \emph{codziennego} ruchu.
Które i dlaczego?\challengedif{sixaspmot} Zaraz to zbadamy.
\vignette{classical}
%
%
%
%
% Impr. Jul 2016
\chapter{From motion measurement to continuity}
% Index OK
\markboth{\thesmallchapter\ from motion measurement to continuity}%
{\thesmallchapter\ from motion measurement to continuity}
\begin{quote}
\selectlanguage{german}Physic ist wahrlich das
eigentliche Studium des Menschen.\selectlanguage{british}%
%
\footnote{`Physics truly is the proper study of man.' Georg~Christoph
Lichtenberg \livedplace(1742 Ober\hbox{}Ramstadt1799 Göttingen) was an
important physicist and essayist.} %
%
\\
\iinns{Georg~Christoph Lichtenberg}
\end{quote}
%
% in a letter to F.F. Wolff, as cited in Phys Blatt, p5456,
%title like the sentence, by H. Zehe, Feb 1999
\csini{T}{he} simplest description of motion is the one we all, like cats or
monkeys, use\linebreak hroughout our
%unconsciously in
everyday life: \emph{only one thing can be at a given spot at a given
time}.\linebreak his general description\index{impenetrability!of
matter}\index{matter!impenetrability of} can be separated into three
assumptions: matter is \emph{impenetrable} and \emph{moves}, time is made of
\ii[instant]{instants}, and space is made of \ii[point!in space]{points}.
Without these three assumptions (do you agree with them?)\challengenor{galmot}
it is not even possible to define velocity. We thus need points embedded in
continuous space and time to talk about motion. This description of nature is
called \ii[Galilean physics]{Galilean physics}, or sometimes \ii{Newtonian
physics}.
\cssmallepsfnb{igalileo}{scale=0.12}{Galileo Galilei \livedfig(15641642).}
\iinns{Galileo Galilei} \lived(15641642), Tuscan professor of mathematics,
was the central founder of modern physics. He became famous for advocating the
importance of observations as checks of statements about nature. By requiring
and performing these checks throughout his life, he was led to continuously
increase the accuracy in the description of motion. For example, Galileo
studied motion by measuring change of position with a selfconstructed
stopwatch. Galileo's experimental aim was to measure all what is measurable
about motion. His approach changed the speculative description of ancient
Greece into the experimental physics of Renaissance Italy.%
%
\footnote{The best and most informative book on the life of Galileo and his
times is by \iinn{Pietro Redondi} (see the section on
\cspageref{redondipage}). % this vol I
Galileo was born in the year the \iin[pencil!invention of]{pencil} was
invented. Before his time, it was impossible to do paper and pencil
calculations. For the curious, the \url{www.mpiwgberlin.mpg.de} website
allows you to read an original manuscript by Galileo.} %
% He is therefore regarded
% as the founder of modern physics.
% Impr. Jan 2015
After Galileo, the English alchemist,\index{alchemy} occultist, theologian,
physicist and politician \iinns{Isaac Newton} \lived(16431727) continued to
explore with vigour the idea that different types of motion have the same
properties, and he made important steps in constructing the concepts necessary
to demonstrate this idea.%
%
% Impr. Jun 2015
\footnote{Newton was born a year after Galileo died. For most of his life
Newton searched for the philosopher's stone. Newton's hobby, as head of the
English mint, was to supervise personally the hanging of counterfeiters.
About Newton's lifelong infatuation with alchemy, see the books by
Dobbs.\cite{a14b} A misogynist throughout his life, Newton believed himself
to be chosen by god;\index{gods!and Newton} he took his Latin name, \emph{Isaacus
Neuutonus}, and formed the anagram \emph{Jeova sanctus unus}. About
Newton and his importance for classical mechanics, see the text by
\iinn{Clifford Truesdell}.\cite{a14}} %
%
% The way that motion, points, instants, and impenetrability are described
% with precision only needs a short overview.
% Sep 2008, Jan 2015
Above all, the explorations and books by Galileo popularized the fundamental
experimental statements on the properties of speed, space and time.
%
%
% Improved Dec 2010, reread July 2016
\subsection{What is velocity?}
% Index OK
\begin{quote}
There is nothing else like it.\\
\iinns{Jochen Rindt}%
%
\footnote{Jochen Rindt \lived(19421970), famous Austrian {Formula One
racing} car driver, speaking about speed.}
\end{quote}
% On certain mornings it can be dangerous to walk in forests. When the
% hunting season opens, men armed with rifles ramble through the landscape,
% eager to use again  at last  their beloved weapon. They shoot, as
% malicious tongues pretend, on everything which moves, including falling
% leaves or fellow hunters. Here, we are going to do something similar. Like
% a hunter, we will concentrate on everything that moves; unlike one however,
% we will not seek to stop it moving, but try to follow it and to understand
% its motion in detail.
\np % Apr 2005
Velocity fascinates.\index{velocity} To physicists, not only car races are
interesting, but any moving entity is. Therefore, physicists first measure as
many examples as possible. A selection of measured speed values is given in
\tableref{velmetab}. The units and prefixes %
used are explained in detail in \appendixref{units1}.\seepageone{units1}
Some speed measurement devices are shown in \figureref{ispeedmeasurement}.
%
% {Table of velocities}
%
{\small
\begin{table}[p]
\small
\centering
\caption{Some measured\protect\index{velocity!values,
table}\protect\index{speed!values, table} velocity values.}
\label{velmetab}
\vbox to 45\baselineskip{%\leavevmode
\dirrtabularstar
\begin{tabular*}{\textwidth}{@{}>{\columncolor{hks152}[0pt][2cm]} l
@{\extracolsep{\fill}} l @{}}
%
\toprule
%
\tabheadf{Observation} & \tabhead{Velocity} \\[0.5mm]
%
\midrule
%
\iin[crust!growth of deep sea manganese]{Growth of deep sea manganese crust} &
\csd{80}{am/s}\\
% 1 million years for 2.5 mm, see \cite{supernovahomo} % is from C3A!
%
Can you find something slower? & \challengenor{slowspeed} \\
%
\iin[stalagmites]{Stalagmite growth} & \csd{0.3}{pm/s}\\
%
\iin[lichen growth]{Lichen growth} & down to \csd{7}{pm/s}\\ % 50 years for 1 cm
%
Typical \iin[motion!of continents]{motion of continents} & $\csd{10}{mm/a}=\csd{0.3}{nm/s}$\\
%
\iin[growth!human]{Human growth} during \iin{childhood}, \iin[hair!growth]{hair growth} & \csd{4}{nm/s}\\
%
\iin[tree!growth]{Tree growth} & up to~\csd{30}{nm/s}\\
%
\iin[electron!speed]{Electron drift in metal wire} & \csd{1}{\muunit m/s}\\
%
\iin[sperm!motion]{Sperm motion} & {60} to \csd{160}{\muunit m/s}\\
%
% May 2013
\iin[speed!of light at Sun centre]{Speed of light at Sun's
centre} \cite{insidesunlightspeed} & \csd{1}{mm/s}\\
%
\iin[ketchup!motion]{Ketchup motion} & \csd{1}{mm/s}\\
%
%\iin[snail speed]{Speed of snail} & \csd{5}{mm/s}\\
%
Slowest speed of light measured in matter on Earth \cite{slowlight}&
\csd{0.3}{m/s}
\\
%
\iin[snowflake!speed]{Speed of snowflakes} & \csd{0.5}{m/s} to
\csd{1.5}{m/s}\\
% from literature : measured values ; almost independent of radius
%
Signal\index{nerve!signal speed} speed in human {nerve cells}
\cite{bioinzahlen} & \csd{0.5}{m/s} to \csd{120}{m/s} \\
%
Wind speed at 1 and 12 Beaufort (light air and hurricane) &
\csd{< 1.5}{m/s}, \csd{> 33}{m/s} \\
%
\iin[rain speed]{Speed of rain} drops, depending on radius & \csd{2}{m/s} to
\csd{8}{m/s}\\
% from literature : measured values
%
Fastest swimming fish, \iin{sailfish} (\iie{Istiophorus platypterus}) &
\csd{22}{m/s}\\
%
% Jun 2006
%2006 Speed sailing record over \csd{500}{m} (by windsurfer \iinn{Finian Maynard})
%& \csd{25.1}{m/s}\\
% Oct 2008
%2008 Speed sailing record over \csd{500}{m} (by kitesurfer \iinn{Alex
%Caizergues}) & \csd{26.0}{m/s}\\
% Sep 2009
2009 Speed sailing record over \csd{500}{m} (by trimaran
\iin{Hydroptère}) & \csd{26.4}{m/s} \\
%
Fastest running animal, \iin{cheetah} (\iie{Acinonyx jubatus}) &
\csd{30}{m/s}\\
%
% Wind speed at 12 \iin{Beaufort} (hurricane) & above \csd{33}{m/s}\\ % 64 knots
%
Speed of air in throat when sneezing & \csd{42}{m/s}\\
%
\iin[throwing!speed record]{Fastest throw: a cricket ball thrown with
baseball
technique while running} & \csd{50}{m/s}\\
%%%%%%% Penny Sucharov wanted ``bowl'' changed into ball %%%%%%%%%
%
%\iin[throwing speed]{Fastest measured throw: baseball pitch} &
% \csd{45}{m/s}\\
%
%
% Fastest Cricket Bowler The highest electronically measured speed for a ball
% bowled by any bowler is 100.23mph (161.3km/h) by Shoaib Akhtar (Pakistan)
% against England on 22 February 2003 in a World Cup match at Newlands, Cape
% Town, South Africa.
%
% Nicknamed the 'Rawalpindi Express' Shoaib burst onto the cricket scene in
% 1999, but has since struggled to cement a first choice place in the Pakistan
% lineup. He has played for many teams, including: Agriculture Development
% Bank of Pakistan, Pakistan International Airlines, Rawalpindi Cricket
% Association, Somerset, Khan Research Labs, Durham, Lashings and Pakistan.
%
%
Freely falling human,\index{free fall!speed of} depending on clothing & 50 to
\csd{90}{m/s}\\
%
\iin[bird!fastest]{Fastest bird}, diving \iie{Falco peregrinus}& \csd{60}{m/s}\\
%
\iin[badminton smash!record]{Fastest badminton smash} &
\csd{70}{m/s}\\
%
Average speed of oxygen molecule in air at room temperature& \csd{280}{m/s}\\
%
Speed of sound\index{sound!speed} in dry air at sea level and standard
temperature&
\csd{330}{m/s}\\
%
%Record car speed & \csd{340}{m/s}\\
%
% Jan 2015
Speed of the equator & \csd{434}{m/s}\\
%
Cracking \iin[whip!speed of]{whip}'s end & \csd{750}{m/s}\\
%
\iin[bullet!speed]{Speed of a rifle bullet} & \csd{1}{km/s}\\
%
Speed of crack propagation in breaking silicon & \csd{5}{km/s}\\
%
\iin[speed, highest]{Highest macroscopic speed} achieved by man  the
\emph{Helios II} satellite & \csd{70.2}{km/s}\\ % Corrected May 2014
%
\iin[Earth!speed]{Speed of Earth} through universe & \csd{370}{km/s}\\
%
\iin[lightning speed]{Average speed (and peak speed) of lightning} tip &
\csd{600}{km/s} (\csd{50}{Mm/s}) \\ % new in Jun 2005
% % from ``Küssen müssen wir noch lernen'', p143
%
Highest macroscopic speed measured in our galaxy \cite{galrec} &
\csd{0.97 \cdot10^{8}}{m/s} \\
%
Speed of electrons inside a colour TV tube & \csd{1 \cdot10^{8}}{m/s}\\
%
Speed of radio messages in space\index{telephone speed}\index{radio speed} &
\csd{299\,792\,458}{m/s}\\
%
Highest ever measured group velocity of light & \csd{10 \cdot10^{8}}{m/s}\\
%
Speed of light spot from a \iin[lighthouse]{lighthouse} when passing over the Moon &
\csd{2 \cdot10^{9}}{m/s}\\
%
Highest \ii[velocity!proper]{proper} velocity
ever achieved for electrons by man & \csd{7\cdot10^{13}}{m/s} \\
%
Highest possible velocity for a light spot or a shadow & no limit \\
\bottomrule
%
\end{tabular*}
\vss % !.!4 not the perfect solution for a long table
}
\end{table}
}
Everyday life teaches us a lot about motion: objects can overtake each other,
and they can move in different directions. We also observe that velocities
can be added or changed smoothly. The precise list of these properties, as
given in \tableref{veltab}, is summarized by mathematicians in a special term;
they say that velocities form a \ii{Euclidean vector space}.%
%
\footnote{It is named after \iname[Euclid, or Eukleides]{Euclid}, or
{Eukleides}, the great Greek mathematician who lived in Alexandria around
300~{\bce}. Euclid wrote a monumental treatise of geometry, the
\csgreekok{Stoiqe\~ia} % checked on internet
or \bt Elements/ which is one of the milestones of human thought. The text
presents the whole knowledge on geometry of that time. For the first time,
Euclid introduces two approaches that are now in common use: all statements
are deduced from a small number of basic \ii{axioms} and for every statement
a \ii{proof} is given. The book, still in print today, has been the
reference geometry text for over 2000 years. On the web, it can be found at
\url{aleph0.clarku.edu/~djoyce/java/elements/elements.html}.}
%
More\index{mathematics}\index{precision} details about this strange term will
be given shortly.\seepageone{eulc} For now we just note that in describing
nature, mathematical concepts offer the most accurate vehicle.
%
% This is an example of a general connection: every time one aims for the
% highest precision in describing nature, mathematical concepts are adopted.
\csepsfnb{ispeedmeasurement}{scale=1}{Some speed measurement devices: an
anemometer, a tachymeter for inline skates, a sport radar gun and a
\protect\iin{PitotPrandtl tube} in an aeroplane
({\textcopyright}~\protect\iinn{Fachhochschule Koblenz},
\protect\iname{Silva}, \protect\iname{Tracer}, \protect\iname{Wikimedia}).}
%
% add more, e.g.:
% door opener above sliding door (NO)
% acoustic systems,
% optical systems,
% mechanical systems
%
% {Properties of Galilean velocity}
{\small
\begin{table}[t]
\small
\centering
\caption{Properties\protect\index{velocity!properties, table} of everyday 
or Galilean  velocity.}
\label{veltab}
\dirrtabularstarnolines
\begin{tabular*}{\textwidth}{@{} p{31mm} @{\extracolsep{\fill}} p{32mm}
@{\extracolsep{\fill}} p{41mm} @{\extracolsep{\fill}} p{25mm}@{}}
%
\toprule
% % !.!3 Das hat zuviel Durchschuss:
% \tabheadf{Velocities}& \tabhead{Physical} & \tabhead{Mathematical}
% & \tabhead{Definition}\\
% \tabheadf{can} & \tabhead{property} & \tabhead{name} & \\[0.5mm]
% %
\tabhead{Velocities can}& \tabhead{Physical \ \ \ \ \ \ property} &
\tabhead{Mathematical name} & \tabhead{Definition}\\[0.5mm]
%
\midrule
%
% Jan 2006: Changed order to match text below
%
Be distinguished &\iin{distinguishability}& \iin{element of set} &
\leavevmode\seepagethree{setdefi} \\
%
Change gradually & \iin{continuum}& real vector space &
\leavevmode\seepageone{vecspde}, \seepagefive{topocont} \\
%
Point somewhere & \iin{direction}& vector space, \iin{dimensionality} &
\leavevmode\seepageone{vecspde} \\
%
Be compared & \iin{measurability}& \iin{metricity}&
\leavevmode\seepagefour{mespde4} \\
%
Be added & \iin{additivity}& {vector space}&
\leavevmode\seepageone{vecspde} \\
%
Have defined angles & \iin{direction}& \iin{Euclidean vector space} &
\leavevmode\seepageone{eulc} \\
%
Exceed any limit & \iin{infinity}& \iin{unboundedness}&
\leavevmode\seepagethree{settab} \\
%
\bottomrule
\end{tabular*}
\end{table}
}
% Impr. Dec 2010
When velocity is assumed to be an Euclidean vector, it is called \ii[Galilean
velocity]{Galilean} velocity.\index{velocity!Galilean} Velocity is a profound
concept. For example, velocity does
% It seems that velocity is a simple and almost boring concept. Well, it is
% not. The first mistake: one is usually brought up with the idea that
not need space and time measurements to be defined.
%But this is utterly wrong.
Are you able to find a means of measuring velocities without measuring space
and time?\challengedif{velme} If so, you probably want to skip to
%\cspageref{specialrelat},
the next volume, jumping 2000 years of enquiries. If you cannot do so,
consider this: whenever we measure a quantity we assume that everybody is able
to do so, and that everybody will get the same result. In other words, we
define \ii[measurement!definition]{measurement} as a comparison with a
standard. We thus implicitly assume that such a standard exists, i.e.,{} that
an example of a `perfect' velocity can be found. Historically, the study of
motion did not investigate this question first, because for many centuries
nobody could find such a standard velocity.
% , and nobody discovered this measurement method.
You are thus in good company.
%
% {Table of speed measurement methods}
%
{\small
\begin{table}[t]
\small
\centering
\caption{Speed measurement\protect\index{velocity!measurement devices, table}
devices in biological and engineered systems.}
\label{speedmeastab}
\dirrtabularstar % Nov 08: added columncolor
\begin{tabular*}{\textwidth}{@{}
>{\PBS\raggedright\columncolor{hks152}[0pt][1.5cm]} p{55mm}
@{\extracolsep{\fill}}
>{\PBS\raggedright\columncolor{hks152}[0pt][1.5cm]} p{42mm}
@{\extracolsep{\fill}}
>{\PBS\raggedright\columncolor{hks152}[0pt][0cm]} p{22mm}
% @{\extracolsep{\fill}} p{12mm}
@{}}
%
\toprule
%
\tabheadf{Measurement} & \tabhead{Device} & \tabhead{Range}
% & \tabhead{Precision}
\\[0.5mm]
%
\midrule
%
Own running speed in insects, mammals and humans & leg beat frequency measured
with internal clock & 0 to \csd{33}{m/s}
% & 20\,\%
\\
%
Own car speed & tachymeter attached to wheels & 0 to
\csd{150}{m/s}
% & 7\,\%
\\
%
Predators and hunters measuring prey speed & vision system & 0 to
\csd{30}{m/s}
% & 15\,\%
\\
%
Police measuring car speed & radar or laser gun & 0 to
\csd{90}{m/s}
% & 3\,\%
\\
%
Bat measuring own and prey speed at night & doppler sonar & 0 to \csd{20}{m/s}
% & 10\,\%
\\
%
Sliding door measuring speed of approaching people & doppler radar & 0 to
\csd{3}{m/s}
% & 30\,\%
\\
%
Own swimming speed in fish and humans & friction and deformation of skin & 0
to \csd{30}{m/s}
% & 20\,\%
\\
%
Own swimming speed in dolphins and ships & sonar to sea floor & 0 to
\csd{20}{m/s} \\
%
Diving speed in fish, animals, divers and submarines & pressure change & 0 to
\csd{5}{m/s} \\
%
Water predators and fishing boats measuring prey speed & sonar & 0 to
\csd{20}{m/s} \\
%
Own speed relative to Earth in insects & often none (grasshoppers) & n.a. \\
%
Own speed relative to Earth in birds & visual system & 0 to \csd{60}{m/s}\\
%
Own speed relative to Earth in aeroplanes or rockets & radio goniometry, radar
& 0 to \csd{8000}{m/s} \\
%
Own speed relative to air in insects and birds & filiform hair deflection,
feather deflection & 0 to \csd{60}{m/s} \\
%
Own speed relative to air in aeroplanes & PitotPrandtl tube
& 0 to \csd{340}{m/s}\\
%
Wind speed measurement in meteorological stations & thermal, rotating or
ultrasound
anemometers & 0 to \csd{80}{m/s} \\
%
Swallows measuring prey speed & visual system & 0 to \csd{20}{m/s} \\
%
Bats measuring prey speed & sonar & 0 to \csd{20}{m/s} \\
%
Macroscopic motion on Earth & Global Positioning System, Galileo, Glonass & 0 to
\csd{100}{m/s} \\
%
Pilots measuring target speed & radar & 0 to \csd{1000}{m/s} \\
%
Motion of stars & optical Doppler effect & 0 to
\csd{1000}{km/s} \\
%
Motion of star jets & optical Doppler effect & 0 to
\csd{200}{Mm/s} \\
% %
% Own motion in interstellar space &\leavevmode{\challengenor{ownmotion}} &
% \\
%
\bottomrule
\end{tabular*}
\end{table}
}
%
% Nov 2008, Jul 2016
How is velocity measured in everyday life? Animals and people estimate their
velocity in two ways: by estimating the frequency of their own movements, such
as their steps, or by using their eyes, ears, sense of touch or sense of
vibration to deduce how their own position changes with respect to the
environment. But several animals have additional capabilities: certain snakes
can determine speeds with their infraredsensing organs, others with their
magnetic field sensing organs. Still other animals emit sounds that create
echoes in order to measure speeds to high precision. Other animals use the
stars to navigate. A similar range of solutions is used by technical devices.
\tableref{speedmeastab} gives an overview.
% May 2007, Mar 2012
Velocity is not always an easy subject. Physicists like to say, provokingly,
that what cannot be measured does not exist.\index{velocity!in space} Can you
measure your own velocity in empty interstellar space?\challengenor{ownmotion}
% Jun 2007, Oct 2014, Nov 2014
% \csepsftwfull{icruisespeed}{scale=1}{How wing load and sealevel cruise
%
\csepsfnb[p]{icruisespeed}{scale=1}{How wing load and sealevel cruise speed
scales with weight in flying objects,\protect\index{bird!speed, graph
of}\protect\index{aeroplane!speed, graph of}\protect\index{insect!speed,
graph of} compared with the general trend line (after a graph
{\textcopyright}~\protect\iinn{Henk Tennekes}).}
% No need for rights, my graph
% Jun 2007
Velocity is of interest to both engineers and evolution
scientist.\index{velocity!of birds} In general, selfpropelled systems are
faster the larger they are. As an example, \figureref{icruisespeed} shows
how this applies to the cruise speed of flying things. In general, cruise
speed scales with the sixth root of the weight, as shown by the trend line
drawn in the graph. (Can you find out why?)\challengedif{cruisespeed} By the
way, similar \ii[scaling!allometric]{allometric scaling} relations hold for
many other properties of moving systems, as we will see later on.
% Jan 2006
Some researchers have specialized in the study of the lowest velocities found
in nature: they are called geologists.\cite{geomorph} Do not miss the
opportunity to walk across a landscape while listening\challengn to one of
them.
Velocity is a profound\index{velocity!is not Galilean} subject for an
additional reason: we will discover that all its seven properties of
\tableref{veltab} are only approximate; \emph{none} is actually correct.
Improved experiments will uncover exceptions for every property of Galilean
velocity. The failure of the last three properties of \tableref{veltab} will
lead us to special and general relativity, the failure of the middle two to
quantum theory and the failure of the first two properties to the unified
description of nature. But for now, we'll stick with Galilean velocity, and
continue with another Galilean concept derived from it: time.
% There is a second mistake in thinking that velocity is a boring subject: the
% latter stages of our walk will show that every single property mentioned in
% \tableref{veltab} is only approximate; \emph{none} is actually correct.
% That is one reason that our hike is so exciting. But for the moment, we
% continue with the next aspect of Galilean states.
% Improved Dec 2010
\begin{quoteunder}
Without the concepts \emph{place}, \emph{void} and \emph{time}, change
cannot be.\ [\ldots] It is therefore clear [\ldots] that their investigation
has to be carried out, by studying each of them separately.\\
%
\inames{Aristotle}%
%
\footnote{Aristotle \livedplace(384/3 Stageira322~{\bce} Euboea), important
Greek philosopher and scientist, founder of the \emph{Peripatetic
school} located at the Lyceum, a gymnasium dedicated to Apollo Lyceus.} %
%
\bt Physics/ Book III, part 1.
\end{quoteunder}
%
%
% Reread Jul 2016
\subsection{What is time?}
% Index OK
% Dec 2013
\begin{quote}
Time is an accident of motion.\\
\inameq{Theophrastus}\footnote{Theophrastus of Eresos \lived(c. 371  c.
287) was a revered Lesbian philosopher, successor of Aristoteles at the
Lyceum.}
\end{quote}
\begin{quote}
Time does not exist in itself, but only through the perceived objects,
from which the concepts of past, of present and of future ensue.\\
%
% My translation from the German
%
%he is talking about the epicurean doctrine
%
\inames{Lucretius},\footnote{\iinn{Titus Lucretius~Carus}
\livedca(\circa95\circa55 {\bce}), Roman scholar and poet.} \bt De rerum
natura/ lib. 1, v. 460 ss.
\end{quote}
\label{timclodef}
%
\np In their first years of life, children spend a lot of time throwing
objects around. The term `object' is a Latin word meaning `that which has
been thrown in front.'\index{object!definition} Developmental psychology has
shown experimentally that from this very experience\cite{childev} children
extract the concepts of time and space. Adult physicists do the same when
studying motion at university.
%, with the difference that they repeat it consciously, using language.
\csepsfnb{iparabola}{scale=1}{A typical path followed by a stone thrown
through the air  a parabola  with photographs (blurred and stroboscopic)
of a table tennis ball rebounding on a table (centre) and a stroboscopic
photograph of a water droplet rebounding on a strongly hydrophobic surface
(right, {\textcopyright}~\protect\iinn{Andrew Davidhazy},
% SENT EMAIL FEB 2008  andpph@rit.edu  OK!
\protect\iinn{Max Groenendijk}).}
% he is ok for a book as well, he is in business
When we throw a \iin[stones]{stone} through the air, we can define a
\emph{sequence} of observations.\index{observation!sequence and time}
\figureref{iparabola} illustrates how. Our memory and our senses give us
this ability. The sense of hearing registers the various sounds during the
rise, the fall and the landing of the stone. Our eyes track the location of
the stone from one point to the next. All observations have their place in a
sequence, with some observations preceding them, some observations
simultaneous to them, and still others succeeding them. We say that
observations are perceived to happen at various
\ii[instant!definition]{instants}  also called `points in time'  and we
call the sequence of all instants \ii[time!definition]{time}.
An observation that is considered the smallest part of a sequence, i.e.,{} not
itself a sequence, is called an \ii[event!definition]{event}. Events are
central to the definition of time; in particular, starting or stopping a
stopwatch are events. (But do\challengenor{events} events really exist? Keep
this question in the back of your head as we move on.)
%{}From the exploration of the many types of change and their sequences, both
%children at the age of about one year, as well as physicists, extract the
%concept of time. The term `time' expresses the fact that one can line up
%observations  events  in a row.
Sequential phenomena have an additional property known as stretch, extension
or duration. Some measured values are given in
\tableref{durmetab}.%
%
\footnote{A year is abbreviated a\indexs{a@a (year)}
(Latin `annus').} %
%
\ii[duration!definition]{Duration} expresses the idea that sequences
\emph{take} time. We say that a sequence takes time to express that other
sequences can take place in parallel with it.
% \comment{% 1st. not interesting intellectually, said Serge, and he is
% right
% % 2nd. he is not sure that benveniste knows more about it than anybody else
% \footnote{As shown by the French linguist Emile Benveniste, the term `time'
% is
% derived from Latin `tempus', a term coined from temestus, tempestas and
% temerare, from a root meaning `...'.{where? ask Serge} It does not stem,
% as thought for a long time, from the Greek teino (tendere, stirare) nor
% from
% temno (to cut). }}%
%
% time: Germanic timon: stretch extend; English tide: fixed time
%
% In other languages the etymology is similarly interesting and similarly
% unhelpful.
%
% {Table of times}
{\small
\begin{table}[t]
\small
\centering
\caption{Selected time\protect\index{time!values, table} measurements.}
\label{durmetab}
\dirrtabularstar
\begin{tabular*}{\textwidth}{@{}%
>{\PBS\raggedright\hspace{0.0em}\columncolor{hks152}[0pt][0.5cm]}p{93mm}
@{\extracolsep{\fill}} >{\columncolor{hks152}[0pt][0cm]} p{37mm} @{}}
%
\toprule
\tabheadf{Observation} & \tabhead{Time}\\[0.5mm]
%
\midrule
%
%Electron gravitational time & \csd{2.3 \cdot10^{66}}{s}\\
%
Shortest measurable time & $\csd{10^{44}}{s}$\\
%
Shortest time ever measured\index{shortest measured time} &
$\csd{10}{ys}$\\
%
Time for light to cross a typical atom & 0.1 to $\csd{10}{as}$\\
%
Shortest laser light pulse produced so far & $\csd{200}{as}$\\
%
Period of \iin{caesium} ground state hyperfine transition&
$\csd{108.782\,775\,707\,78}{ps}$\\
%
Beat of wings of fruit fly & $\csd{1}{ms}$\\
%
Period of pulsar\index{pulsar period} (rotating neutron star) \csaciin{PSR
1913+16} & $\csd{0.059\,029\,995\,271(2)}{s}$\\
%
Human `instant'\index{instant, human} & $\csd{20}{ms}$\\
%
Shortest lifetime of living being\index{life!shortest} & $\csd{0.3}{d}$\\
%
Average length of day 400 million years ago\index{day!length in the past} &
$\csd{79\,200}{s}$\\
%
Average length of day today & $\csd{86\,400.002(1)}{s}$\\
%
From birth to your 1000 million seconds anniversary & $\csd{31.7}{a}$\\
%
Age of oldest living tree & $\csd{4600}{a}$\\
%
Use of human language & $\csd{0.2}{Ma}$\\
%
Age of Himalayas\index{Himalaya age} & 35 to $\csd{55}{Ma}$\\
% Science news, p189, 24.3.2001
%
% March 2007
Age of oldest rocks, found in \iin{Isua Belt},
\iin{Greenland}\break % Added Nov 2008
and in
\iin{Porpoise Cove}, \iin{Hudson Bay} & $\csd{3.8}{Ga}$\\
% "Science" (Bd. 315, S. 1.704, 2007).
%
Age of Earth\index{Earth!age} & $\csd{4.6}{Ga}$\\
%
Age of oldest stars\index{star!age} & $\csd{13.8}{Ga}$ \\
%
Age of most protons\index{proton!age} in your body & $\csd{13.8}{Ga}$ \\
%
Lifetime of \iin{tantalum} nucleus ${}^{180m}{\rm Ta}$ & $\csd{10^{15}}{a}$\\
%
Lifetime of \iin{bismuth} ${}^{209}$Bi nucleus & $\csd{1.9(2) \cdot
10^{19}}{a}$\\
\bottomrule
%
\end{tabular*}
\end{table}
}
How exactly is the concept of time, including sequence and duration, deduced
from observations? Many people have looked into this question: astronomers,
physicists, watchmakers, psychologists and philosophers. All find:
%
\begin{quotation}
\noindent \csrhd {Time\index{time!deduction} is deduced by comparing
motions.}
\end{quotation}
\np This is even the case for children and animals. Beginning at a very young
age, they develop the concept of `time' from the comparison of motions in
their surroundings.\cite{childev}
%
Grownups take as a standard the motion of the Sun and call the resulting type
of time \ii{local time}. From the Moon they deduce a \ii{lunar calendar}. If
they take a particular village clock on a European island they call it the
\ii[time! coordinate, universal]{universal time coordinate} (\csaciin{UTC}),
once known as `Greenwich mean time.'%
%
\footnote{Official \csaciin{UTC} is used to determine the phase of the
power grid, phone and internet companies' bit streams and the signal
to the \csaciin{GPS} system. The latter is used by many navigation
systems around the world, especially in ships, aeroplanes and mobile
phones. For more information, see the \url{www.gpsworld.com}
website. The timekeeping infrastructure is also important for
other parts of the modern economy. Can you spot the
most\challengenor{timeind} important ones?}%
%
Astronomers use the movements of the stars and call the result \ii{ephemeris
time} (or one of its successors). An observer who uses his personal watch
calls the reading his \ii[time!proper]{proper time}; it is often used in the
theory of relativity.
Not every movement is a good standard for time. In the year 2000, an Earth
rotation did\seepageone{secondsday} not take 86\,400 seconds any more, as it did
in the year 1900, but 86\,400.002 seconds. Can you deduce in which year your
birthday will have shifted by a whole day from the time predicted with 86\,400
seconds?\challengenor{birthshift}
All methods for the definition of time are thus based on comparisons of
motions. In order to make the concept as precise and as useful as possible, a
\emph{standard} reference motion is chosen, and with it a standard sequence
and a standard duration is defined. The device that performs this task is
called a \ii[clock!definition]{clock}. We can thus answer the question of the
section title:
\begin{quotation}
\noindent \csrhd \iin[time!deduced from clocks]{Time is what we read from a
clock}.
\end{quotation}
\np Note that all definitions of time used in the various branches of physics
are equivalent to this one; no `deeper' or more fundamental definition is
possible.%
%
\footnote{The oldest clocks are \iin{sundials}. The science of making them
is called \ii{gnomonics}.\cite{zenkert}} %
%
Note that the word `\iin{moment}' is indeed derived from the word `movement'.
Language follows physics in this case. Astonishingly, the definition of time
just given is final; it will never be changed, not even at the top of Motion
Mountain. This is surprising at first sight, because many books have been
written on the nature of time. Instead, they should investigate the nature of
motion!
\begin{quotation}
\noindent \csrhd Every clock reminds us that in order to understand time, we
need to understand motion.
\end{quotation}
% Cheap literature often suggests the opposite, in contrast to the facts.
\np But this is the aim of our walk anyhow. We are thus set to discover all
the secrets of time as a side result of our adventure.
Time is not only an aspect of observations, it is also a facet of personal
experience. Even in our innermost private life, in our thoughts, feelings and
dreams, we experience sequences and durations. Children learn to relate this
internal experience of time with external observations, and to make use of the
sequential property of events in their actions. Studies of the origin of
psychological time show that it coincides  apart from its lack of accuracy
 with clock time.%
%
\footnote{The brain contains numerous clocks.\seepagefive{humanclocks} The
most precise clock for short time intervals, the internal interval timer of
the brain, is more accurate than often imagined, especially when trained. For
time periods between a few tenths of a second,\cite{persclock} as necessary
for music, and a few minutes, humans can achieve timing accuracies of a few
per cent.} %
%
Every living human necessarily uses in his daily life the concept of time as a
combination of sequence and duration; this fact has been checked in numerous
investigations. For example, the term `when' exists in all human
languages.\cite{when}
Time is a concept \emph{necessary} to\index{time!is necessary} distinguish
between observations.
% of our thinking; we introduce it
% automatically when we distinguish between observations which are part of a
% sequence. There is no way to avoid time when talking about life. This
% seems
% to contradict our aim to go beyond time. In fact it doesn't, as we
% will find out in the third part of our mountain ascent.
%
% All experiences collected in everyday life with the help of clocks can be
% summarized in a few sentences.
In any sequence of observations, we observe that events succeed each other
smoothly, apparently without end. In this context, `smoothly' means that
observations that are not too distant tend to be not too different. Yet
between two instants, as close as we can observe them, there is always room
for other events. %\label{classtime}
Durations, or \ii[time!interval]{time intervals}, measured by different people
with different clocks agree in everyday life; moreover, all observers agree on
the order of a sequence of events. Time is thus \emph{unique} in everyday
life.\index{time!is unique} One also says that time is \emph{absolute} in
everyday life.\index{time!is absolute}
% May 2014, spelling ok, index ok
Time is necessary to distinguish between observations. For this reason, all
observing devices that distinguish between observations, from brains to
dictaphones and video cameras, have internal clocks. In particular, all
animal brains have internal clocks. These brain clocks allow their users to
distinguish between present, recent and past data and observations.
% Moved here in May 2014, spelling ok, index ok
When \iname[Galilei, Galileo]{Galileo} studied motion in the seventeenth
century, there were as yet no stopwatches.
%\cite{galwatch}
He thus had to build one himself, in order to measure times in the range
between a fraction and a few seconds. Can you imagine how he did
it?\challengenor{galclock}
%
% {Properties of Galilean time}
%
{\small
\begin{table}[t]
\small
\caption{Properties\protect\index{time!properties, table} of Galilean time.}
\label{galt}
\centering
\dirrtabularstarnolines
\begin{tabular*}{\textwidth}{@{} l
@{\extracolsep{\fill}} l
@{\extracolsep{\fill}} l
@{\extracolsep{\fill}} l @{}}
%
\toprule
%
\tabheadf{Instants of time}
& \tabhead{Physical} & \tabhead{Mathematical} & \tabhead{Definition}\\
& \tabhead{property} & \tabhead{name} & \\[0.5mm]
%
\midrule
%
Can be distinguished &\iin{distinguishability} & \iin{element of set} &
\seepagethree{setdefi} \\
Can be put in order &\iin{sequence} & \iin{order} &
\seepagefive{topocont} \\
Define duration &\iin{measurability} & \iin{metricity}&
\seepagefour{mespde4} \\
Can have vanishing duration &\iin{continuity} & \iin{denseness},
\iin{completeness} & \seepagefive{topocont} \\
Allow durations to be added&\iin{additivity} & \iin{metricity}&
\seepagefour{mespde4} \\
Don't harbour surprises &\iin{translation invariance} & \iin{homogeneity} &
\seepageone{surpdet}\\
%don't end & \iin{infinity} & \iin{openness}&
% \seepagethree{settab} \\
Don't end &\iin{infinity} & \iin{unboundedness} & \seepagethree{settab} \\
Are equal for all observers & \iin[time!absoluteness of]{absoluteness} &
\iin{uniqueness} & \\
%
\bottomrule
\end{tabular*}
\end{table}
}
% Impr. May 2014, spellchecked
If we formulate with precision all the properties of time that we experience
in our daily life, we are lead to \tableref{galt}. This concept of time is
called \ii[Galilean time!definition]{Galilean time}. All its properties can
be expressed simultaneously by describing time with the help of
\ii[numbers!and time]{real numbers}. In fact,\index{time!measured with real
numbers} real numbers have been constructed by mathematicians to have
exactly the same properties as Galilean time,\index{time!absolute} as
explained in the chapter on the brain.\seepagethree{realnu} In the case of
Galilean time, every instant of time can be described by a real number, often
abbreviated $t$. The duration of a sequence of events is then given by the
difference between the time values of the final and the starting event.
% Impr. May 2014, spellchecked
We will have quite some fun with Galilean time in this part of our adventure.
However, hundreds of years of close scrutiny have shown that \emph{every
single} property of Galilean time listed in \tableref{galt} is approximate,
and none is strictly correct.\index{Galilean time!limitations} This story is
told in the rest of our adventure.
% Making these
% discoveries, with all the surprises that follow, is part of our journey.
\csepsfnb[p]{iclocks}{scale=1}{Different types of clocks: a hightech
\protect\iin{sundial} (size c.\,30\,cm), a naval pocket chronometer (size
c.\,6\,cm), a caesium \protect\iin{atomic clock} (size c.\,4\,m), a group of
cyanobacteria and the Galilean satellites of Jupiter
({\textcopyright}~\protect\iinn{Carlo Heller} at
\protect\url{www.heliosuhren.de}, \protect\iname{Anonymous},
\protect\iname{INMS}, \protect\iname{Wikimedia}, \protect\iname{NASA}).}
%
% Reread Jul 2016
\subsection{Clocks}
% Index OK
% Dec 2013
\begin{quote}
The most valuable thing a man can spend is time.\\
\inameq{Theophrastus}
\end{quote}
% Jun 2007
A \emph{clock} is\index{clock(} a moving system whose position can be read.
There are many types of clocks:\label{livingclocktable} stopwatches,
twelvehour clocks, sundials, lunar clocks, seasonal clocks, etc. A few are
shown in \figureref{iclocks}. Most of these clock types are also found in
plants\cite{plantclocks} and animals, as shown in \tableref{biorhy}.
Interestingly, there is a strict rule in the animal kingdom: large clocks go
slow.\cite{morgado} How this happens is shown in \figureref{iageing},
another example of an \emph{allometric} \ii[scaling!`law']{scaling `law'}.
% Moved in May 2014, spelling ok, index ok
A {clock} is a moving system whose position can be read. Of course, a
\emph{precise} clock\index{clock!precision} is a system moving as regularly as
possible, with as little outside disturbance as possible. Clock makers are
experts in producing motion that is as regular as possible. We
will\seepageone{ipendulum} discover some of their tricks below. We will also
explore, later on, the limits for the precision of clocks.\seepagefive{kkkk}
Is there a perfect clock in nature? Do clocks exist at all?
% If one goes into the details, these turn out
% to be tricky questions.
We will continue to study these questions throughout this work and eventually
reach a surprising conclusion. At this point, however, we state a simple
intermediate result: since clocks do exist, somehow there is in nature an
intrinsic, natural and \ii[time!measurement, ideal]{ideal} way to measure
time. Can you see it?\challengenor{timenat}
%
% {Biological rhythms} % also appears in vol 5
%
{\small
\begin{table}[p]
\small
\caption{Examples of biological\protect\index{clock!types, table} rhythms and
clocks.}
\label{biorhy}
\centering
\dirrtabularstar %nolines
\begin{tabular*}{\textwidth}{@{}
>{\PBS\raggedright\columncolor{hks152}[0pt][2cm]} p{46mm}
@{\extracolsep{\fill}} >{\PBS\raggedright\columncolor{hks152}[0pt][2cm]}
p{57mm}
@{\extracolsep{\fill}} >{\PBS\raggedright} p{20mm}
@{}}
%
\toprule
%
\tabheadf{Living being } & \tabhead{Oscillating system} & \tabhead{Period} \\
%
\midrule
%
Sand hopper (\iie{Talitrus saltator}) & knows in which direction to flee from
the position of the Sun or Moon & circadian \\
%
Human (\iie{Homo sapiens})
% & highest sound and ear hair ocillations &
% \csd{50}{\muunit s} & \seepage{} \\
%
& gamma waves in the brain & 0.023 to \csd{0.03}{s} \\
%
& alpha waves in the brain & 0.08 to \csd{0.13}{s} \\
%
& heart beat & 0.3 to \csd{1.5}{s} \\
%
& delta waves in the brain & 0.3 to \csd{10}{s} \\
%
& blood circulation & \csd{30}{s} \\
%
& cellular circhoral rhythms & 1 to \csd{2}{ks} \\
%
& rapideyemovement sleep period & \csd{5.4}{ks} \\
%
& nasal cycle & 4 to \csd{9}{ks} \\
%
& growth hormone cycle & \csd{11}{ks} \\
%
& suprachiasmatic nucleus (SCN), circadian hormone
concentration,
temperature, etc.; leads to jet lag & \csd{90}{ks} \\
% Nov 2012
& skin clock & circadian \\
%
& monthly period & \csd{2.4(4)}{Ms}\\
%
& builtin aging & \csd{3.2(3)}{Gs}\\
%
Common fly (\iie{Musca domestica}) & wing beat & \csd{30}{ms} \\
%
Fruit fly (\iie{Drosophila melanogaster}) & wing beat for courting &
\csd{34}{ms} \\
%
% boring: & circadian cycle & \csd{86}{ks} \\
%
Most insects (e.g. wasps, fruit flies) & winter approach detection (diapause)
by length of day measurement; triggers metabolism changes & yearly \\
%
% !.!2 finish Algae (\iie{Gonyaulax polyhedra}) & & \\
Algae (\iie{Acetabularia}) & Adenosinetriphosphate (ATP) concentration & \\
%
%Cyanobacteria \\
%
Moulds (e.g. \iie{Neurospora crassa}) & conidia formation & circadian \\
%
Many flowering plants & flower opening and closing & circadian \\
%
Tobacco plant
% boring: & & circadian & \csd{86}{ks}\\
%
& flower opening clock; triggered by length
of days, discovered in 1920 by Garner and Allard & annual \\
%
\iie{Arabidopsis} & circumnutation & circadian \\
& growth & a few hours \\
%
Telegraph plant (\iie{Desmodium gyrans}) & side leaf rotation &
\csd{200}{s} \\
%
\iie{Forsythia europaea}, \iie{F. suspensa}, \iie{F.
viridissima}, \iie{F. spectabilis} & Flower petal oscillation,
discovered by \iinn{Van Gooch} in 2002 & \csd{5.1}{ks} \\
%
\bottomrule
\end{tabular*}
\end{table}
}
% \cssmallepsfnb{ilifetimescaling}{scale=1}{How %lifetime and}
% SENT EMAIL FEB 2008  emorgado@usach.cl
% New drawing in Mar 2012  filename is wrong...
\csepsfnb[p]{iageing}{scale=1}{How biological rhythms scale with size in
mammals: all scale more or less with a quarter power of the mass (after data
from the \protect\iname{EMBO} and \protect\iinn{Enrique Morgado}).}
%
\newpage % is needed here
% Reread Jul 2016
\subsection{Why do clocks go clockwise?}
% Index OK
\begin{quote}
What time is it\challengenor{north} at the \iin{North Pole} now?\\
\end{quote} %
\np Most rotational motions in our society,\index{clockwise!rotation} such as
athletic races, horse,\index{rotation!sense in athletic
stadia}\index{rotation!clockwise} bicycle or ice skating races, turn
anticlockwise.%
%
\footnote{Notable exceptions are most, but not all, Formula 1
races.} %
%
Mathematicians call this the positive rotation sense. Every \iin{supermarket}
leads its guests anticlockwise through the hall. Why? Most people are
righthanded, and the right hand has more freedom at the outside of a
circle. Therefore thousands of years ago chariot races in \iin{stadia} went
anticlockwise. As a result, all stadium races still do so to this day, and
that is why runners move anticlockwise. For the same reason, helical
\iin{stairs} in\index{helicity!of stairs} castles are built in such a way that
defending righthanders, usually from above, have that hand on the outside.
On the other hand, the clock imitates the shadow of \iin[sundial!story
of]{sundials}; obviously,\index{shadow!of sundials} this is true on the
northern hemisphere only, and only for sundials on the ground, which were the
most common ones. (The old trick to determine south by pointing the hour hand
of a horizontal watch to the Sun and halving the angle between it and the
direction of 12 o'{\kern 0.1mm}clock % (OK) check typesetting
does not work on the southern hemisphere  but there you can determine north
in this way.) So every clock implicitly continues to state on which hemisphere
it was invented. In addition, it also tells us that sundials on walls came in
use much later than those on the floor.\index{clock)}
%
% Reread Jul 2016, Dec 2016
\subsection{Does time flow?}
% Index OK
% \begin{quote}
% Eins, zwei drei, im Sauseschritt \\
% es eilt die Zeit, wir eilen mit.\\
% Wilhelm Busch \lived(18321908)\index{Busch, Wilhelm}
% \end{quote}
\begin{quote}
\selectlanguage{german}Wir können keinen Vorgang mit dem `Ablauf der Zeit'
vergleichen  diesen gibt es nicht , sondern nur mit einem anderen
Vorgang (etwa dem Gang des Chronometers).\selectlanguage{british}%
%
\footnote{`We cannot compare any process with `the passage of time'  there
is no such thing  but only with another process (say, with the working
of a chronometer).'}\\ % Odgen translation
%
Ludwig Wittgenstein, \bt Tractatus/ 6.3611\indname{Wittgenstein, Ludwig}
\end{quote}
\begin{quote}
\selectlanguage{french}Si le temps est un fleuve, quel est son
lit?\selectlanguage{british}%
%
\footnote{`If time is a river, what is his bed?'}\\
\end{quote}
\np The\label{tfno} expression\index{time!flow of(} `the \iin[flow!of
time]{flow of time}' is often used to convey that in nature change follows
after change, in a steady and continuous manner. But though the hands of a
clock `flow', time itself does not. Time is a concept introduced specially to
describe the flow of events around us; it does not itself flow, it
\emph{describes} flow. Time does not advance. Time is neither linear nor
cyclic. The idea that time flows is as hindering to understanding nature as is
the idea that mirrors exchange right and\seepagethree{mirr1}
left.\index{time!flow of}\index{flow!of time}
The misleading\label{arrownonsense} use of the incorrect expression `flow of
time' was propagated first by some flawed Greek %\serge
thinkers\cite{aristo} and then again by \iname[Newton, Isaac]{Newton}. And it
still continues. \iname{Aristotle}, careful to think logically, pointed out
its misconception, and many did so after him. Nevertheless, expressions such
as `time reversal',\index{time!reversal} the `irreversibility of time', and
the muchabused `time's arrow'\index{arrow of time}\index{time!arrow of} are
still common. Just read a popular science magazine chosen at random.\challengn
The fact is: time cannot be reversed, only motion can, or more precisely, only
velocities of objects; time has no arrow, only motion has; it is not the flow
of time that humans are unable to stop, but the motion of all the objects in
nature. Incredibly, there are even books written by respected
physicists\cite{arrowbook} that study different types of `time's arrows' and
compare them with each other. Predictably, no tangible or new result is
extracted.
\begin{quotation}
\npcsrhd Time does \emph{not} flow. Only bodies flow.
\end{quotation}
Time has no direction. Motion has. For the same reason, colloquial
expressions such as `the start (or end) of time' should be avoided. A motion
expert translates them straight away into `the start (or end) of
motion'.\index{time!flow of)}
% Confused expressions can lead reason astray in many ways; we must avoid them
% because they render the ascent of Motion Mountain unnecessarily difficult.
% They even prevent it beyond a certain stage, located about halfway to the
% top. With a clear understanding of time we now can continue with the next
% aspect of motion states.
%
% Reread Jul 2016
\subsection{What is space?}
% Index OK
\begin{quote}
The introduction of numbers\label{weylbio} as coordinates [...] is an act of
violence [...].\\
\iinns{Hermann Weyl}, \btsim Philosophie der Mathematik und
Naturwissenschaft/.%
%\yrend 1918/
\footnote{Hermann Weyl \lived(18851955) was one of the most important
mathematicians of his time, as well as an important theoretical physicist.
He was one of the last universalists in both fields, a contributor to
quantum theory and relativity, father of the term `gauge'
theory,\index{gauge!theory} and author of many popular texts.}
% there is also André Weil (19061998)
\end{quote}
% Why can we distinguish one tree from another? We see that they are in
% different positions.
\np Whenever we distinguish two objects from each other, such as two stars, we
first of all distinguish their positions. We distinguish positions with our
senses of sight, touch, proprioception and hearing. Position is therefore an
important aspect of the physical state of an object. A position is taken by
only one object at a time. Positions are limited. The set of all available
positions, called \ii[space!physical]{(physical) space}, acts as both a
\iin{container} and a \iin{background}.
Closely related to space and position is \ii{size}, the set of positions an
object occupies. Small objects occupy only subsets of the positions occupied
by large ones. We will discuss size in more detail shortly.\seepageone{BANT}
How do we deduce space from observations? During \iin{childhood}, humans (and
most higher animals) learn to bring together the various \emph{perceptions} of
space, namely the visual, the tactile, the auditory, the kinaesthetic, the
vestibular etc., into one selfconsistent set of experiences and description.
The result of this learning process is a certain concept of space in the
brain.
%
%Everybody who lives uses these properties of space.
% Among others,
Indeed, the question `where?' can be asked and answered in all languages of
the world. Being more precise, adults derive space from distance measurements.
The concepts of length, area, volume, angle and solid angle are all deduced
with their help. Geometers, surveyors, architects, astronomers, carpet
salesmen and producers of metre sticks base their trade on distance
measurements.
\begin{quotation}
\npcsrhd Space is formed from all the position and distance relations
between objects using metre
sticks.\index{space!definition}\index{metre!stick}\index{stick!metre}
\end{quotation}
Humans developed metre sticks to specify distances, positions and sizes as
accurately as possible.
Metre sticks work well only if they are straight.\index{straightness} But when
humans lived in the jungle, there were no straight objects around them. No
straight rulers, no straight tools, nothing. Today, a cityscape is essentially
a collection of straight lines. Can you\challengenor{lightline} describe how
humans achieved this?
Once humans came out of the jungle with their newly built metre sticks, they
collected a wealth of results. The main ones are listed in \tableref{galsp};
they are easily confirmed by personal experience. In particular, objects can
take positions in an apparently \ii[continuity]{continuous} manner: there
indeed are more positions than can be counted.%
\footnote{For a definition of uncountability, see \cspageref{uncounta} in
Volume III.} %
%
Size is captured by defining the distance between various positions, called
\ii{length}, or by using the field of view an object takes when touched,
called its \ii[surface!area]{surface area}. Length and area can be measured
with the help of a metre stick. (Selected measurement results are given in
\tableref{dismetab};
%In daily life, all length measurements performed by different people
% coincide.
some length measurement devices are shown in
\figureref{ilengthmeasurementdevices}.)
%
The length of objects is independent of the person measuring it, of the
position of the objects and of their orientation. In daily life the sum of
angles in any triangle is equal to two right angles. There are no limits to
distances, lengths and thus to space.
\cssmallepsfnb{iear3}{scale=1.25}{Two proofs that space has three dimensions:
the vestibular labyrinth in the inner ear of mammals (here a human) with three
canals and a knot ({\textcopyright}~\protect\iname{Northwestern University}).}
% SENT FEB 2008  thain@northwestern.edu
Experience shows us that space has three \iin{dimensions}; we can define
sequences of positions in precisely three independent ways. Indeed, the inner
\iin{ear} of (practically) all vertebrates has three semicircular canals that
sense the body's acceleration in the three dimensions of space, as shown in
\figureref{iear3}.%
%
\footnote{Note that saying that space has three dimensions \emph{implies} that
space is continuous; the %Dutch
mathematician and philosopher \iinn{Luitzen Brouwer} \livedplace(1881
Overschie1966 Blaricum) showed that dimensionality is only a useful concept
for continuous sets.} %
%
Similarly, each human eye is moved by three pairs of muscles.\index{eye
motion} (Why three?)\challengenor{eyemus}
%
Another proof that space has three dimensions is provided by \iin{shoelaces}:
if space had more than three dimensions, shoelaces would not be useful,
because knots exist only in threedimensional space. But why does space have
three dimensions? This is one of the most difficult question of physics.
We leave it open for the time being.
% attempt to an answer appears in the very last part of our walk.
It is often said that thinking in four dimensions is impossible. That is
wrong. Just try.\challengenor{fourknot} For example, can you confirm that in
four dimensions knots are impossible?
%
% {Properties of Galilean space}
{\small
\begin{table}[t]
\small
\caption{Properties of\protect\index{space!properties, table} Galilean space.}
\label{galsp}
\centering
\dirrtabularstarnolines
\begin{tabular*}{\textwidth}{@{}>{\PBS\raggedright\hspace{0.0em}} p{42mm}
@{\extracolsep{\fill}}>{\PBS\raggedright\hspace{0.0em}} p{34mm}
@{\extracolsep{\fill}}>{\PBS\raggedright\hspace{0.0em}} p{34mm}
@{\extracolsep{\fill}}>{\PBS\raggedright\hspace{0.0em}} p{20mm} @{}}
%
\toprule
%
\tabheadfp{Points, or positions in space}
& \tabhead{Physical property} & \tabhead{Mathematical name} &
\tabhead{Definition}\\[.5mm]
%
% & \tabhead{property} & \tabhead{name} & \\[0.5mm]
%
\midrule
%
Can be distinguished &\iin{distinguishability} & \iin{element of set} &
\leavevmode\seepagethree{setdefi} \\
Can be lined up if on one line &\iin{sequence} & \iin{order} &
\leavevmode\seepagefive{topocont} \\
Can form shapes &\iin{shape} & \iin{topology} & \leavevmode\seepagefive{topo}
\\
Lie along three independent directions
&\iin{possibility of knots} & 3\iin{dimensionality}
& \leavevmode\seepageone{eulc}, \seepagefour{vector4} \\ %
Can have vanishing distance &\iin{continuity} &
%\iin{completeness} &
\iin{denseness}, \iin{completeness} &
\leavevmode\seepagefive{topocont} \\
%
Define distances &\iin{measurability} & \iin{metricity} &
\leavevmode\seepagefour{mespde4} \\
% not so good:
Allow adding translations&\iin{additivity} & \iin{metricity}&
\leavevmode\seepagefour{mespde4} \\
%
Define angles &\iin{scalar product} & \iin{Euclidean space} &
\leavevmode\seepageone{eulc}\\
%
Don't harbour surprises &\iin{translation invariance} & \iin{homogeneity}\\
%don't end & \iin{infinity} & \iin{openness}&
% \seepagethree{settab} \\
Can beat any limit &\iin{infinity} & \iin{unboundedness} &
\leavevmode\seepagethree{settab} \\
Defined for all observers& \iin[space!absoluteness of]{absoluteness} &
\iin{uniqueness} & \leavevmode\seepageone{arstiiio}\\
\bottomrule
%
\end{tabular*}
\end{table}
}
Like time intervals, length intervals can be described most precisely with the
help of \ii[number!real]{real numbers}. In order to simplify communication,
standard \ii{units} are used, so that everybody uses the same numbers for the
same length. Units allow us to explore the general properties of
\ii[space!Galilean]{Galilean space} experimentally: space, the container of
objects, is continuous, threedimensional, isotropic, homogeneous, infinite,
Euclidean and unique  or `absolute'. In mathematics, a structure or
mathematical concept with all the properties just mentioned is called a
threedimensional \ii{Euclidean space}.\index{space!Euclidean} Its elements,
\ii[point!mathematical]{(mathematical) points}, are described by three real
parameters. They are usually written as
\begin{equation}
(x,y,z)
\label{eq:popop}
\end{equation}
and are called \ii{coordinates}. They specify and order the location of a
point in space. (For the precise definition of Euclidean spaces,
see\seepageone{eulc} below.) % this vol I
What is described here in just half a page actually took 2000 years to be
worked out, mainly because the concepts of `real number' and `coordinate' had
to be discovered first. The first person to describe points of space in this
way was the famous mathematician and philosopher \iinns{René
Descartes}\footnote{René Descartes or \inames{Cartesius} \livedplace(1596 La
Haye1650 Stockholm), % French
mathematician and philosopher, author of the famous statement `je pense,
donc je suis', which he translated into `cogito ergo sum'  I think
therefore I am. In his view this is the only statement one can be sure of.},
after whom the coordinates of expression (\ref{eq:popop}) are named
\ii{Cartesian}.%
%
% \comment{(He also was the first who wrote exponents the way used today, as
% explained in \appendixref{notation}.)}
\cssmallepsfnb{idescartes}{scale=0.25}{René Descartes \livedfig(15961650).}
Like time, space is a \emph{necessary} concept to describe the
world.\index{space!is necessary} Indeed, space is automatically introduced
when we describe situations with many objects. For example, when many spheres
lie on a billiard table, we cannot avoid using space to describe the relations
between them. There is no way to avoid using spatial concepts when talking
about nature.
%
%Second, since space can be measured, it \iin{exists}. We discuss
%this in detail in the intermezzo following this chapter.
Even though we need space to talk about nature, it is still interesting to ask
{why} this is possible. For example, since many length measurement methods do
exist  some are listed in \tableref{lengthmeastab}  and since they all
yield consistent results, there must be a \emph{natural} or \emph{ideal} way
to measure distances, sizes and straightness. Can you find
it?\challengenor{ll}
%
%light rays, of course
As in the case of time, each of the properties of space just listed has to be
checked. And again, careful observations will show that each property is an
approximation. In simpler and more drastic words, \emph{all} of them are
wrong. This confirms Weyl's statement at the beginning of this section. In
fact, his statement about the violence connected with the introduction of
numbers is told by every forest in the world.
%
% OUT IN JULY 2016:
% , and of course also by the one at the foot of Motion Mountain.
%
%
% Oct 2006
% To hear it, we need only listen carefully to what
% the trees have to tell.
% Oct 2006
% More about this soon.
%
The rest of our adventure will show this.
%
% {Table of lengths}
%
% fill in more, if possible
{\small
\begin{table}[t]
\small
\centering
\caption{Some measured\protect\index{distance!values,
table}\protect\index{length!values, table} distance values.}
\label{dismetab}
\dirrtabularstar
%\begin{tabular}{@{\hspace{0em}} p{87mm} p{45mm} @{\hspace{0em}}}
\begin{tabular*}{\textwidth}{@{}>{\columncolor{hks152}[0pt][0.5cm]}p{87mm}
@{\extracolsep{\fill}}>{\columncolor{hks152}[0pt][0cm]} p{45mm} @{}}
%
\toprule
%
\tabheadf{Observation} & \tabhead{Distance} \\[0.5mm]
%
\midrule
%
Galaxy Compton wavelength & $\csd{10^{85}}{m}$ (calculated only)\\
%
Planck length, % & $\csd{10^{35}}{m}$\\
%
the shortest measurable length & $\csd{10^{35}}{m}$\\
%
Proton diameter & $\csd{1}{fm}$\\
%
Electron Compton wavelength & $\csd{2.426\,310\,215(18)}{pm}$ \\
%
Smallest air oscillation detectable by human \iin{ear} & $\csd{11}{pm}$\\
% Aug 2011, calculated with M. Zecherle from Scanlab
% 1 khz, 0 db, i.e. 1 pW, amp = (1/pi f) sqrt(Power/2 rho v_{sound})
%
%
{Hydrogen atom} size & $\csd{30}{pm}$\\
%
Size of small bacterium & $\csd{0.2}{\muunit m}$\\
% average is 5 um, said a reader
%
Wavelength of visible light & 0.4~to~$\csd{0.8}{\muunit m}$\\
%
% Sep 2012
Radius of sharp razor blade & $\csd{5}{\muunit m}$\\
%
Point: diameter of smallest object visible with naked eye &
$\csd{20}{\muunit m}$\\
%
Diameter of human hair\index{hair!diameter} (thin to thick) & 30 to
$\csd{80}{\muunit m}$\\
%
Total length of \csaciin{DNA} in each human cell& \csd{2}{m}\\
%
{Largest living thing},\index{living thing, largest} the\indexe{Armillaria
ostoyae} fungus \emph{Armillaria ostoyae} & \csd{3}{km}\\
%
{Longest human throw with any object},\index{throw!record}
using a boomerang &
\csd{427}{m}\\
%
{Highest humanbuilt structure},\index{structure!highest humanbuilt}
Burj Khalifa &
\csd{828}{m}\\
%
{Largest spider webs}\index{web!largest spider}
in Mexico &
\circa\csd{5}{km}\\
%
Length of Earth's Equator & $\csd{40\,075\,014.8(6)}{m}$\\
%
% Impr. Oct 2010
Total length of human blood vessels (rough estimate) &$ 4 to \csd{16\cdot10^{4}}{km}$\\
% 40 billion capillaries of 1 mm length; my own estimate
% interenet gives between 100 000 and 160 000 km
%
% Impr. Oct 2010
Total length of human nerve cells (rough estimate) &$ 1.5 to
\csd{8\cdot10^{5}}{km}$\\
% Brain: 150 000 to 180 000 km, but higher numbers ``millions of miles'' are
% often quoted
%
Average distance to Sun & $\csd{149\,597\,870\,691(30)}{m}$\\
%
Light year&$\csd{9.5}{Pm}$\\
%
Distance to typical star at night & $\csd{10}{Em}$\\
%
Size of\index{galaxy!size} galaxy & $\csd{1}{Zm}$\\ % 10 Zm too large, says Lothar Beyer
%
Distance to Andromeda galaxy & $\csd{28}{Zm}$\\ % 2.9 mio light years
%
Most distant visible object & $\csd{125}{Ym}$\\
% Mar 2004, redshift 10, = 13.2 Mrd al, french suiss
%
\bottomrule
\end{tabular*}
\end{table}
}
\begin{quoteunder}
\csgreekok{M'etron >'ariston.}\footnote{`Measure is the best (thing).'
Cleobulus (\csgreekok{Kleoboulos}) of Lindos, \lived(\circa620550
{\csac{BCE}}) was another of the proverbial seven sages.}\\
\inames{Cleobulus}
\end{quoteunder}
\csepsfnb{ilengthmeasurementdevices}{scale=1}{Three mechanical (a vernier
caliper,\protect\index{caliper} a micrometer screw, a
moustache)\protect\index{moustache} and three optical (the eyes, a laser
meter, a light curtain) length and distance measurement devices
({\textcopyright}~\protect\url{www.medienwerkstatt.de},
% SENT EMAIL FEB 2008  bogusch@medienwerkstatt.de
\protect\iname{Naples Zoo},
% SENT EMAIL FEB 2008  tim@napleszoo.com
\protect\iname{Keyence},
% SENT EMAIL FEB 2008  keyence@keyence.com
and \protect\iname{Leica Geosystems}).}
% SENT EMAIL FEB 2008  Alessandra.Doell@leicageosystems.com
%
% {Table of length measurement methods}
%
%
{\small
\begin{table}[t]
\small
\centering
\caption{Length measurement\protect\index{length!measurement devices,
table}\protect\index{distance!measurement devices, table} devices in
biological and engineered systems.}
\label{lengthmeastab}
\dirrtabularstarnolines
\begin{tabular*}{\textwidth}{@{}
>{\PBS\raggedright\hspace{0mm}} p{68mm} @{\extracolsep{\fill}}
>{\PBS\raggedright\hspace{0mm}} p{28mm} @{\extracolsep{\fill}}
>{\PBS\raggedright\hspace{0mm}} p{28mm}
% @{\extracolsep{\fill}} p{12mm}
@{}}
%
\toprule
%
\tabheadf{Measurement} & \tabhead{Device} & \tabhead{Range}
% & \tabhead{Precision}
\\[0.5mm]
%
\midrule
%
\emph{Humans} \\
%
Measurement of body shape, e.g.~finger distance, eye position, teeth distance
& muscle sensors & \csd{0.3}{mm} to \csd{2}{m}\\
%
Measurement of object distance & stereoscopic vision & 1 to \csd{100}{m}\\
%
Measurement of object distance & sound echo effect & 0.1 to \csd{1000}{m}\\
%
\emph{Animals}\cstabhlineup \\
%
% (NO) more LENGTH (not distance) sensors in animals?
%
Measurement of hole size & moustache & up to \csd{0.5}{m}\\
%
Measurement of walking distance by desert ants & step counter & up to
\csd{100}{m}\\
%
Measurement of flight distance by honey bees & eye & up to
\csd{3}{km}\\
%
Measurement of swimming distance by sharks & magnetic field map & up to
\csd{1000}{km}\\
%
Measurement of prey distance by snakes & infrared sensor & up
to \csd{2}{m}\\
%
Measurement of prey distance by bats, dolphins, and hump whales & sonar & up
to \csd{100}{m}\\
%
Measurement of prey distance by raptors & vision & 0.1
to \csd{1000}{m}\\
%
\emph{Machines}\cstabhlineup \\
%
Measurement of object distance by laser & light reflection & \csd{0.1}{m}
to \csd{400}{Mm}\\
%
Measurement of object distance by radar & radio echo & 0.1
to \csd{50}{km}\\
%
Measurement of object length & interferometer & \csd{0.5}{\muunit m}
to \csd{50}{km}\\
%
Measurement of star, galaxy or quasar distance & intensity decay & up to
\csd{125}{Ym}\\
% 10 billion al * 9 Pm =
%
Measurement of particle size & accelerator & down to
\csd{10^{18}}{m}\\
%
\bottomrule
\end{tabular*}
\end{table}
}
%
% Reread July 2016
\subsection{Are space and time absolute or relative?}
% Index OK
% \begin{quote}
% Die Lösung des Rätsels des Lebens in Raum und Zeit\\
% liegt \emph{au\ss erhalb} von Raum und Zeit.%
% %
% \footnote{The solution of the riddle of life in space and time
% lies \emph{outside} space and time.}
% %
% \\
% Ludwig Wittgenstein, \bt Tractatus/ 6.4312\indname{Wittgenstein, Ludwig}
% \end{quote}
\np In everyday life,\label{arstiiio} the\index{space!relative or
absolute}\index{time!relative or absolute} concepts of Galilean space and
time include two opposing aspects; the contrast has coloured every discussion
for several centuries. On the one hand, space and time express something
invariant and permanent; they both act like big \ii[space!as
container]{containers} for all the objects and events found in nature. Seen
this way, space and time have an existence of their own. In this sense one
can say that they are {fundamental} or \ii[spacetime!relative or
absolute]{absolute}. On the other hand, space and time are tools of
description that allow us to talk about relations between objects. In this
view, they do not have any meaning when separated from objects, and only
result from the relations between objects; they are derived, {relational} or
\emph{relative}. Which of these viewpoints do you prefer?\challengn The
results of physics have alternately favoured one viewpoint or the other. We
will repeat this alternation throughout our adventure, until we find the
solution.\cite{ar} And obviously, it will turn out to be a third option.
%
% Reread Jul 2016
\subsection{Size  why length and area exist, but volume does not}
% Index OK
\cssmallepsfnb{imaptool}{scale=1}{A\protect\index{curvimeter} curvi\meter or
odometer (photograph {\textcopyright}~\protect\iinn{Frank Müller}).}
A\label{BANT} central aspect of objects is their size. As a small child,
% All children learn the details of the shape and size of their own body.
% During this development stage, which takes place mainly
under school age, every human learns how to use the properties of size and
space in their actions. As adults seeking precision, %it seems obvious that
with the definition of \ii{distance} as the difference between coordinates
allows us to define \ii{length} in a reliable way. It took hundreds of years
to discover that this is \emph{not} the case. Several investigations in
physics and mathematics led to complications.
The physical issues started with an astonishingly simple question asked by
{Lewis Richardson}:%
%
\footnote{\iinns{Lewis~Fray Richardson} \lived(18811953),
English physicist and psychologist.} %
%
How long is the western coastline\index{coastline!length} of Britain?
% Or was it R.L. Richardson, as Peitgen says? This is cited there:
% R.L. Richardson,
% The problem of contiguity: an appendix of statistics of deadly quarrels,
% general systems Yearbook 6 (1961) 139187
%
%
Following the coastline on a map using an \iin{odometer}, a device shown in
\figureref{imaptool}, Richardson found that the length $l$ of the
coastline depends on the scale $s$ (say 1\,:\,10\,000 or 1\,:\,500\,000)
% !.!4 typeset correctly ? (3 times)
of the map used:
\begin{equation}
l = l_{0} \; s^{0.25}
\end{equation}
(Richardson found other exponentials for other coasts.) The number $l_{0}$ is
the length at scale 1\,:\,1. The main result is that the larger the map, the
longer the coastline. What would happen if the scale of the map were
increased even beyond the size of the original? The length would increase
beyond all bounds. Can a coastline really have \emph{infinite} length? Yes,
it can. \index{infinite!coastline}\index{coastline!infinite} In fact,
mathematicians have described many such curves; nowadays, they are called
\ii{fractals}. An infinite number of them exist, and \figureref{ifrac} shows
one example.%
%
\label{frcft}%
%
\footnote{Most of these curves are \ii[selfsimilarity]{selfsimilar}, i.e.,{}
they follow scaling `laws' similar to the abovementioned. The term
`fractal' is due to the %Polish
mathematician \iinn{Benoît Mandelbrot} and refers to a strange property: in
a certain sense, they have a nonintegral number $D$ of dimensions, despite
being onedimensional by construction. Mandelbrot saw that the noninteger
dimension was related to the exponent $e$ of Richardson by $D=1+e$, thus
giving $D=1.25$ in the\cite{saupefn} example above.
% Nov 2012
The number $D$ varies from case to case. Measurements yield a value
$D=1.14$ for the land frontier of Portugal, $D=1.13$ for the Australian
coast and $D=1.02$ for the South African coast.}
%
%
Can you\label{d1frac} construct another?\challengn
\csepsf{ifrac}{scale=1}{An example of a fractal: a selfsimilar curve of
\emph{infinite} % is now ok (\it does not work)
length (far right), and its construction.}
% This fractal has log4/log3 as fractal dimension
%
% (NO) add more examples of fractals
Length has other strange properties. The % Italian
mathematician \iinn{Giuseppe Vitali} was the first to
% NOT the great mathematician \iinn{Felix Hausdorff}
discover that it is possible to cut a line segment of length 1 into pieces
that can be reassembled  merely by shifting them in the direction of the
segment  into a line segment of length 2. Are you able to find such a
division using the hint\index{length!puzzle}\index{puzzle!length} that it is
only possible using %(countably)
infinitely many pieces?\challengedif{vitali}
To sum up
\begin{quotation}
\npcsrhd Length exists. But length\index{length!issues} is well defined only
for lines that are straight or nicely curved, but not for intricate lines,
or for lines that can be cut into infinitely many pieces.
\end{quotation}
We therefore avoid fractals and other strangely shaped curves in the
following, and we take special care when we talk about infinitely small
segments. These are the central assumptions in the\index{length!assumptions}
first five volumes of this adventure, and we should never forget them! We
will come back to these assumptions in the last part of our adventure.
In fact, all these problems pale when compared with the following problem.
Commonly, area and volume are defined using length. You think that it is
easy? You're wrong, as well as being a victim of prejudices spread by schools
around the world. To define area and volume with precision, their definitions
must have two properties: the values must be \ii[additivity!of area and
volume]{additive}, i.e.,{} for finite and infinite sets of objects, the total
area and volume\index{area!additivity}\index{volume!additivity} must be the
sum of the areas and volumes of each element of the set; and the values must
be \emph{rigid}, i.e.,{} if we cut an area or a volume into pieces and then
rearrange the pieces, the value must remain the same. Are such definitions
possible? In other words, do such concepts of volume and area exist?
For areas in a plane, we proceed in the following standard way: we define the
area $A$ of a rectangle of sides $a$ and $b$ as $A=ab$; since any polygon can
be rearranged into a rectangle with a finite number of straight
cuts,\challengenor{exer} we can then define an area value for all polygons.
Subsequently, we can define area for nicely curved shapes as the limit of the
sum of infinitely many polygons. This method is called \ii{integration}; it
is\seepageone{intexpla} introduced in detail in the section on physical
action.
% as well as in the appendix.\seepage{integr}
However, integration does not allow us to define area for arbitrarily bounded
regions. (Can you imagine such a region?)\challengenor{arar} For a complete
definition, more sophisticated tools are needed. They were discovered in 1923
by the famous mathematician \iinns{Stefan Banach}.%
%
\footnote{Stefan Banach \livedplace(1892 Krakow1945 Lvov), important % Polish
mathematician.} %
%
He proved that one can indeed define an area for any set of points whatsoever,
even if the border is not nicely curved but extremely complicated, such as the
fractal curve previously mentioned. Today this generalized concept of area,
technically a `finitely additive isometrically invariant measure,' is called a
\ii{Banach measure} in his honour.
%
% It extends the Lebesgue measure, in the case of 2 dimensions
%
% \footnote{Defining a Banach measure means to be able to assign a finite
% positive
% value to any set of points, however weird,
% with the properties of being \emph{rigid}, i.e.,{} invariant under
% translations,
% and \emph{additive} for disjunct sets.} %
%
% !.!4 {Put precise definition of Banach measure here.}
%
Mathematicians sum up this discussion by saying that since in two dimensions
there is a Banach measure, there is a way to define the concept of area  an
additive and rigid measure  for any set of points whatsoever.%
%
\footnote{Actually, this is true only for sets on the plane. For curved
surfaces, such as the surface of a sphere, there are complications that will
not be discussed here. In addition, the problems mentioned in the
definition of length of fractals also reappear for area if the surface to be
measured is not flat. A typical example is the area of the human lung:
depending on the level of details examined, the area values vary from a few
up to over a hundred square metres.} %
%
% \comment{Thus no Banach Tarski paradox in two dimensions. However, on the
% surface of a sphere one can change areas; thus there must be no Banach
% measure.}%
%
In short,
\begin{quotation}
\npcsrhd Area exists.\index{area!existence of} Area is well defined for
plane and other nicely behaved surfaces, but not for intricate shapes.
\end{quotation}
\cssmallepsfnb{ipoliedro}{scale=1}{A polyhedron with one of its dihedral
angles ({\textcopyright}~\protect\iinn{Luca Gastaldi}).}
%\cssmallepsf{imaxdehn}{scale=1}{A polyhedron with one of its dihedral
%angles}
% EMAILED FEB 2008 gastaldi.luca@libero.it
What is the situation in \emph{three} dimensions, i.e.,{} for \iin{volume}?
We can start in the same way as for area, by defining the volume $V$ of a
rectangular polyhedron with sides $a$, $b$, $c$ as $V=abc$. But then we
encounter a first problem: a general polyhedron cannot be cut into a cube by
straight cuts! The limitation was discovered in 1900 and 1902 by \iinns{Max
Dehn}.%
%
\footnote{Max Dehn \livedplace(1878 Hamburg1952 Black Mountain), % German
mathematician, student of David Hilbert.} %
%
He found that the possibility depends on the values of the edge angles, or
\iin{dihedral angles}, as the mathematicians call them. (They are defined in
\figureref{ipoliedro}.) If one ascribes to every edge of a general
polyhedron a number given by its length $l$ times a special function
$g(\alpha)$ of its dihedral angle $\alpha$, then Dehn found that the sum of
all the numbers for all the edges of a solid does not change under dissection,
provided that the function fulfils $g(\alpha+\beta)=g(\alpha)+g(\beta)$ and
$g(\pi)=0$. An example of such a strange function $g$ is the one assigning
the value 0 to any rational multiple of $\pi$ and the value 1 to a basis set
of irrational multiples of $\pi$. The values for all other dihedral angles of
the polyhedron can then be constructed by combination of rational multiples of
these basis angles.
%
Using this function, you may then deduce for yourself\challengenor{dehn} that
a cube cannot be dissected into a regular tetrahedron because their respective
Dehn invariants are different.%
%
\footnote{This is also told in the beautiful book by \asi M. Aigler,
G.M. Ziegler/ \bt Proofs from the Book/ Springer Verlag, \yrend 1999/ The
title is due to the famous habit of the great mathematician \iinn{Paul Erd\H
{o}s} to imagine that all beautiful mathematical proofs can be assembled
in the `book of proofs'.}
Despite the problems with Dehn invariants, a rigid and additive concept of
volume for polyhedra does exist, since for all polyhedra and, in general, for
all `nicely curved' shapes, the volume can be defined with the help of
integration. %\seepage{integr}
Now\label{btparax} let us consider general shapes and general cuts in {three}
dimensions, not just the `nice' ones mentioned so far.\index{dissection!of
volumes} We then stumble on the famous \ii[BanachTarski paradox or
theorem]{BanachTarski theorem} (or paradox). In 1924, \iinn{Stefan Banach}
%wwwgroups.dcs.standrews.ac.uk/~history/Mathematicians/Banach.html
and \iinns{Alfred Tarski}%
%
\footnote{Alfred Tarski \livedplace(1902 Warsaw1983 Berkeley),
influential % Polish
mathematician.} %
%
proved\cite{doughf}
%wwwgroups.dcs.standrews.ac.uk/~history/Mathematicians/Tarski.html
that it is possible to cut one sphere into five % not six
pieces that can be recombined to give \emph{two} spheres, each the size of the
original. This counterintuitive result is the BanachTarski theorem. Even
worse, another version of the theorem states: take any two sets not extending
to infinity and containing a solid sphere each; then it is always possible to
dissect one into the other with a \emph{finite} number of cuts. In particular
it is possible to dissect a {pea}\index{pea!dissection} into the
{Earth},\index{Earth!dissection} or vice versa. Size does not count!%
%
\footnote{The proof of the result does not need much mathematics; it is
explained beautifully by \iinn{Ian Stewart} in \ti Paradox of the spheres/
\jo New Scientist/, 14 January 1995, \ppend 2831/ The proof is based on the
axiom of choice, which is presented later on.\seepagethree{settab}
% !.!4 Read also his book: From here to infinity
% The proof could also be in there (this note appears twice in this text)
The BanachTarski paradox also exists in four dimensions, as it does in any
higher dimension. More mathematical detail can be found in the beautiful
book by \iinn{Stan Wagon}.\cite{wagonbook}}
%
In short, volume is thus not a useful concept at all!%
% in hyperbolic space, one does not even need the axiom of choice; one even
% gets a decomposition in Borel sets (said Lieven Marchand
% mal@bewoner.dma.be from the internet in July 1998)
%In 1994, Foreman and Dougherty expanded the result by showing that there is
% a
%finite collection of (disjoint open) sets in the unit cube which can be
% moved
%(by isometries) in such a way that the result (a disjoint open union) fills
% a
%cube of size 2.\cite{doughf}
\csepsfnb{icrystals}{scale=1}{Straight lines found in nature:
cerussite\protect\index{cerussite} (picture width approx. 3$\,$mm,
{\textcopyright}~\protect\iinn{Stephan Wolfsried})
% SENT EMAIL FEB 2008  stephan.wolfsried@tonline.de
%
%and hercynite,\protect\index{hercynite}
% ({\textcopyright}~\protect\iinn{Stephan Wolfsried})
% both pictures with a width
% of approximately 3$\,$mm
%
and selenite\protect\index{selenite} (picture width approx.~15$\,$m,
{\textcopyright}~Arch.{} \protect\iname{Speleoresearch \& Films/La Venta} at
\protect\url{www.laventa.it} and
\protect\url{www.naica.com.mx}).}
% SENT EMAIL FEB 2008  tux@tulliobernabei.it
% Improved in Oct 2004
The BanachTarski theorem raises two questions: first, can the result be
applied to gold or bread? That would solve many problems. Second, can it be
applied to empty space? In other words, are matter and empty space
continuous?\challengenor{contsp} Both topics will be explored later in our
walk; each issue will have its own, special consequences. For the moment, we
eliminate this troubling issue by restricting our interest  again  to
smoothly curved shapes (and cutting knives). With this restriction, volumes
of matter and of empty space do behave nicely: they are additive and rigid,
and show no paradoxes.%
%
\footnote{Mathematicians say that a socalled \ii{Lebesgue measure}
is sufficient in physics. This countably additive isometrically invariant
measure provides the most general way to define a volume.} %
%
Indeed, the cuts required for the BanachTarski paradox are not smooth; it is
not possible to perform them with an everyday knife, as they require
(infinitely many) infinitely sharp bends performed with an infinitely sharp
knife. Such a knife does not exist. Nevertheless, we keep in the back of our
mind that the size of an object or of a piece of empty space is a tricky
quantity  and that we need to be careful whenever we talk about it.
% Impr. July 2016
In summary,
\begin{quotation}
\npcsrhd Volume only exists as an approximation.\index{area!existence of}
Volume is well defined only for regions with smooth surfaces. Volume does
not exist in general, when infinitely sharp cuts are allowed.
\end{quotation}
We avoid strangely shaped volumes, surfaces and curves in the following, and
we take special care when we talk about infinitely small entities. We can talk
about length, area and volume \emph{only} with this restriction. This
avoidance is a central assumption in the\index{volume!assumptions} first five
volumes of this adventure. Again: we should never forget these restrictions,
even though they are not an issue in everyday life.
% Oct 2016  double with above, but I leave it
We will come back to the assumptions at the end of our adventure.
%
% Reread July 2016
\subsection{What is straight?}
% Index OK
\label{strrl}
%
When you see a solid object with a straight edge,\index{straight!lines in
nature}\index{line!straight, in nature} it is a 99\,\%safe bet that it is
manmade. Of course, there are exceptions,\index{crystal!and straight line} as
shown in \figureref{icrystals}.%
%
\footnote{Why do crystals have straight edges?\challenge % !!!5
Another example of straight lines in nature, unrelated to atomic structures,
is the wellknown Irish geological formation called the \iin{Giant's
Causeway}.\seepageone{istarchcolumns} Other candidates that might come
to mind, such as certain bacteria which have (almost) square or (almost)
triangular shapes are not\cite{bacshape2} counterexamples, as the shapes
are only approximate.} %
%
The largest crystals ever found are \csd{18}{m} in length.\cite{mycrystalbook}
%
But in general, the contrast between the objects seen in a city  buildings,
furniture, cars, electricity poles, boxes, books  and the objects seen in a
forest  trees, plants, stones, clouds  is evident: in the forest no object
is straight or flat, whereas in the city most objects are.
% OUT ; DOUBLE
% How is it possible for humans to produce straight objects while there are
% almost none to be found in nature?
% How can man make objects which are more
% straight than the machine tools one finds in nature?
Any forest teaches us the origin of straightness;\seepageone{isunbeams} it
presents tall tree trunks and rays of daylight entering from above through the
leaves. For this reason\indexs{straightness} we call a line \emph{straight}
if it touches either a \iin{plumbline} or a light ray along its whole length.
In fact, the two definitions are equivalent. Can you confirm this?
% (This is not an easy question.)
Can you find another definition?\challengenor{list} Obviously, we call a
surface \emph{flat} if for any chosen orientation and position the surface
touches a plumbline or a light ray along its whole extension.
\csepsfnb{iearthfull}{scale=0.755}{A photograph of the Earth  seen from the
direction of the Sun (NASA).}
% 0.788 for 140 mm ok, 758 gerade zu groß für hier
In summary, the concept of \iin{straightness}  and thus also of
\iin{flatness}  is defined with the help of bodies or radiation. In fact,
all spatial concepts, like all temporal concepts, require motion for their
definition.
%
% Reread Jul 2016
\subsection{A hollow Earth?}
% Index OK
\csepsfnb{finnenwe}{scale=0.7}{A model illustrating the hollow Earth theory,
showing how day and night appear ({\textcopyright}~\protect\iinn{Helmut
Diehl}).}
% SENT EMAIL FEB 2008  helmut.diehl@gmx.de
Space and straightness pose subtle challenges.\label{a123456} Some strange
people maintain that all humans live on the \emph{inside} of a sphere; they
call this the \emph{hollow Earth model}.\index{Earth!hollow} They claim that
the Moon, the Sun and the stars are all near the centre of the hollow
sphere,\cite{hollowearth} as illustrated in \figureref{finnenwe}. They also
explain that light follows curved paths in the sky and that when conventional
physicists talk about a distance $r$ from the centre of the Earth, the real
hollow Earth distance is $r_{\rm he}=R_{\rm Earth}^2/r$. Can you show that
this model is wrong?\challengenor{hollow} \iinns{Roman
Sexl}%
%
\footnote{Roman Sexl, \lived(19391986), important Austrian physicist,
author of several influential textbooks on gravitation and relativity.} %
%
used to ask this question to his students and fellow physicists.
The answer is simple: if you think you have an argument to show that the
hollow Earth model is wrong, you are mistaken! There is \emph{no way} of
showing that such a view is wrong. It is possible to explain the horizon, the
appearance of day and night, as well as the satellite photographs of the round
Earth, such as \figureref{iearthfull}.\challengn To explain what happened
during a flight to the Moon is also fun. A consistent hollow Earth view is
fully \emph{equivalent} to the usual picture of an infinitely extended space.
We will come back to this problem in the section\seepagetwo{hollowea} on
general relativity.
%
% Impr. Mar 2016
\subsection{Curiosities and fun challenges about everyday space and time}
% Including geometry
% Index OK
% Space and time lead to many thoughtprovoking questions.
% (NO!) Add Augustine citation on time here. ``what is time?''
%
% Quid est ergo tempus? Si meno ex me quaeret,scio, si quaerenti explicare
% velim, nescio. Augustinus
\begin{curiosity}
% Jan 2006
\item[] How does\label{bulletmeas} one measure the speed of a gun
bullet\index{bullet!speed measurement} with a stop watch, in a space of
\csd{1}{m^3}, without electronics?\challengenor{bulletspeed} Hint: the same
method can also be used to measure the speed of light.\index{light
speed!measurement}
% Apr 2013
\item For a striking and interactive way to zoom through all length scales in
nature, from the Planck length to the size of the universe, see the website
\url{www.htwins.net/scale2/}.
% Mar 2008
\item What is faster: an arrow or a motorbike?\challengenor{arrowbike}
% Aug 2015
\item Why are manholes always round?\challengenor{roundmanhole}
% Jul 2016
\item Can you show to a child that the sum of the angles in a triangle equals
two right angles? What about a triangle on a sphere or on a
saddle?\challengn
% Dec 2015
\item Do you own a glass\index{glass!and $\pi$} whose height is
larger\index{p@$\pi$ and glasses} than its maximum circumference?
% Jan 2014
\item A gardener wants to plant nine trees in such a way that they form ten
straight lines with three trees each. How does he do it?\challengn
% * * *
% * * *
% * * *
% Oct 2012
\item How fast does the grim reaper walk?\index{reaper, grim}\index{grim
reaper} This question is the title of a publication in the British Medial
Journal from the year 2011. Can you imagine how it is
answered?\challengedif{grimreaper}
% Feb 2012
\item Time measurements require periodic phenomena. Tree rings are traces of
the seasons. Glaciers also have such traces, the \emph{ogives}. Similar
traces are found in teeth. Do you know more examples?
% Oct 2009
\item A man wants to know how many stairs he would have to climb if the
escalator in front of him, which is running upwards, were standing still.
He walks up the escalator and counts 60 stairs; walking down the same
escalator with the same speed he counts 90 stairs. What is the
answer?\challengenor{escalpu}
% Oct 2009, Mar 2016
\item You have two \iin[hourglass puzzle]{hourglasses}: one needs 4 minutes
and one needs 3 minutes.\index{puzzle!hourglass} How can you use them to
determine when 5 minutes are over?\challengn
% May 2016
\item You have two fuses of different length that each take one minute to
burn. You are not allowed to bend them nor to use a ruler. How do you
determine that \csd{45}{s} have gone by?\challengn Now the tougher puzzle:
How do you determine that \csd{10}{s} have gone by with a single fuse?
% Der Spiegel, 2016
% Sie zünden eine Schnur gleichzeitig an beiden Enden an und im selben Moment
% die andere Schnur an nur einem Ende. Nach 30 Sekunden ist Schnur Nummer eins
% vollständig abgebrannt, Schnur Nummer zwei zur Hälfte. In diesem Augenblick
% zünden Sie bei der zweiten Schnur das noch freie Ende an. 15 Sekunden später
% ist auch diese Schnur abgebrannt  macht insgesamt 45 Sekunden.
% Wie aber misst man zehn Sekunden mit einer 60SekundenSchnur? Ganz einfach:
% Es müssen permanent sechs Flammen brennen  dann ist die Schnur sechs Mal
% schneller abgebrannt als bei nur einer Flamme. Sie zünden die Schnur
% zugleich an den beiden Enden und an zwei beliebigen Punkten dazwischen
% an. Aus den beiden mittleren Zündpunkten entstehen vier Flammen, weil diese
% jeweils in beide Richtungen wandern. Sobald ein Segment abgebrannt ist,
% müssen Sie aber sofort in einem anderen Segment einen beliebigen inneren
% Punkt anzünden, damit weiterhin sechs Flammen brennen. Am Ende beträgt die
% gesamte Brenndauer der Lunte zehn Sekunden.
% Oct 2009, Jan 2014, Mar 2016
\item You have three wine containers: a full one of 8 litres, an empty one of
5 litres, and another empty one of 3
litres.\index{wine!puzzle}\index{puzzle!five litre} How can you use them to
divide the wine evenly into two?\challengn
% Nov 2008
\item How can you make a hole in a \iin{postcard} that allows you to step
through it?\challengenor{postcard}
% Dec 2008
\cssmallepsf{iglassfifth}{scale=1}{At what height is a conical glass half
full?}
% Dec 2008
\item What fraction of the height of a conical glass, shown in
\figureref{iglassfifth}, must be filled to make the glass half
full?\challengenor{fifth}
% Dec 2008
\item How many \iin[pencil!puzzle]{pencils} are needed to draw a line as long
as the Equator of the Earth?\challengenor{penc}
% Aug 2015
\item Can you place five\index{coin!puzzle}\index{puzzle!coin} equal coins so
that each one touches the other four?\challengn Is the stacking of two
layers of three coins, each layer in a triangle, a solution for six coins?
Why?
What is the smallest number of coins that can be laid flat on a table so
that every coin is touching exactly three other coins?\challengn % 16
% Jan 2014
\item Can you find three crossing points on a chessboard that lie on an
equilateral triangle?\challengn % No
% Feb 2007
\item The following bear puzzle is well known: A hunter leaves his home, walks
\csd{10}{km} to the South and \csd{10}{km} to the West, shoots a bear, walks
\csd{10}{km} to the North, and is back home. What colour is the bear? You
probably know the answer straight away. Now comes the harder question,
useful for winning money in bets.\index{bear!puzzle} The house could be on
{several} \emph{additional} spots\index{puzzle!bear} on the Earth; where are
these less obvious spots from which a man can have \emph{exactly} the same
trip (forget the bear now) that was just described and be at home
again?\challengenor{whitebear}
% Aug 2007
\cssmallepsf{isnailhorse}{scale=0.9}{Can the snail reach the horse once it
starts galloping away?}
% Aug 2007
\item Imagine a rubber band that is attached to a wall on one end and is
attached to a horse at the other end, as shown in \figureref{isnailhorse}.
On the rubber band, near the wall,\index{puzzle!snail and horse} there is a
snail. Both the snail and the horse start moving, with typical speeds 
with the rubber being infinitely stretchable. Can the snail reach
the\challengenor{snailhorse} horse?
% Jun 2007
\item For a mathematician, \csd{1}{km} is the same as \csd{1000}{m}. For a
physicist the two are different! Indeed, for a physicist, \csd{1}{km} is a
measurement lying between \csd{0.5}{km} and \csd{1.5}{km}, whereas
\csd{1000}{m} is a measurement between \csd{999.5}{m} and \csd{1000.5}{m}.
So be careful when you write down measurement values. The professional way
is to write, for example, \csd{1000(8)}{m} to mean \csd{1000\pm 8}{m}, i.e.,
a value that lies between 992 and \csd{1008}{m} with a probability of
68.3\,\%.\seepageone{accuprec1}
% Jun 2005
\item Imagine a black spot on a white surface. What is the colour of the line
separating the spot from the background?\challengenor{piercech} This
question is often called \iin{Peirce's puzzle}.
% May 2005
\item Also bread is an (approximate) fractal, though an irregular one. The
fractal dimension of bread is around 2.7. Try to
measure\challengenor{breadfrac} it!
% May 2007
\item How do you find the centre of a \iin{beer mat} using paper and
pencil?\challengn
\item How often in 24 hours do the hour and minute hands of a \iin[clock
puzzles]{clock} lie on top of each other?\challengenor{clhands} For clocks
that also have a second hand, how often do all three hands lie on top of
each other?\index{hands of clock}
% July 2014
\item How often in 24 hours do the hour and minute hands of a
\iin[clock!puzzles]{clock} form a right\challengenor{clhandsra}
angle?\index{hands of clock}\index{puzzle!about clocks}
\item How many times in twelve %\cite{fl}
hours can the two hands of a \iin[clock!exchange of hands]{clock} be
\emph{exchanged} with the result that the new situation shows a \emph{valid}
time?\challengenor{clex} What happens for clocks that also have a third hand
for seconds?
\item How many minutes does the Earth rotate in one\challengenor{minu} minute?
\item What is the highest speed\index{throwing!speed record} achieved by
throwing (with and without a racket)? What was the
projectile\challengenor{speedthrow} used?
% improved April 2010
\item A rope is put around the Earth, on the Equator, as tightly as possible.
The rope is then lengthened by \csd{1}{m}. Can a mouse slip under the
rope?\challengenor{mouserope} The original, tight rope is lengthened by
\csd{1}{mm}. Can a child slip under the rope?
\item Jack was rowing his boat on a river. When he was under a bridge, he
dropped a ball into the river. Jack continued to row in the same direction
for 10 minutes after he dropped the ball. He then turned around and rowed
back. When he reached the ball, the ball had floated \csd{600}{m} from the
bridge. How fast was the river flowing?\challengenor{bridge}
% May 2005
\item Adam and Bert are brothers. Adam is 18 years old. Bert is twice as old
as at the time when Adam was the age that Bert is now. How old is
Bert?\challengn
\item `Where am I?' is a common question; `When am I?' is almost never asked,
not even in other languages.\challengenor{wwfi} Why?
\item Is there a smallest time interval in nature? A smallest
distance?\challengenor{smi}
\item Given that you know what \iin{straightness} is, how would you
characterize or define the \iin{curvature} of a curved line using numbers?
And that of a surface?\challengenor{curv}
\item What is the speed of your \iin{eyelid}?\challengenor{eyelid}
% corrected may 2007
\item The surface area of the human body is about \csd{400}{m^2}. Can you say
where this large number comes from?\challengenor{surfacehuman}
% % June 2005  not not put in
% \item Are you able to you lick your \iin[elbow licking]{elbow}?\challengn
\csepsfnb{ivernier}{scale=1.0}{A 9to10 vernier/nonius/clavius and a
19to20 version (in fact, a 38to40 version) in a
caliper\protect\index{caliper} %\break % added Nov 2008, out in June 2011
({\textcopyright}~\protect\url{www.medienwerkstatt.de}).}
% SENT EMAIL FEB 2008  bogusch@medienwerkstatt.de
\item How does a \ii{vernier} work? It is called \ii{nonius} in other
languages. The first name is derived from a French military
engineer\footnote{\iinns{Pierre Vernier} \lived(15801637), French military
officer interested in cartography.} who did not invent it, the second is a
play of words on the Latinized name of the Portuguese inventor of a more
elaborate device\footnote{\iinns{Pedro Nuñes} or \iinns{Peter Nonnius}
\lived(15021578), Portuguese mathematician and cartographer.} and the
Latin word for `nine'. In fact, the device as we know it today  shown in
\figureref{ivernier}  was designed around 1600 by \iinns{Christophonius
Clavius},\footnote{Christophonius Clavius or Schl\"{u}ssel
\lived(15371612), Bavarian astronomer, one of the main astronomers of his
time.} the same astronomer whose studies were the basis of the
\iin{Gregorian calendar} reform of 1582. Are you able to design a
vernier/nonius/clavius that, instead of increasing the precision tenfold,
does so by an arbitrary factor?\challengenor{noni} Is there a limit to the
attainable precision?
\item Fractals in three dimensions bear many surprises. Let us generalize
\figureref{ifrac} to\seepageone{ifrac} three dimensions. Take a regular
\iin{tetrahedron}; then glue on every one of its triangular faces a smaller
regular tetrahedron, so that the surface of the body is again made up of
many equal regular triangles. Repeat the process, gluing still smaller
tetrahedrons to these new (more numerous) triangular surfaces. What is the
shape of the final fractal, after an infinite number of
steps?\challengenor{tetrafrac}
\cssmallepsf{icarpark}{scale=1.0}{Leaving a parking space.}[%
\psfrag{w}{\small $w$}%
\psfrag{d}{\small $d$}%
\psfrag{b}{\small $b$}%
\psfrag{L}{\small $L$}%
]
\item Motoring poses many mathematical problems. A central one is the
following \ii[parking!parallel]{parallel parking}
challenge:\index{car!parking} what is the shortest distance $d$ from the car
in front necessary to leave a parking spot without using reverse
gear?\challengenor{carpark1} (Assume that you know the geometry of your car,
as shown in \figureref{icarpark}, and its smallest outer turning radius
$R$, which is known for every car.) Next question: what is the smallest gap
required when you are allowed to manoeuvre back and forward as often as you
like?\challengenor{carpark2} Now a problem to which no solution seems to be
available in the literature:\index{parking!challenge} How does the gap
depend on the number, $n$, of times you use reverse
gear?\challengenor{carpark3} (The author had offered 50 euro for the first
wellexplained solution; the winning solution by \iinn{Daniel Hawkins} is
now found in the appendix.)
\item Scientists use a special way\index{exponential notation} to write large
and small numbers, explained in \tableref{exponot}.
% Apr 2005
%
% {Table of exponential notation}
%
{\small
\begin{table}[t]
\sbox{\cshelpbox}{\small\dirrtabularnolines
%
\begin{tabular}{@{}l@{\hspace{5mm}}l%
@{\extracolsep{\fill}}%
l
@{\extracolsep{\fill}}%
l@{\hspace{5mm}}l@{}}
%
\cmidrule(){12}\cmidrule(){45}
\tabheadf{Number} & \tabhead{Exponential} & \ \hspace{20mm}\ &
\tabhead{Number} & \tabhead{Exponential} \\
& \tabhead{notation} & \ \hspace{3mm}\ &
& \tabhead{notation} \\[0.5mm]
%
\cmidrule(){12}\cmidrule(){45}
%
1 & $10^{0}$ & & & \\
0.1 & $10^{1}$ & & 10 & $10^{1}$ \\
0.2 & $2 \cdot 10^{1}$ & & 20 & $2 \cdot 10^{1}$ \\
0.0324 & $3.24 \cdot 10^{2}$ & & 32.4 & $3.24 \cdot 10^{1}$ \\
0.01 & $10^{2}$ & & 100 & $10^{2}$ \\
0.001 & $10^{3}$ & & 1000 & $10^{3}$ \\
0.000\,1 & $10^{4}$ & & 10\,000 & $10^{4}$ \\
0.000\,056 & $5.6\cdot 10^{5}$ & & 56\,000 & $5.6 \cdot 10^{4}$ \\
0.000\,01\ \ \ \ \ \ & $10^{5}$ \ \ \ \ etc.
& & 100\,000\ \ \ \ \ \ & $10^{5}$ \ \ \ \ etc. \\
%
\cmidrule(){12}\cmidrule(){45}
\end{tabular}%
}% end of sbox
%
\centering
\begin{minipage}{\wd\cshelpbox}
\captionsetup{width=\wd\cshelpbox} % needed since Dec 2015
\caption{The exponential notation:\protect\index{exponents!notation, table}
how to write small and large
numbers.}\label{exponot}\noindent\usebox{\cshelpbox}
\end{minipage}
\end{table}
}
\item In 1996 the \iin{smallest experimentally probed distance} was\cite{qufe}
\csd{10^{19}}{m}, achieved between quarks at \iin{Fermilab}. (To savour
the distance value, write it down without the exponent.) What does this
measurement mean for the continuity of space?\challengenor{spacef}
\item \iname[Zeno of Elea]{Zeno}, the Greek philosopher, discussed in detail
what happens to a moving object at a given instant of time. To discuss with
him, you decide to build the fastest possible \iin{shutter} for a
photographic \iin{camera} that you can imagine. You have all the money you
want. What is the shortest \iin{shutter time} you would
achieve?\challengenor{shut2}
\item Can you prove \iin{Pythagoras' theorem} by geometrical means alone,
without using \hbox{coordinates}?\challengenor{pytageo} (There are more than
30 possibilities.)
\item Why are\seepageone{iearthfull} most planets and moons, including ours,
(almost) spherical (see, for example,
\figureref{iearthfull})?\challengenor{plan}
\item A rubber band connects the tips of the two hands of a clock. What is
the path followed by the midpoint of the band?\challengenor{flower}
\csepsf{iangles}{scale=1}{The definition of plane and solid angles.}%
[\psfrag{al}{\small$\alpha$}%
\psfrag{af}{\small$\alpha=\frac{a}{r}$}%
\psfrag{ar}{\small$a$}%
\psfrag{ra}{\small$r$}%
%
\psfrag{Wx}{\small$\Omega$}%
\psfrag{Wf}{\small$\Omega=\frac{A}{r^2}$}%
\psfrag{A}{\small$A$}%
\psfrag{rb}{\small$r$}%
]
% Apr 2005
\item There are two important quantities connected to angles. As shown in
\figureref{iangles}, what is usually called a \ii[angle!plane]{(plane)
angle} is defined as the ratio between the lengths of the arc and the
radius. A right angle is $\pi/2$ \ii{radian} (or \csd{\pi/2}{rad}) or
\csd{90}{\csdegrees}.
% Apr 2005
The \ii[angle!solid]{solid angle} is the ratio between area and the square
of the radius. An eighth of a sphere is $\pi/2$ \ii{steradian} or
\csd{\pi/2}{sr}. (Mathematicians, of course, would simply leave out the
steradian unit.) As a result, a small solid angle shaped like a cone and
the angle of the cone tip are \emph{different}. Can you find the
relationship?\challengenor{solidangle}
% Apr 2005, Impr. Nov 2011
\item The definition of angle helps to determine the size of a \iin{firework}
display. Measure the time $T$, in seconds, between the moment that you see
the rocket explode in the sky and the moment you hear the explosion, measure
the (plane) angle $\alpha$  pronounced `alpha'  of the ball formed by
the firework with your hand. The diameter $D$ is
\begin{equation}
D \approx \frac{6 \, \rm m}{s\, \, \csdegrees} T \, \alpha \cp
\end{equation}
Why?\challengn For more information about fireworks, see the
\url{cc.oulu.fi/~kempmp} website.
% Moved here Apr 2005:
By the way, the angular distance between the knuckles of an extended fist are
about \csd{3}{\csdegrees}, \csd{2}{\csdegrees} and \csd{3}{\csdegrees}, the
size of an extended hand
\csd{20}{\csdegrees}.\index{knuckle!angles}\index{angle!and knuckles} Can you
determine the other angles related to your hand?\challengenor{angulos}
\csepsfnb{itrigonometry}{scale=1}{Two equivalent definitions of the {sine},
{cosine}, {tangent}, {cotangent}, {secant} and {cosecant} of an angle.}
\csepsfnb{imoonsize}{scale=1}{How the apparent size of the Moon and the Sun
changes during a day.}
% Jun 2008
\item Using angles, the \emph{sine}, \emph{cosine}, \emph{tangent},
\emph{cotangent}, \emph{secant} and \emph{cosecant} can be defined, as shown
in \figureref{itrigonometry}. You should remember this from secondary
school. Can you confirm that\challengn
$\sin 15\csdegrees = (\sqrt{6}  \sqrt{2})/4$,
$\sin 18\csdegrees = (1 + \sqrt{5})/4$,
$\sin 36\csdegrees = \sqrt{10  2\sqrt{5}}/4$,
$\sin 54\csdegrees = (1 + \sqrt{5})/4$ and that
$\sin 72\csdegrees = \sqrt{10 + 2\sqrt{5}}/4$?
%
Can you show also that
\begin{equation}
\frac{\sin x }{x}= \cos\frac{x}{2}\cos\frac{x}{4}\cos\frac{x}{8}\ldots
\end{equation}
is correct?\challengn
\item Measuring angular size with the eye only is tricky. For example, can
you say whether the {Moon} is larger\index{Moon!size, angular} or\index{Sun
size, angular} smaller than the nail of your \iin{thumb} at the end of
your extended arm?\challengn Angular size is not an intuitive quantity; it
requires measurement instruments.
A famous example, shown in \figureref{imoonsize}, illustrates the
difficulty of estimating angles. Both\label{moonsizeillus} the Sun and the
Moon seem larger when they are on the horizon. In ancient times,
\iname{Ptolemy} explained this socalled \ii[Moon!illusion]{Moon illusion}
by an unconscious apparent distance change induced by the human
brain.\cite{moonillu} Indeed, the Moon illusion disappears when you look at
the Moon through your legs. In fact, the Moon is even \emph{further away}
from the observer when it is just above the horizon, and thus its image is
\emph{smaller} than it was a few hours earlier, when it was high in the
sky.\index{Moon!size illusion} Can you confirm this?\challengenor{moonsi}
The Moon's angular size changes even more due to another effect: the orbit
of the Moon round the Earth is elliptical. An example of the consequence is
shown in \figureref{imoondiffsize}.
\cssmallepsfnb{imoondiffsize}{scale=0.4}{How the size of the Moon actually
changes during its orbit ({\textcopyright}~\protect\iinn{Anthony
Ayiomamitis}).} % SENT EMAIL FEB 2008  anthony@perseus.gr % ALL OK!
\item \iname[Galilei, Galileo]{Galileo} also made mistakes. In his famous
book, the \bt Dialogues/ he says that the curve formed by a thin
\iin[chain!hanging shape]{chain} hanging between two nails is a
\iin{parabola}, i.e.,{} the curve defined by $y=x^2$. That is not correct.
What is the correct curve?\challengedif{cate} You can observe the shape
(approximately) in the shape of suspension bridges.
\cstftlepsf{ifourcircles}{scale=1}{A famous puzzle: how are the four radii
related?}[15mm]{itetrahedronpuzzle}{scale=1}{What is the area ABC, given
the other three areas and three right angles at O?}
\item Draw three circles, of different sizes, that touch each other, as shown
in \figureref{ifourcircles}. Now draw a fourth circle in the space
between, touching the outer three. What simple relation do the inverse
radii of the four circles obey?\challengenor{invrad}
\item Take a tetrahedron {OABC} whose triangular sides OAB, OBC and OAC are
rectangular in O, as shown in \figureref{itetrahedronpuzzle}. In other
words, the edges OA, OB and OC are all perpendicular to each other. In the
tetrahedron, the areas of the triangles OAB, OBC and OAC are respectively 8,
4 and~1. What is the area of triangle ABC?\challengenor{tetrasol}
\cssmallepsf{iladderpuzzle}{scale=1}{Two ladder puzzles: a moderately
difficult (left) and a difficult one (right).}
% Nov 2006
\item There are many puzzles about\cite{ladderpage}
ladders.\index{ladder!puzzles} Two are illustrated in
\figureref{iladderpuzzle}. If a \csd{5}{m} ladder is put against a wall
in such a way that it just touches a box with \csd{1}{m} height and depth,
how high does the ladder reach?\challengenor{ladderheight} If two ladders
are put against two facing walls, and if the lengths of the ladders and the
height of the crossing point are known, how distant are the
walls?\challengedif{ladderwalldist}
\item With two rulers, you can add and subtract numbers by lying them side by
side. Are you able to design rulers that allow you to multiply and divide
in the same manner?\challengenor{log}
\item How many days would a year have if the Earth turned the other way with
the same rotation frequency?\challengenor{ydays}
\item The Sun is hidden in the spectacular situation shown in
\figureref{ianticrepuscular} Where is it?\challengenor{anticre}
\cssmallepsfnb{ianticrepuscular}{scale=0.5}{Anticrepuscular rays  where is
the Sun in this situation? ({\textcopyright}~\protect\iinn{Peggy
Peterson})}
% EMAILED FEB 2008  krakcanyon@aol.com
% Put here in Feb 2008
\item A slightly different, but equally fascinating situation  and useful
for getting used to perspective drawing  appears when you have a
\iin{lighthouse} in your back. Can you draw the rays you see in the sky up
to the horizon?\challengn
% this is pretty, but different: http://www.tylerwestcott.com/2010Mar13/
\item Two cylinders of equal radius intersect at a right angle. What is the
value of the intersection volume? (First make a
drawing.)\challengenor{cyl2int}
% Aug 2007
\item Two sides of a hollow cube with side length \csd{1}{dm} are removed, to
yield a tunnel with square opening. Is it true that a cube with edge length
\csd{1.06}{dm} can be made to pass through the hollow cube with side length
\csd{1}{dm}?\challengenor{cubecu}
\item Could a twodimensional universe exist? \iinn{Alexander Dewdney}
imagined\cite{dewd} such a universe in great detail and wrote a wellknown
book about it.\index{universe!twodimensional} He describes houses, the
transportation system, digestion, reproduction and much more. Can you
explain why a twodimensional universe is impossible?\challengedif{dewdney}
% Aug 2011
\csepsfnb{iropetstz}{scale=1}{Ideal configurations of ropes made of two,
three and four strands. In the ideal configuration, the specific pitch angle
relative to the equatorial plane  \csd{39.4}{\csdegrees},
\csd{42.8}{\csdegrees} and \csd{43.8}{\csdegrees}, respectively  leads to
zerotwist structures. In these ideal configurations, the rope will neither
rotate in one nor in the other direction under vertical strain
({\textcopyright}~\protect\iinn{Jakob Bohr}).}
% Aug 2011
\item \ii[rope!geometry of]{Ropes} are wonderful structures. They are
flexible, they are helically woven, but despite this, they do not unwind or
twist, they are almost inextensible, and their geometry depends little on
the material used in making them.\cite{artrope} What is the origin of all
these properties?
Laying rope is an old craft; it is based on a purely geometric result: among
all possible helices of $n$ strands of given length laid around a central
structure of fixed radius, there is one helix for which the number of turns
is \emph{maximal}. For purely geometric reasons, ropes with that specific
number of turns and the corresponding inner radius have the mentioned
properties that make ropes so useful. The geometries of ideal ropes made of
two, three and four strands are shown in \figureref{iropetstz}.
\item Some researchers are investigating whether time could be
twodimensional. Can this\challengenor{time2d} be?
\item Other researchers are investigating whether space could have more than
three dimensions. Can this\challengenor{moret3dime} be?
% Mar 2012
\item One way to compare speeds of animals and machines is to measure them in
`body lengths per second'. The click beetle achieves a value of around 2000
during its jump phase, certain Archaea (bacterialike) cells a value of 500,
and certain hummingbirds 380. These are the recordholders so far. Cars,
aeroplanes, cheetahs, falcons, crabs, and all other motorized systems are
much slower.\cite{oly}
% Jun 2016
\item Why is the cross section of a tube usually circular?\challengn
% Jun 2016
\item What are the dimensions of an open rectangular water tank that contains
\csd{1}{m^3} of water and uses the smallest amount of wall
material?\challengn
% April 2010
\cssmallepsfnb{itightknot}{scale=1}{An open research problem: What is the
ropelength of a tight knot? ({\textcopyright}~\protect\iinn{Piotr Pieranski},
from \protect\citen{pieranskivol1})}
% Feb 2012
\item Draw a square consisting of four equally long connecting line segments
hinged at the vertices. Such a structure may be freely deformed into a
rhombus if some force is applied. How many additional line interlinks of
the same length must be supplemented to avoid this freedom and to prevent
the square from being deformed? The extra line interlinks must be in the
same plane as the square and each one may only be pegged to others at the
endpoints.\challengenor{sqhinge}
% April 2014
\item Area measurements can be difficult. In 2014 it became clear that the
area of the gastrointestinal tract of adult health humans is between 30 and
\csd{40}{m^2}. For many years, the mistaken estimate for the area was
between 180 and \csd{300}{m^2}.
% Scandinavian Journal of Gastroenterology, 2 April 2014
% Feb 2018
\item If you never explored plane geometry,\index{geometry!plane} do it once
in your life. An excellent introduction is \asi Claudi Alsina, Roger
B. Nelsen/ \bt Icons of Mathematics: An Exploration of Twenty Key Images/
\pu MAA Press/ \yrend 2011/ This is a wonderful book with many simple and
surprising facts about geometry that are never told in school or
university. You will enjoy it.
% Feb 2018
\item Triangles\index{triangle!geometry} are full of surprises.
Together, \iinn{Leonhard Euler}, \iinn{Charles Julien Brianchon} and
\iinn{Jean Victor Poncelet} discovered that in any triangle, nine
points lie on the same circle: the midpoints of the sides, the feet of
the altitude lines, and the midpoints of the altitude segments
connecting each vertex to the orthocenter. Euler also discovered that
in every triangle, the orthocenter, the centroid, the circumcenter and
the center of the ninepointcircle lie on the same line, now called
the Euler line.
For the most recent recearch results on plane triangles, see the
wonderful \ti Encyclopedia of Triangle Centers/ available at
\url{faculty.evansville.edu/ck6/encyclopedia/ETC.html}.
% April 2010
\item Here is a simple challenge on length that nobody has solved yet.
Take a piece of ideal rope: of constant radius, ideally flexible, and
completely slippery. Tie a tight \iin{knot} into it, as shown in
\figureref{itightknot}. By how\index{puzzle!knot} much did the two
ends of the rope come closer together?\challengeres{knottightshort}
\end{curiosity}
%
% Nov 2008, impr. April 2010
\subsection{Summary about everyday space and time}
% Index OK
Motion defines speed, time and length. Observations of everyday life and
precision experiments are conveniently and precisely described by describing
velocity as a vector, space as threedimensional set of points, and time as a
onedimensional real line, also made of points. These three definitions form
the everyday, or \emph{Galilean}, description of our
environment.\indexs{Galilean spacetime!summary}\indexs{spacetime!Galilean,
summary}
Modelling velocity, time and space as \emph{continuous} quantities is precise
and convenient. The modelling works during most of the adventures that
follows. However, this common model of space and time \emph{cannot} be
confirmed by experiment. For example, no experiment can check distances
larger than \csd{10^{25}}{m} or smaller than \csd{10^{25}}{m}; the continuum
model is likely to be incorrect at smaller scales.\index{continuity!limits of}
We will find out in the last part of our adventure that this is indeed the
case.
\vignette{classical}
%
%
%
%
% Reread July 2016
\chapter{How to describe motion  kinematics}
% Index OK
\markboth{\thesmallchapter\ how to describe motion  kinematics}%
{\thesmallchapter\ how to describe motion  kinematics}
\begin{quote}
% checked and translated by me
\selectlanguage{italian}La filosofia è scritta in questo grandissimo libro
che continuamente ci sta aperto innanzi agli occhi (io dico l'universo)
{\ldots} Egli è scritto in lingua
matematica.\selectlanguage{british}\footnote{Science is written in this huge
book that is continuously open before our eyes (I mean the universe)
{\ldots} It is written in mathematical language.}\\
\iinn{Galileo Galilei}, \btsim Il saggiatore/ VI.
\end{quote}
\csini{E}{xperiments} show that the properties of
% Dec 2016:
% Galilean
motion,
%
time and space
are\linebreak xtracted from the environment both by children and animals.
This\linebreak xtraction has been confirmed for cats, dogs, rats, mice, ants
and fish, among others. They all find the same results.
% Later, when children learn to speak, they put these experiences into
% concepts, as was just done above.
First of all, \ii[motion!is change of position with time]{motion is change of
position with time}. This description is illustrated by rapidly flipping
the lower left corners of this book, starting at \cspageref{leasa}.\index{flip
film!explanation of} % this vol I
Each page simulates an instant of time, and the only change that takes place
during motion is in the position of the object, say a stone, represented by
the dark spot. The other variations from one picture to the next, which are
due to the imperfections of printing techniques, can be taken to simulate the
inevitable measurement errors.
Stating that `motion is the change of position with time' is \emph{neither} an
explanation \emph{nor} a definition, since both the concepts of time and
position are deduced from motion itself. It is only a \ii{description} of
motion. Still, the statement is useful, because it allows for high
\iin{precision}, as we will find out by exploring gravitation and
electrodynamics. After all, precision is our guiding principle during this
promenade. Therefore the detailed description of changes in position has a
special name: it is called \ii{kinematics}.
%\cssmallepsf{ifirework}{scale=1}{Trajectories}%
The idea of change of positions implies that the object can be \emph{followed}
during its motion. This is not obvious; in the section on quantum theory we
will find examples where this is impossible. But in everyday life, objects
can always be tracked. The set of all positions taken by an object over time
forms its \ii{path} or \ii{trajectory}. The origin of this concept is evident
when one watches fireworks\cite{fireworksintro} or again the flip film in the
lower left corners starting at \cspageref{leasa}.\index{flip film!explanation
of} % this vol I
In everyday life, animals and humans agree on the Euclidean properties of
velocity, space and time. In particular, this implies that a {trajectory} can
be described by specifying three numbers, three \ii{coordinates} $(x,y,z)$ 
one for each dimension  as continuous functions of time $t$. (Functions are
defined in detail later on.)\seepagethree{func} This is usually written as
\begin{equation}
{\bm x}= {\bm x}(t) = (x(t),y(t),z(t)) \cp
\end{equation}
For example, already Galileo found, using stopwatch and ruler, that the height
$z$ of any thrown or falling \iin[stones]{stone} changes as
\begin{equation}
z(t)= z_{0} + v_{z0}\, (tt_{0}){\te \frac{1}{2}} g\,(tt_{\rm
0})^2
\label{kin}
\end{equation}
where $t_{0}$ is the time the fall starts, $z_{0}$ is the initial
height, $v_{z0}$ is the initial velocity in the vertical direction and
$g=\csd{9.8}{m/s^2}$ is a constant that is found to be the same, within about
one part in 300, for all falling bodies on all points of the surface of the
Earth. Where do the value \csd{9.8}{m/s^2} and its slight variations come
from?\cite{gravimetry} A preliminary answer will be given shortly, but the
complete elucidation will occupy us during the larger part of this hike.
% Oct 2009
The special case with no initial velocity is of great interest. Like a few
people before him, Galileo made it clear that $g$ is the same for all bodies,
if air resistance can be neglected. He had many arguments for this
conclusion;\seepageone{galmimg} can you find one? And of course, his famous
experiment at the \iin[tower!leaning, in Pisa]{leaning tower in Pisa}
confirmed the statement. (It is a \emph{false} urban legend that Galileo
never performed the experiment. He did it.)\cite{stillmandrake}
Equation (\ref{kin}) therefore allows us to determine the depth of a well,
given the time a stone takes to reach its bottom.\challengenor{well} The
equation also gives the speed $v$ with which one hits the ground after jumping
from a tree, namely
\begin{equation}
v = \sqrt{2 g h} \cp
\end{equation}
A height of \csd{3}{m} yields a velocity of \csd{27}{km/h}. The velocity is
thus proportional only to the square root of the height. Does this mean that
one's strong fear of falling results from an overestimation of its actual
effects?\challengenor{no}
\csepsf{igalileocannon}{scale=1}{Two ways to test that the time of free fall
does not depend on horizontal velocity.}
Galileo\indname{Galilei, Galileo} was the first to state an important result
about free fall:\index{fall!and flight are independent} the motions in the
horizontal and vertical directions are \emph{independent}. He showed that the
time it takes for a cannon ball that is shot exactly horizontally to fall is
\emph{independent} of the strength of the gunpowder, as shown in
\figureref{igalileocannon}. Many great thinkers did not agree with this
statement even after his death: in 1658 the \iin{Academia del Cimento} even
organized an experiment\cite{frova1} to check this assertion, by comparing the
flying cannon ball with one that simply fell vertically. Can you imagine how
they checked the simultaneity?\challengenor{cannon}
\figureref{igalileocannon} shows how you can check this at home. In
this experiment, whatever the spring load of the cannon, the two bodies will
always collide in midair (if the table is high enough), thus proving the
assertion.
In other words,\label{parabolaxyz} a flying cannon ball is not accelerated in
the horizontal direction. Its horizontal motion is simply unchanging  as
long as air resistance is negligible. By extending the description of
equation (\ref{kin}) with the two expressions for the horizontal coordinates
$x$ and $y$, namely
\begin{align}
x(t)&= x_{0} + v_{\rm x0} (tt_{0}) \non
y(t)&= y_{0} + v_{\rm y0} (tt_{0}) \cvend
%\label{kin2}
\end{align}
a \emph{complete} description\index{fall!is parabolic} for the path followed
by thrown stones results. A path of this shape is called a \ii{parabola}; it
is shown in Figures~\ref{iparabola},\seepageone{iparabola}
\ref{igalileocannon} and \ref{iparaother}. (A parabolic shape is also used
for light reflectors inside pocket lamps or car headlights. Can you show
why?)\challengenor{parash}
\csepsf{iparaother}{scale=1}{Various types of graphs describing the
same\protect\index{configuration space}\protect\index{spacetime
diagram}\protect\index{hodograph}\protect\index{phase space!diagram}
path of a thrown stone.}[%
\psfrag{x}{\small $x$}%
\psfrag{t}{\small $t$}%
\psfrag{z}{\small $z$}%
%
\psfrag{vz}{\small $v_{z}$}%
\psfrag{mvz}{\small $m v_{z}$}%
\psfrag{mvx}{\small $m v_{x}$}%
]
Physicists enjoy generalizing the idea of a {path}.\cite{animalpaths} As
\figureref{iparaother} shows, a path is a trace left in a diagram by a moving
object. Depending on what diagram is used, these paths have different names.
Spacetime diagrams are useful to make the theory of relativity accessible.
The \ii{configuration space} is spanned by the coordinates of all particles of
a system. For many particles, it has a high number of dimensions and plays an
important role in selforganization.\seepageone{selforgt} The difference
between chaos and order can be described as a difference in the properties of
paths in configuration space. \ii[hodograph]{Hodographs}, the paths in
`velocity space', are used in weather forecasting. The phase space diagram is
also called \ii[state space]{state space diagram}.\index{diagram!state space}
It plays an essential role in thermodynamics.
%
% Reread July 2016
\subsection{Throwing, jumping and shooting}
% Index OK
The kinematic description of motion is useful for answering a whole range of
questions.
\begin{curiosity}
\item What is the upper limit for the \iin{long jump}?\index{speed
record!human running} The running peak speed world record in 2008 was over
\csd{12.5}{m/s}\csd{\;\approx 45}{km/h} by \iinn{Usain Bolt},\cite{usain}
and the 1997 women's record was \csd{11}{m/s}\csd{\;\approx
40}{km/h}.\cite{faz97} However, male long jumpers never run much faster
than about \csd{9.5}{m/s}. How much extra jump distance could they achieve
if they could run full speed? How could they achieve that? In addition,
long jumpers\index{record!human running speed} take off at angles of about
\csd{20}{\csdegrees}, as\cite{longjump} they are not able to achieve a
higher angle at the speed they are running. How much would they gain if
they could achieve \csd{45}{\csdegrees}?\challengenor{longj} Is
\csd{45}{\csdegrees} the optimal angle?
% Oct 2009
\item What do the athletes \iinn{Usain Bolt} and \iinn{Michael Johnson}, the
last two world record holders on the \csd{200}{m} race at time of this
writing,{\present} have in common? They were tall, athletic, and had many
fast twitch fibres in the muscles. These properties made them good
sprinters. A last difference made them world class sprinters: they had a
flattened spine, with almost no Sshape. This abnormal condition saves them
a little bit of time at every step, because their spine is not as flexible
as in usual people. This allows them to excel at short distance races.
% This was told to me by Juergen Freiwald, Prof für Bewegungswissenschaft
% in Wuppertal
%%%% Er sagte auch: Hürdenöäufer habe immer ein blockiertes Iliosacralgelenk,
%%%% nämlich das vom Fuß der nach der Hürde auftritt.
\item Athletes continuously improve speed records. Racing \iin[horse!speed
of]{horses} do not. Why? For racing horses, breathing rhythm is related to
gait; for humans, it is not. As a result, racing horses cannot change or
improve their technique, and the speed of racing horses is essentially the
same since it is measured.
% Jan 2015
\item What is the highest height achieved by a human throw of any object? What
is the longest distance achieved by a human throw? How would you clarify the
rules?\challengenor{throwrecord} Compare the results with the record
distance with a \iin{crossbow},
% \csd{1,871.84}{m}, achieved in 1988 by \iinn{Harry Drake}, the
% record distance with a \iin{footbow}, \csd{1854.40}{m}, achieved in
% 1971 also by \iinn{Harry Drake}, and with a handheld \iin[bow,
% record distance]{bow}, \csd{1,222.01}{m}, achieved in 1987 by
%
\csd{1,871.8}{m}, achieved in 1988 by \iinn{Harry Drake}, the
record distance with a \iin{footbow}, \csd{1854.4}{m}, achieved in
1971 also by \iinn{Harry Drake}, and with a handheld \iin[bow,
record distance]{bow}, \csd{1,222.0}{m}, achieved in 1987 by
\iinn{Don Brown}.
% http://www.usaarcheryrecords.org/FlightPages/2007/worldrecords07.htm
\item How can the speed of falling rain be measured using an
\iin{umbrella}?\challengenor{umbr} The answer is important: the same method
can also be used to measure the speed of light, as we will find out later.
(Can you guess how?)\seepagetwo{aberrrr}
% June 2007
\item When a dancer\index{dancer!rotations}\index{rotation!in dance} jumps in
the air, how many times can he or she rotate around his or her vertical axis
before arriving back on earth?\challengenor{dancerot}
\cssmallepsfnb{ikotcut}{scale=1}{Three superimposed images of a frass
pellet shot away by a caterpillar inside a rolledup leaf
({\textcopyright}~\protect\iinn{Stanley Caveney}).}
\item Numerous species of moth and butterfly \iin{caterpillars} shoot away
their \iin{frass}  to put it more crudely: their \iin{faeces}  so
that\cite{shootshit} its smell does not help predators to locate them.
\iinn{Stanley Caveney} and his team took photographs of this process.
\figureref{ikotcut} shows a caterpillar (yellow) of the \iin{skipper}
\iie{Calpodes ethlius} inside a rolled up green leaf caught in the act.
Given that the record distance observed is \csd{1.5}{m} (though by another
species, \iie{Epargyreus clarus}), what is the ejection
speed?\challengenor{caterpillarshit} How do caterpillars achieve it?
\item What is the horizontal distance one can reach by throwing a stone, given
the speed and the angle from the horizontal at which it is
thrown?\challengenor{stonet}
\item What is the maximum numbers of balls that could be juggled at the same
time?\challengenor{jug}
\item Is it true that \iin{rain drops} would kill if it weren't for the air
resistance of the atmosphere?\challengenor{rain} What about hail?
\item Are bullets, fired into the air from a gun, dangerous when they fall
back down?\challengenor{gun} % Changed answer in Aug 2007
% Nov 2008
\item Police finds a dead human body at the bottom of cliff with a height of
\csd{30}{m}, at a distance of \csd{12}{m} from the cliff. Was it suicide or
murder?\challengenor{forensic}
% Apr 2010, Figure edited Apr 2014
\csepsfnb{ianimaljumps}{scale=1}{The height achieved by jumping land
animals.}
% Elephant data point is mine, cited from memory from a paper which I never
% found again. Human data point is also mine, so are cats, dogs.
% Improved Apr 2010
% Index ok
\item All land animals,\label{animaljumpheight} regardless\index{jump!height
of animals} of their size, achieve jumping heights of at most
\csd{2.2}{m},\cite{animaljumpref} as shown in \figureref{ianimaljumps}.
The explanation of this fact takes only two lines. Can you find
it?\challengenor{anijump}
\np The last two issues arise because the equation (\ref{kin}) describing
free fall does not hold in all cases. For example, leaves or potato crisps
do not follow it. As Galileo already knew, this is a consequence of
\iin[air!resistance]{air resistance}; we will discuss it shortly. Because
of air resistance, the path of a \iin[stones]{stone} is not a parabola.
In fact, there are other situations where the path of a falling stone is not
a parabola, even without air resistance. Can you find
one?\challengenor{stonepa}
\end{curiosity}
%
% Reread in July 2016  not enough fun!
\subsection{Enjoying vectors}
% Index OK
% Rewritten in Oct 2007
%
Physical quantities with a defined direction, such as speed, are described
with three numbers, or three components, and are called
\ii[vector!definition]{vectors}.\label{vecspde} Learning to calculate with
such multicomponent quantities is an important ability for many sciences.
Here is a summary.
% Oct 2007
Vectors can be pictured by small arrows.
% Mar 2006
Note that vectors do not have specified points at which they start: two arrows
with same direction and the same length are the \emph{same} vector, even if
they start at different points in space.
%
% Oct 2007
Since vectors behave like arrows, vectors can be added and they can be
multiplied by numbers. For example, stretching an arrow
${\bm a} = (a_{x}, a_{y}, a_{z})$ by a number $c$ corresponds, in component
notation, to the vector $c {\bm a}= (c a_{x}, c a_{y}, c a_{z}) $.
In precise, mathematical language, a vector is an element of a set, called
\ii[vector!space]{vector space}, in which the following properties hold for
all vectors $\bm a$ and $\bm b$ and for all numbers\label{eulc} $c$ and~$d$:
\begin{equation}
c ({\bm a} + {\bm b}) = c {\bm a} + c {\bm b}
\qhbox{,}
(c + d) {\bm a} = c {\bm a} + d {\bm a}
\qhbox{,}
(cd) {\bm a} = c (d {\bm a})
\qhbox{and}
1 {\bm a} = {\bm a} \cp
\end{equation}
Examples of vector spaces are the set of all \ii{positions} of an object, or
the set of all its possible velocities. Does the set of all rotations form a
vector space?\challengenor{vec}
All vector spaces allow defining a unique \ii{null vector} and a unique
\ii{negative vector} for each vector.\challengn
In most vector spaces of importance when describing nature the concept of
\emph{length}  specifying the `magnitude'  of a vector can be introduced.
This is done via an intermediate step, namely the introduction of the scalar
product of two vectors. The product is called `scalar' because its result is
a scalar; a \ii{scalar} is a number that is the same for all observers; for
example, it is the same for observers with different orientations.%
%
\footnote{We mention that in mathematics, a scalar is a \emph{number}; in
physics, a scalar is an \emph{invariant} number, i.e., a number that is the
same for all observers. Likewise, in mathematics, a vector is an element of
a vector space; in physics, a vector is an \emph{invariant} element of a
vector space, i.e., a quantity whose coordinates, when observed by different
observers, change like the components of velocity.}
%
The \ii{scalar product} between two vectors $\bm a$ and $\bm b$ is a number
that satisfies
\begin{equation}
\begin{split}
{\bm a} {\bm a} \geqslant 0
\ , \\
{\bm a} {\bm b} = {\bm b} {\bm a}
\ , \\
({\bm a} + {\bm a'}) {\bm b} = {\bm a} {\bm b} + {\bm a'} {\bm b}
\ , \\
{\bm a} ({\bm b} + {\bm b'}) = {\bm a} {\bm b} + {\bm a} {\bm b'}
\ \hbox{ and} \\
% \qhbox{ and}
(c {\bm a}) {\bm b} = {\bm a} (c {\bm b}) = c ( {\bm a} {\bm b} ) \cp
\end{split}
\end{equation}
This definition of a scalar product is not unique; however, there is a
\emph{standard} scalar product. In Cartesian coordinate notation, the
standard scalar product is given by
\begin{equation}
{\bm a} {\bm b} = a_{x}b_{x}+a_{y}b_{y}+a_{z}b_{z} \cp
\end{equation}
If the scalar product of two vectors vanishes the two vectors are
\emph{orthogonal}, at a right angle to each other.\index{orthogonality} (Show
it!)\challengn Note that one can write either ${\bm a} {\bm b}$ or
${\bm a} \cdot {\bm b}$ with a central dot.
The \ii{length} or \ii{magnitude} or \ii{norm} of a vector can then be defined
as the square root of the scalar product of a vector with itself:
$a= \sqrt{{\bm a} {\bm a}}$. Often, and also in this text, lengths are
written in \emph{italic} letters, whereas vectors are written in \textbf{bold}
letters. The magnitude is often written as $a= \sqrt{{\bm a}^2}$. A vector
space with a scalar product is called an \ii[vector!space,
Euclidean]{Euclidean} vector space.
The scalar product is especially useful for specifying directions. Indeed,
the scalar product between two vectors encodes the angle between them. Can
you deduce this important relation?\challengenor{scalprod}
%
% Reread July 2016
\subsection{What is rest? What is velocity?}
% Index OK
% Improved figure in Mar 2018
\cssmallepsf{islopenew}{scale=1}{The derivative in a point as the limit of
secants.}[% psfrag bombs when using parbox
\psfrag{y}{\small $y$}%
\psfrag{t}{\small $t$}%
\psfrag{Dt}{\small $\Delta t$}%
\psfrag{Dy}{\small $\Delta y$}%
\psfrag{sp}{\small\textcolor[rgb]{0,0,1}{\small\noindent secant
slope:\ $\Delta y/ \Delta t$}}%
\psfrag{de}{\small\textcolor[rgb]{1,0,0}{\small\noindent derivative
slope:\ $\diffd y/ \diffd t$}}%
]
In the Galilean description of nature, motion and rest are opposites. In other
words, a body is at rest when its position, i.e.,{} its coordinates, do not
change with time. In other words, (Galilean) \ii[rest!Galilean]{rest} is
defined as
\begin{equation}
{\bm x}(t) = {\rm const} \cp
\end{equation}
We recall that ${\bm x}(t)$ is the abbreviation for the three coordinates
$(x(t),y(t),z(t))$. Later we will see that this definition of rest, contrary
to first impressions, is {not} much use and will have to be expanded.
Nevertheless, any definition of rest implies that nonresting objects can be
distinguished by comparing the rapidity of their displacement. Thus we can
define the \ii{velocity} $\bm v$ of an object as the change of its position
$\bm x$ with time $t$. This is usually written as
\begin{equation}
{\bm v}=\frac{ \diffd{\bm x}}{ \diffd t} \cp
\end{equation}
In this expression, valid for each coordinate separately, $\diffd/\diffd t$
means `change with time'. We can thus say that velocity is the
\ii{derivative} of position with respect to time. The \ii{speed} $v$ is the
name given to the magnitude of the velocity $\bm v$. Thus we have
$v=\sqrt{{\bm v}{\bm v}}$. Derivatives are written as fractions in order to
remind the reader that they are derived from the idea of slope. The
expression
\begin{equation}
\frac{\diffd s }{ \diffd t} \qhbox{is meant as an abbreviation of}
\lim_{\Delta t \rightarrow 0} \frac{\Delta s}{\Delta t} \cvend
\end{equation}
a shorthand for saying that the \ii[derivative!definition]{derivative at a
point} is the limit of the secant {slopes} in the neighbourhood of the
point, as shown in \figureref{islopenew}. This definition implies the working
rules\challengn
\begin{equation}
\frac{\diffd (s+r) }{ \diffd t} = \frac{\diffd s}{ \diffd t} + \frac{\diffd
r}{ \diffd t}\qhbox{,} \frac{\diffd (cs) }{ \diffd t} = c \frac{\diffd s}{
\diffd t}\qhbox{,} \frac{\diffd }{ \diffd t}\frac{\diffd s}{ \diffd t} =
\frac{\diffd^{2}s}{ \diffd t^{2}}\qhbox{,} \frac{\diffd (sr) }{ \diffd t} =
\frac{\diffd s}{ \diffd t}r + s\frac{\diffd r}{ \diffd t} \cvend
\end{equation}
$c$ being any number. This is all one ever needs to know about derivatives in
physics. Quantities such as $\diffd t$ and $\diffd s$, sometimes useful by
themselves, are called \ii[differential]{differentials}. These concepts are
due to \inames[Leibniz, Gottfried Wilhelm]{Gottfried Wilhelm Leibniz}.%
%
\footnote{Gottfried Wilhelm Leibniz \livedplace(1646 Leipzig1716
Hannover), % Saxon
lawyer, physicist, mathematician, philosopher, diplomat and historian. He
was one of the great minds of mankind; he invented the differential calculus
(before Newton) and published many influential and successful books in the
various fields he explored, among them \bt De arte combinatoria/ \bt
Hypothesis physica nova/ \bt Discours de métaphysique/ \bt Nouveaux essais
sur l'entendement humain/ the \btsim Théodicée/ and the \btsim Monadologia/.
} %
Derivatives lie at the basis of all calculations based on the continuity of
space and time. Leibniz was the person who made it possible to describe and
use velocity in physical formulae and, in particular, to use the idea of
velocity at a given point in time or space for calculations.
\cssmallepsfnb{ileibniz}{scale=0.30}{Gottfried Wilhelm Leibniz
\livedfig(16461716).}
The definition of velocity assumes that it makes sense to take the limit
${\Delta t \rightarrow 0}$. In other words, it is assumed that
\emph{infinitely small} time intervals do exist in nature. The definition of
velocity with derivatives is possible only\index{velocity!as derivative}
because both space and time are described by sets which are \emph{continuous},
or in\index{set!connected} mathematical language, \emph{connected and
complete}.
% Not `complete', says a mathematician  I put it back in again
In the rest of our walk we shall not forget that from the beginning of
classical physics, \ii[infinity!in physics]{infinities} are present in its
description of nature. The infinitely small is part of our definition of
velocity. Indeed, differential calculus can be defined as the study of
infinity and its uses. We thus discover that the appearance of infinity does
not automatically render a description impossible or imprecise. In order to
remain precise, physicists use only the smallest two of the various possible
types of infinities. Their precise definition and an overview of other types
are introduced\seepagethree{infin} later on.
The appearance of infinity in the usual description of motion was first
criticized in his famous ironical arguments by \iname{Zeno of Elea} (around
445 {\bce}),\cite{zeno} a disciple of \iname[Parmenides of Elea]{Parmenides}.
In his socalled third argument, Zeno explains that since at every instant a
given object occupies a part of space corresponding to its size, the notion of
velocity at a given instant makes no sense; he provokingly concludes that
therefore motion does not exist. Nowadays we would not call this an argument
against the \emph{existence} of motion, but against its usual
\emph{description}, in particular against the use of infinitely divisible
space and time. (Do you agree?)\challengn Nevertheless, the description
criticized by Zeno actually works quite well in everyday life. The reason is
simple but deep: in daily life, changes are indeed continuous.
\emph{Large changes in nature are made up of many small changes.} This
property of nature is not obvious. For example, we note that we have (again)
tacitly assumed that the path of an object is not a \iin[fractals]{fractal} or
some other badly behaved entity. In everyday life this is correct; in other
domains of nature it is not. The doubts of Zeno will be partly rehabilitated
later in our walk, and increasingly so the more we
proceed.\seepagesix{timeaway} The rehabilitation is only partial, as the final
solution will be different from that which he envisaged; on the other hand,
the doubts about the idea of `velocity at a point' will turn out to be
wellfounded. For the moment though, we have no choice: we continue with the
basic assumption that in nature changes happen smoothly.
Why is velocity necessary as a concept? Aiming for precision in the
description of motion, we need to find the complete list of aspects necessary
to specify the state of an object. The concept of velocity is obviously on
this list.
%
% {Table of accelerations}
%
%
{\small
\begin{table}[t]
\small
\caption{Some measured acceleration\protect\index{acceleration!table of
values} values.}
\label{accvltab}
\centering
\dirrtabularstar
\begin{tabular*}{\textwidth}{%
@{}>{\PBS\raggedright\hspace{0.0em}\columncolor{hks152}[0pt][1cm]} p{100mm}
@{\extracolsep{\fill}} %
>{\PBS\raggedright\hspace{0.0em}\columncolor{hks152}[0pt][0cm]} p{32mm} @{}}
%
\toprule
%
\tabheadf{Observation} & \tabhead{Acceleration} \\[0.5mm]
%
\midrule
%
%
What is the lowest you can find? & \relax\leavevmode\challengenor{lowaccrec}\\
%
Backacceleration of the galaxy \iin[M82, galaxy]{M82}
by its ejected jet & $\csd{10}{fm/s^{2}}$ \\ % Fathoumi
%
Acceleration of a young star by an ejected jet & $\csd{10}{pm/s^{2}}$ \\ %
Fathoumi
%
Acceleration of the Sun in its orbit around the Milky Way &
$\csd{0.2}{nm/s^{2}}$
\\
%
Deceleration of the \iin{Pioneer satellites}, due to heat radiation imbalance &
$\csd{0.8}{nm/s^{2}}$ \\
%
Centrifugal acceleration at Equator due to Earth's rotation &
$\csd{33}{mm/s^{2}}$\\
%
%Centrifugal acceleration due to the Earth's rotation & $\csd{33}{mm/s^{2}}$\\
%
Electron acceleration in household electricity wire due to alternating current
& $\csd{50}{mm/s^{2}}$\\
%
Acceleration of fast underground train & $\csd{1.3}{m/s^{2}}$\\
%
Gravitational acceleration on the Moon & $\csd{1.6}{m/s^{2}}$\\
%
Minimum deceleration of a car, by law, on modern dry asphalt &
$\csd{5.5}{m/s^{2}}$\\
%
Gravitational acceleration on the Earth's surface, depending on location
& $\csd{9.8 \pm 0.3}{m/s^{2}}$\\
%
Standard gravitational acceleration\index{gravitational acceleration,
standard}
& $\csd{9.806\,65}{m/s^{2}}$\\
%
Highest acceleration for a car or motorbike with enginedriven wheels&
$\csd{15}{m/s^{2}}$\\
%
Space rockets at takeoff & 20 to $\csd{90}{m/s^{2}}$\\
%
Acceleration of \iin{cheetah} & $\csd{32}{m/s^{2}}$\\
%
Gravitational acceleration on Jupiter's surface & $\csd{25}{m/s^{2}}$\\
%
Flying fly (\iie{Musca domestica}) & \circa$\csd{100}{m/s^{2}}$\\
%
Acceleration of thrown stone & \circa$\csd{120}{m/s^{2}}$\\
%
Acceleration that triggers air bags in cars& $\csd{360}{m/s^{2}}$\\
%
Fastest legpowered acceleration (by the \iin{froghopper}, \iie{Philaenus
spumarius}, an insect) & $\csd{4}{km/s^{2}}$\\
%
% Froghoppers, Philaenus spumarius
%
% new record of july 2003, nature, 424, p 509
%
Tennis ball against wall & $\csd{0.1}{Mm/s^{2}}$\\
%
Bullet acceleration in rifle & $\csd{2}{Mm/s^{2}}$\\ % checked on internet
%
Fastest centrifuges & $\csd{0.1}{Gm/s^{2}}$\\
%
Acceleration of protons in large accelerator & $\csd{90}{Tm/s^{2}}$\\
%
Acceleration of protons inside nucleus & $\csd{10^{31}}{m/s^{2}}$\\
%
Highest possible acceleration in nature & $\csd{10^{52}}{m/s^2}$ \\
\bottomrule
%
\end{tabular*}
\end{table}
}
%
% Impr. 2014
\subsection{Acceleration}
% Index ok
Continuing\index{acceleration(} along the same line, we call
\ii{acceleration} $\bm a$ of a body the change of velocity $\bm v$ with time,
or
\begin{equation}
{\bm a}=
\frac{ \diffd{\bm v}}{ \diffd t}
=\frac{ \diffd^2{\bm x}}{ \diffd t^2} \cp
\end{equation}
Acceleration is what we feel when the Earth trembles, an aeroplane takes off,
or a bicycle goes round a corner. More examples are given in
\tableref{accvltab}. Acceleration is the time derivative of velocity. Like
velocity, acceleration has both a magnitude and a direction. In short,
acceleration, like velocity, is a vector quantity. As usual, this property is
indicated by the use of a \textbf{bold} letter for its abbreviation.
% Feb 2007, Mar 2012
In a usual car, or on a motorbike, we can \emph{feel} being accelerated.
(These accelerations are below 1$g$ and are therefore harmless.)
% Jan 2014
We feel acceleration because some part inside us is moved against some other
part: acceleration deforms us. Such a moving part can be, for example, some
small part inside our ear, or our stomach inside the belly, or simply our
limbs against our trunk.
%
All acceleration sensors, including those listed in \tableref{accelsensors} or
those shown in \figureref{iaccelerationmeasurement}, whether biological or
technical, work in this way.
% May 2005, Jan 2014
Acceleration is felt.\index{acceleration!effects of} Our body is deformed and
the sensors in our body detect it, for example in amusement parks. Higher
accelerations can have stronger effects. For example, when accelerating a
sitting person in the direction of the head at two or three times the value of
usual gravitational acceleration, eyes stop working and the sight is greyed
out, because the blood cannot reach the eye any more. Between 3 and 5$g$ of
continuous acceleration, or 7 to 9$g$ of short time
acceleration,\cite{nasaacc} consciousness is lost, because the brain does not
receive enough blood, and blood may leak out of the feet or lower legs. High
acceleration in the direction of the feet of a sitting person can lead to
haemorrhagic strokes in the brain. The people most at risk are jet pilots;
they have special clothes that send compressed air onto the pilot's bodies to
avoid blood accumulating in the wrong places.
Can you think of a situation where you are accelerated but do \emph{not} feel
it?\challengenor{notfeelacc}
% (NO) Acceleration: scaling?
Velocity is the time derivative of position. Acceleration is the second time
derivative of position.\index{jerk} Higher derivatives than acceleration can
also be defined, in the same manner. They add little to the description of
nature,\challengenor{jerk} because  as we will show shortly  neither these
higher derivatives nor even acceleration itself are useful for the description
of the state of motion of a\index{acceleration)} system.
%
% {Table of acceleration sensors}
%
%
{\small
\begin{table}[t]
\small
\caption{Some acceleration\protect\index{acceleration!sensors, table}
sensors.}
\label{accelsensors}
\centering
\dirrtabularstar
\begin{tabular*}{\textwidth}{%
@{}>{\PBS\raggedright\hspace{0.0em}\columncolor{hks152}[0pt][1cm]} p{50mm}
@{\extracolsep{\fill}}
>{\PBS\raggedright\hspace{0.0em}\columncolor{hks152}[0pt][1cm]} p{50mm}
@{\extracolsep{\fill}} p{25mm} @{}}
%
\toprule
%
\tabheadf{Measurement} & \tabhead{Sensor} & \tabhead{Range} \\[0.5mm]
%
\midrule
%
Direction of gravity in plants (roots, trunk, branches, leaves) & statoliths
in cells & 0 to \csd{10}{m/s^2} \\
%
Direction and value of accelerations in mammals & the membranes in each
semicircular canal, and the utricule and saccule in the inner ear & 0 to
\csd{20}{m/s^2} \\
%
Direction and value of acceleration in modern step counters for hikers &
piezoelectric sensors & 0 to \csd{20}{m/s^2} \\
%
Direction and value of acceleration in car crashes & airbag sensor using
piezoelectric ceramics & 0 to \csd{2000}{m/s^2} \\
%
%
\bottomrule
%
\end{tabular*}
\end{table}
}
\csepsfnb{iaccelerationmeasurement}{scale=1}{Three accelerometers: a
oneaxis\protect\index{accelerometers!photographs of} piezoelectric airbag
sensor, a threeaxis capacitive accelerometer, and the utricule and saccule
near the three semicircular canals inside the human ear
({\textcopyright}~\protect\iname{Bosch},
% SENT FEB 2008  bosch.semiconductors@de.bosch.com
\protect\iname{Rieker Electronics},
% SENT FEB 2008  info@riekerinc.com
\protect\iname{Northwestern University}).}
% SENT FEB 2008  thain@northwestern.edu
%
% Reread Jul 2016
\subsection{From objects to point particles}
% Index OK
\begin{quote}
\selectlanguage{german}Wenn ich den Gegenstand kenne, so kenne ich auch sämtliche
Möglichkeiten seines Vorkommens in Sachverhalten.\selectlanguage{british}%
%
\footnote{`If I know an object, then I also know all the possibilities of its
occurrence in atomic facts.'} % Odgen translation
%
\\
Ludwig Wittgenstein, \bt Tractatus/ 2.0123\indname{Wittgenstein, Ludwig}
\end{quote}
\np One aim of the study of motion is to find a complete and precise
description of both states and objects. With the help of the concept of
space, the description of objects can be refined considerably. In particular,
we know from experience that all objects seen in daily life have an important
property: they can be divided into \ii{parts}.\challengn Often this
observation is expressed by saying that all objects, or {bodies}, have two
properties. First, they are made out of \ii{matter},%
%
\footnote{Matter is a word derived from the Latin `materia', which originally
meant `wood' and was derived via\cite{ety} intermediate steps from
`mater', meaning `mother'.} %
%
defined as that aspect of an object responsible for its impenetrability,
i.e.,{} the property preventing two objects from being in the same place.
Secondly, bodies have a certain form or \ii{shape}, defined as the precise way
in which this impenetrability is distributed in space.
\label{mapode}%
%
In order to describe motion as accurately as possible, it is convenient to
start with those bodies that are as simple as possible. In general, the
smaller a body, the simpler it is. A body that is so small that its parts no
longer need to be taken into account is called a \ii{particle}. (The older
term \ii{corpuscle} has fallen out of fashion.) Particles are thus idealized
small \iin{stones}. The extreme case, a particle whose size is
\emph{negligible} compared with the dimensions of its motion, so that its
position is described completely by a \emph{single} triplet of coordinates, is
called a \ii[point!particle]{point particle} or a \ii[point!mass]{point mass}
or a \ii[mass!point]{mass point}.
%
% or maybe a pearl ...
%
In equation (\ref{kin}), the stone was assumed to be such a point particle.
Do pointlike objects, i.e.,{} objects smaller than anything one can measure,
exist in daily life? Yes and no. The most notable examples are the stars.
At present, angular sizes as small as \csd{2}{\muunit rad} can be
measured, %(0.4 arc seconds)
a limit given by the fluctuations of the air in the atmosphere. In space,
such as for the Hubble telescope orbiting the Earth, the angular limit is due
to the diameter of the telescope and is of the order of
\csd{10}{nrad}. % (0.002 arc seconds).
Practically all stars seen from Earth are smaller than that, and are thus
effectively `pointlike', even when seen with the most powerful telescopes.
\csepsfnb{fbetel}{scale=1}{Orion\protect\index{Orion} in
natural\protect\index{Betelgeuse} colours
({\textcopyright}~\protect\iinn{Matthew Spinelli})
% antwrp.gsfc.nasa.gov/apod/ap030207.html
% SENT EMAIL FEB 2008  c8user@prodigy.net  no answer
% SENT EMAIL FEB 2008 to APOD  he does not know
and Betelgeuse (ESA, NASA).}
% Jan 2014
\csepsfnb{istarsizesvol1}{scale=1}{A comparison of star sizes
({\textcopyright}~\protect\iinn{Dave Jarvis}).}
% Impr. Aug 2013
As an exception to the general rule, the size of a few large or nearby stars,
mostly of red giant type, can be measured with special instruments.%
% 
\footnote{The website \url{stars.astro.illinois.edu/sow/sowlist.html}
%\url{www.astro.uiuc.edu/~kaler/sow/sowlist.html}% broken in Aug 2013
gives an introduction to the different types of \iin{stars}. The
\url{www.astro.wisc.edu/~dolan/constellations}
website provides detailed and interesting information about
\iin{constellations}.
%\isa[pl]
For an overview of the planets, see the beautiful book by \asi Kenneth R.
Lang, Charles A. Whitney/ \bt Vagabonds de l'espace  Exploration et
découverte dans le système solaire/ Springer Verlag, \yrend 1993/ Amazingly
beautiful pictures of the stars can be found in \asi David Malin/ \bt A View
of the Universe/ \pu Sky
Publishing and Cambridge University Press/ \yrend 1993/ } %
% 
Betelgeuse,\index{Betelgeuse} the higher of the two shoulders of Orion shown
in \figureref{fbetel}, Mira in Cetus,
%
% it is 2000 times the size of the Sun, and 1400 light years away (NRC,
% October 1997)
%
\iin{Antares} in Scorpio, \iin{Aldebaran} in Taurus and \iin{Sirius} in Canis
Major are examples of stars whose size has been measured; they are all
% corrected in Aug 2013
%
% Ad 1: You named 5 stars, but Sirius is certainly neither large, nor red giant
% type. Betelgeuse (640 ly) is not nearby star. (Yes, it is arguable  Deneb
% in Cygnus is 3200 ly from Earth, but it is the farthest BRIGHT star. V762 in
% Cassiopeia is even 15000 ly from Earth as probably the farthest star visible
% to the naked eye – albeit barely with its magnitude 5.9).
%
% Ad 2: It is true for Sirius (8.6 ly), but not for the other stars – Betelgeuse
% (640 ly), Mira (200 to 400 ly), Antares (550 ly) and Aldebaran (65 ly).
%
less than two thousand light years from Earth.\cite{a17}
% Aug 2013
For a comparison of dimensions, see \figureref{istarsizesvol1}. Of course,
like the Sun, also all other stars have a finite size, but one cannot prove
this by measuring their dimension in photographs.
(True?)\challengenor{starsize}
% Oct 2011
\cssmallepsfnb{itwinkle}{scale=1}{Regulus and Mars, photographed with an
exposure time of 10\,s on 4 June 2010 with a wobbling camera, show the
difference between a pointlike star\protect\index{star!twinkling} that
twinkles and an extended planet that does not
({\textcopyright}~\protect\iinn{Jürgen Michelberger}).}
% improved in March 2005, Oct 2011
The difference between `pointlike' and\index{star!twinkle}\index{twinkle!of
stars} finitesize sources can be seen with the naked eye: at night, stars
twinkle, but planets do not. (Check it!)\challengn A beautiful visualization
is shown in \figureref{itwinkle}. This effect is due to the turbulence of
air. Turbulence has an effect on the almost pointlike stars because it
deflects light rays by small amounts. On the other hand, air turbulence is
too weak to lead to twinkling of sources of larger angular size, such as
planets or artificial satellites,%
%
\footnote{A \ii[satellite!definition]{satellite} is an object circling a
planet, like the Moon; an \ii[satellite!artificial]{artificial satellite} is
a system put into orbit by humans, like the Sputniks.} %
%
because the deflection is averaged out in this case.
An object is \ii[pointlikeness]{pointlike for the naked eye} if its angular
size is smaller than about \csd{2}{'}\csd{=0.6}{mrad}.
% From the article on the model of the eye.
Can you estimate the size of a `pointlike' dust particle?\challengenor{dust}
By the way, an object is \ii[invisibility!of objects]{invisible} to the naked
eye if it is pointlike \emph{and} if its luminosity, i.e.,{} the intensity of
the light from the object reaching the eye,\index{object!invisibility} is
below some critical value. Can you estimate whether there are any manmade
objects visible from the Moon, or from the space
shuttle?\challengenor{moonvis}
The above definition of `pointlike' in everyday life is obviously misleading.
Do proper, real point particles exist? In fact, is it at all possible to show
that a particle has vanishing size?
% This question will be central in the last part of our walk.
In the same way, we need to ask and check whether points in
space do exist.\index{point!in space}\index{space!point} Our walk will lead us
to the astonishing result that all the answers to these questions are
negative. Can you imagine why?\challengenor{nopoi} Do not be disappointed if
you find this issue difficult; many brilliant minds have had the same problem.
However, many particles, such as electrons, quarks or photons are pointlike
for all practical purposes. Once we know how to describe the motion of point
particles, we can also describe the motion of extended bodies, rigid or
deformable: we assume that they are made of parts. This is the same approach
as describing the motion of an animal as a whole by combining the motion of
its various body parts. The simplest description, the
\ii[approximation!continuum]{continuum approximation},\index{continuum!as
approximation} describes extended bodies as an infinite collection of point
particles. It allows us to understand and to predict the motion of milk and
honey, %as well as that of water and of any other fluid,
the motion of the air in hurricanes and of perfume in rooms. The motion of
fire and all other gaseous bodies, the bending of bamboo in the wind, the
shape changes of chewing gum, % and of all other deformable solids
and the growth of plants and animals can also be described in this
way.\cite{bentstuff}
%
% , as well as all combinations of them, from whipped
% cream to cigarette smoke and jugglers.
All observations so far have confirmed that the motion of large bodies can be
described to full precision as the result of the motion of their parts. All
machines that humans ever built are based on this idea. A description that is
even more precise than the continuum approximation is given later
on.\seepagefour{facsal} Describing body motion with the motion of body parts
will guide us through the first five volumes of our mountain ascent; for
example, we will understand life in this way. Only in the final volume will
we discover that, at a fundamental scale, this decomposition is impossible.
%
% Reread July 2016
\subsection{Legs and wheels}
% Index OK
The parts\index{wheel!in living beings(} of a body determine its shape.
Shape\label{legwh1} is an important aspect of bodies:\index{legs!in nature}
among other things, it tells us how to count them. In particular, living
beings are always made of a single body. This is not an empty statement: from
this fact we can deduce that animals cannot have large wheels or large
\iin[propeller!in living beings]{propellers}, but only legs, fins, or wings. Why?
Living beings have only one surface; simply put, they have only one piece of
\iin{skin}. Mathematically speaking, animals are
\ii[body!connected]{connected}.\seepagefive{manidef} This is often assumed to
be obvious, and it is often mentioned that\cite{fakew} the
\iin[blood!supply]{blood supply}, the nerves and the lymphatic connections to
a rotating part would get tangled up. However, this argument is not so
simple, as \figureref{ifakewheel3} shows. The figure proves that it is
indeed possible to rotate a body continuously against a second one, without
tangling up the connections. Three dimensions of space allow
\ii[rotation!tethered]{tethered rotation}. Can you find an example for this
kind of motion, often called \emph{tethered
rotation},\index{tether}\index{rotation!tethered} in your own
body?\challengenor{arm} Are you able to see how many cables may be attached to
the rotating body of the figure without hindering the
rotation?\challengenor{arm2}
\csepsftw{ifakewheel3}{scale=0.9212}{Tethered rotation: How an object can
rotate continuously without tangling up the connection to a second object.}
% New in Mar 2016
\csmovfilmrepeat{dodecatwist}{scale=0.5}{Tethered
rotation:\protect\index{rotation!tethered} the continuous rotation of an
object attached to its environment (QuickTime film
{\textcopyright}~\protect\iinn{Jason Hise}).}
Despite the possibility of animals having rotating parts, the method of
\figureref{ifakewheel3} or \figureref{dodecatwist} still cannot be used to
make a practical wheel or propeller. Can you see why?\challengenor{wheel}
Therefore, evolution had no choice: it had to avoid animals with (large) parts
rotating around axles. That is the reason that propellers and wheels do not
exist in nature. Of course, this limitation does not rule out that living
bodies move by rotation as a whole: \iin{tumbleweed},\cite{tumble} seeds from
various trees, some insects, several\index{spider!rolling} spiders, certain
other animals, children and dancers occasionally move by rolling or rotating
as a whole.
% (OK) make as wide as page
\csepsfnb{ilegwheel}{scale=1.0}{Legs and `wheels' in living
beings:\protect\index{rolling!motion} the red millipede
\protect\iie{Aphistogoniulus erythrocephalus} (15$\,$cm body length), a
gecko on a glass pane (15$\,$cm body length), an {amoeba}
\protect\iie{Amoeba proteus} (1$\,$mm size), the rolling shrimp
\protect\iie{Nannosquilla decemspinosa} (2$\,$cm body length, 1.5 rotations
per second, up to 2$\,$m, can even roll slightly uphill slopes) and the
rolling caterpillar \protect\iie{Pleurotya ruralis} (can only roll downhill,
to escape predators), ({\textcopyright}~\protect\iinn{David Parks},
% SENT EMAIL FEB 2008  drparks@stanford.edu
\protect\iinn{Marcel Berendsen},
\protect\iinn{Antonio Guillén Oterino}, % added double name in March 2014
\protect\iinn{Robert Full},
% SENT EMAIL FEB 2008  rjfull@berkeley.edu
\protect\iinn{John Brackenbury} / \protect\iname{Science Photo Library}
% SENT EMAIL FEB 2008  jhb1000@cam.ac.uk
).}
% Mar 2014
\csepsf{icebrennussomersault}{scale=1}{Two of the rare lifeforms that are
able to roll \emph{uphill} also on steep slopes: the desert spider
\protect\iie{Cebrennus villosus} and \emph{Homo sapiens}
({\textcopyright}~\protect\iinn{Ingo Rechenberg}, \protect\iinn{Karva Javi}).}
% www.bionik.tuberlin.de
%
% www.flickr.com/photos/karvajavi/3726129366 with permission
% (no, not locomotion in general) one day, maybe add photo of snail at water
% air interface, and of snake on sand
\emph{Large single bodies}, and thus all large living beings, can thus only
move through \ii[shape!deformation and motion]{deformation} of their
shape:\cite{jgrayloc} therefore they are limited to walking, running, jumping,
rolling, gliding, crawling, flapping fins, or flapping wings. Moving a leg is
a common way to deform a body.
% Impr. Mar 2014
Extreme examples of leg use\cite{leguse} in nature are shown in
\figureref{ilegwheel} and \figureref{icebrennussomersault}. The most
extreme example of rolling spiders  there are several species  are
\iie{Cebrennus villosus} and live in the sand in Morocco.\cite{rechenberg}
They use their legs to accelerate the rolling, they can steer the rolling
direction and can even roll uphill slopes of 30\,\%  a feat that humans are
unable to perform. Films of the rolling motion can be found at
\url{www.bionik.tuberlin.de}.%
%
% Oct 2009
\footnote{Rolling is also known for the Namibian wheel spiders of the
\iie{Carparachne} genus; films of their motion can be found on the internet.}
% \url{www.youtube.com/watch?v=5XwIXFFVOSA} and
% \url{www.youtube.com/watch?v=ozn31QBOHtk}.
%
Walking on water is shown in \figureref{iwaterstrider} on
\cspageref{iwaterstrider}; % this vol I
examples of wings\seepagefive{iwings} are given later on,
% Apr 2010
as are the various types of deformations that allow swimming in
water.\seepagefive{extmot}
In contrast, \emph{systems of several bodies}, such as bicycles, pedal boats
or other machines, can move \emph{without} any change of shape of their
components, thus enabling the use of axles with wheels, propellers and other
rotating devices.%
%
\footnote{Despite the disadvantage of not being able to use rotating parts and
of being restricted to one piece only, nature's moving constructions,
usually called animals, often outperform human built machines. As an
example, compare the size of the smallest flying systems built by evolution
with those built by humans. (See, e.g., \url{pixelito.reference.be}.)
There are two reasons for this discrepancy. First, nature's systems have
integrated repair and maintenance systems. Second, nature can build large
structures inside containers with small openings. In fact, nature is very
good at what people do when they build sailing ships inside glass bottles.
The human body is full of such examples; can you name
a\challengenor{bodyship} few?}
% Mar 2014
In short, whenever we observe a construction in which some part is turning
continuously (and without the `wiring' of \figureref{ifakewheel3}) we know
immediately that it is an \iin{artefact}: it is a \iin{machine}, not a living
being (but built by one). However, like so many statements about living
creatures, this one also has exceptions.\label{rotbact}
% April 2014, Impr. Mar 2015
\csepsfnb[p]{iflagellarmotorvarieties}{scale=0.9}{Some types of flagellar
motors found in nature; the images are taken by cryotomography. All
yellow scale bars are 10\,nm long ({\textcopyright}~\protect\iinn{S. Chen} \& al.,
\protect\iname{EMBO Journal, Wiley \& Sons}).}
% I have got permission for both figures from the Wiley Website in 2014
% Mar 2014
\csmovfilmrepeat{MotorReversal}{scale=1}{The rotational motion of a bacterial
flagellum, and its reversal (QuickTime film
{\textcopyright}~\protect\iname{Osaka University}).}
% http://www.fbs.osakau.ac.jp/labs/namba/npn/movies/MotorReversal.mpeg
% Sent email 16 Mar 2014, got pemission!
% Mar 2014
\csmovfilmrepeat{FlagellarAssembly}{scale=1}{The growth of a bacterial
flagellum, showing the molecular assembly (QuickTime film
{\textcopyright}~\protect\iname{Osaka University}).}
The distinction between one and two bodies is poorly defined if the whole
system is made of only a few molecules. This happens most clearly inside
bacteria.\index{wheel!in bacteria} Organisms such as \iie{Escherichia coli},
the wellknown \iin[bacterium]{bacterium} found in the human gut, or bacteria
from the \iie{Salmonella} family, all swim using flagella.
\ii[flagella]{Flagella} are thin filaments, similar to tiny hairs that stick
out of the cell membrane. In the 1970s it was shown that each flagellum, made
of one or a few long molecules with a diameter of a few tens of nanometres,
does in fact turn about its axis.\seepagefive{extmot}
% Mar 2014
Bacteria are able to rotate their flagella in both clockwise and anticlockwise
directions, can achieve more than 1000 turns per second, and can turn all its
{flagella} in perfect synchronization.\cite{a16} These wheels are so tiny that
they do not need a mechanical connection;
%
\figureref{iflagellarmotorvarieties} shows a number of motor models found
in bacteria.\cite{flagmotxx}
%
The motion and the construction of these amazing structures is shown in more
details in the films \figureref{MotorReversal} and
\figureref{FlagellarAssembly}.
% Mar 2014
In summary, wheels actually do exist in living beings, albeit only tiny ones.
The growth and motion of these wheels are wonders of nature. Macroscopic
wheels in living beings are not possible, though rolling
motion\index{wheel!in living beings)} is.
% We now continue with our study of simple objects.
% Macroscopic creatures have no turning parts; however,
% In this promenade, we want to achieve the highest precision possible for
% the
% description of motion. We will therefore concentrate on the simplest
% examples, namely the motion of one or a few particles, and leave aside the
% description of motion of large bodies, so beautiful it often is, as the
% picture on the front cover shows. It turns out that the colour, the
% temperature, the shape, the smoothness of bodies are all consequences of
% differences in position or velocity of its constituents.
% Improved Apr 2013
\csepsfnb{icometmcnaught}{scale=1}{Are comets, such as the beautiful comet
McNaught seen in 2007, images or bodies? How can you show it? (And why is
the tail curved?) ({\textcopyright}~\protect\iinn{Robert McNaught})}
% SENT EMAIL FEB 2008  rmn@murky.anu.edu.au
%
% Impr. Jul 2016
\subsection{Curiosities and fun challenges about kinematics}
% Index OK
\begin{curiosity}
% Aug 2007
\item[] What is the biggest wheel\index{wheel!biggest ever} ever
made?\challengenor{biggestwheel}
% Aug 2007
\item A football\index{soccer ball} is shot, by a goalkeeper, with around
\csd{30}{m/s}. Use a video to calculate the distance it should fly and
compare it with the distances realized in a soccer match. Where does the
difference come\challengenor{soccerz} from?
% Sep 2005
\item A train starts\index{train!puzzle}\index{puzzle!train} to travel at a
constant speed of \csd{10}{m/s} between two cities A and B, \csd{36}{km}
apart. The train will take one hour for the journey. At the same time as
the train, a fast dove starts to fly from A to B, at \csd{20}{m/s}. Being
faster than the train, the dove arrives at B first. The dove then flies
back towards A; when it meets the train, it turns back again, to city B. It
goes on flying back and forward until the train reaches B. What distance did
the dove cover?\challengn
% Jul 2016
\csepsfnbcenter{isafetyparabola}{scale=1}{The {parabola of safety} around a
cannon,\protect\index{parabola!of safety} shown in red. The highest points
of all trajectories form an ellipse, shown in blue.
({\textcopyright}~\protect\iname{Theon})} % from the french wikipedia
% Jul 2016
\item \figureref{isafetyparabola} illustrates that around a cannon, there is
a line %  more precisely, a surface 
outside which you cannot be hit. Already in the 17th century,
\iinn{Evangelista Torricelli} showed, without algebra, that the line is a
parabola, and called it the \emph{parabola of safety}. Can you show this as
well?\challengn Can you confirm that the highest points of all trajectories
lie on an ellipse?
% Nov 2016
The parabola of safety also appears in certain water
fountains.\index{fountain!water}
% June 2005
\item Balance a pencil\index{pencil} vertically (tip upwards!) on a piece of
paper near the edge of a table. How can you pull out the paper without
letting the pencil fall?\challengn
\item Is a return flight by aeroplane\index{aeroplane!flight puzzle}  from a
point A to B and back to A  faster if the wind blows or if it does
not?\challengn
\item The level of acceleration\index{acceleration!dangers of} that a human
can survive depends on the duration over which one is subjected to it. For
a tenth of a second, \csd{30}{}$g$ \csd{=300}{m/s^{2}}, as generated by an
ejector seat in an aeroplane, is acceptable. (It seems that the record
acceleration a human has survived is about \csd{80}{}$g$
\csd{=800}{m/s^{2}}.) But as a rule of thumb it is said that accelerations
of \csd{15}{}$g$ \csd{=150}{m/s^{2}} or more are fatal.
\cssmallepsfnb{fsonolu}{scale=0.6}{Observation of sonoluminescence with a
simple setup that focuses ultrasound in water
({\textcopyright}~\protect\iinn{Detlef Lohse}).}
% EMAILED FEB 2008  D.Lohse@tnw.utwente.nl
% add a diagram of the experimental setup (NO) for sonoluminescence
% A diagram is here physics.open.ac.uk/~swebb/ach.htm (look at from
% archive.org; title is ``Achieving Sonoluminescence'')
\item The highest \emph{microscopic} accelerations\index{acceleration!highest}
are observed in particle collisions, where values up to
\csd{10^{35}}{m/s^{2}} are achieved.
% seems ok: v^2/2x with c^2 and 10^18 (I hope I did not forget any
% relativistic effect ...)
%
% a = v/t = v^{2}/2x therefore, using x = 0.1, v > 13 m/s^{2} = 50 km/h
% is v_{final} for jump from 10m
%
The highest \emph{macroscopic} accelerations are probably found in the
collapsing interiors of \ii{supernovae}, exploding stars which can be so
bright as to be visible in the sky even during the daytime. A candidate on
Earth is the interior of collapsing bubbles in liquids, a process called
\ii{cavitation}. Cavitation often produces light, an effect discovered by
\iname[Frenzel, H.]{Frenzel} and \iname[Schultes, H.]{Schultes} in 1934 and
called \ii{sonoluminescence}. (See \figureref{fsonolu}.)\cite{lohse} It
appears most prominently when air bubbles in water are expanded and
contracted by underwater loudspeakers at around \csd{30}{kHz} and allows
precise measurements of bubble motion. At a certain threshold intensity,
the bubble radius changes at \csd{1500}{m/s} in as little as a few
\csdunit{\muunit m}, giving an acceleration of several
\csd{10^{11}}{m/s^{2}}.\cite{wenni}
% intensity, the bubble radius changes by a \csdunit{\muunit m} in as little
% as
% \csd{10}{ps}, giving an acceleration of a few \csd{10^{16}}{m/s^{2}}.
% this result was from simulations only
% Aug 2007
\item Legs are easy to build.\index{leg!number record} Nature has even
produced a millipede, \iie{Illacme plenipes}, that has 750 legs. The animal
is 3 to \csd{4}{cm} long and about \csd{0.5}{mm} wide. This seems to be the
record so far. In contrast to its name, no millipede actually has a
thousand legs.
% % % Jun 2008 (NO)
% % \item So far, we have discovered ..
\end{curiosity}
%
% Impr. Jul 2016
\subsection{Summary of kinematics}
% Index OK
The description\index{kinematics!summary} of everyday motion of mass points
with three coordinates as $(x(t), y(t), z(t))$ is simple, precise and
complete. This description of paths is the basis of kinematics. As a
consequence, space is described as a threedimensional Euclidean space and
velocity and acceleration as Euclidean vectors.
The description of motion with paths assumes that the motion of objects can be
\emph{followed} along their paths. Therefore, the description often does not
work for an important case: the motion of images.
\vignette{classical}
%
%
%
%
\newpage
% Reread July 2016
\chapter{From objects and images to conservation}
% Index OK
\markboth{\thesmallchapter\ from objects and images to conservation}%
{\thesmallchapter\ from objects and images to conservation}
\label{oim}
%
\csini{W}{alking} through a forest\index{object!difference from
image}\index{image!difference from object}
%here at the base of Motion Mountain,\seefig{imiland9}
we observe two rather different types of motion:\linebreak e see the breeze
move the leaves, and at the same time, on the ground,\linebreak e see their
\iin[shadow!motion]{shadows} move. Shadows are a simple type of
image.\cite{itshad} Both objects and images are able to move; both change
position over time. Running tigers, falling snowflakes, and material ejected
by volcanoes, but also the shadow following our body, the beam of light
circling the tower of a lighthouse on a misty night, and the rainbow that
constantly keeps the same apparent distance from us are examples of motion.
Both objects and images differ from their environment in that they have
\ii{boundaries} defining their size and shape. But everybody who has ever
seen an animated cartoon knows that images can move in more surprising ways
than objects. Images can change their size and shape, they can even change
colour, a feat only few objects are able to\label{minnae} perform.%
%
\footnote{Excluding very slow changes such as the change of colour of leaves
in the Autumn, in nature only certain crystals, the octopus and other
cephalopods, the chameleon and a few other animals achieve this. Of
manmade objects, television, computer displays, heated objects and certain
lasers can do it. Do you know more examples?\challengenor{colourc} An
excellent source of information on the topic of colour is the book by \asi
K. Nassau/ \bt The Physics and Chemistry of Colour  the fifteen causes of
colour/ J.~Wiley \& Sons, \yrend 1983/ In the popular science domain, the
most beautiful book is the classic work by the Flemish astronomer \asi
Marcel G.J. Minnaert/ \bt Light and Colour in the Outdoors/ \pu Springer/
\yr 1993/ an updated version based on his wonderful book series, \bt De
natuurkunde van `t vrije veld/ Thieme \& Cie, % Zutphen
\yrend 1937/ Reading it is a must for all natural scientists.\cite{colorbk}
On the web, there is also the  simpler, but excellent 
\url{webexhibits.org/causesofcolour} website.} %
%
Images can appear and disappear without trace, multiply, interpenetrate, go
backwards in time and defy gravity or any other force. Images, even ordinary
\iin{shadows}, can move faster than light. Images can float in space and keep
the same distance from approaching objects. Objects can do almost none of
this.\cite{cartphys} In general, the `laws of cartoon physics' are rather
different from those in nature.\index{cartoon physics, `laws' of} In fact, the
motion of images does not seem to follow any rules, in contrast to the motion
of objects. We feel the need for precise criteria allowing the two cases to
be distinguished.
Making a clear distinction between images and objects is performed using the
same method that children or animals use when they stand in front of a mirror
for the first time: they try to \ii{touch} what they see. Indeed,
\begin{quotation}
\npcsrhd If we are
able to touch what we see  or more precisely, if we are able to move it
with a collision 
we call it an \ii{object}, otherwise an \ii{image}.%
%
\footnote{One could propose including the requirement that objects may be
rotated; however, this requirement, surprisingly, gives difficulties in the
case of atoms, as explained on \cspageref{atomrot} in Volume IV.}
\end{quotation}
Images cannot be touched, but objects can.\seepagefour{fermmat} Images cannot
hit each other, but objects can. And as everybody knows, touching something
means feeling that it resists movement. Certain bodies, such as butterflies,
pose little resistance and are moved with ease, others, such as ships, resist
more, and are moved with more difficulty.
\begin{quotation}
\npcsrhd The resistance to motion  more precisely, to change of motion 
is called \ii{inertia}, and the difficulty with which a body can be moved is
called its \ii[mass!inertial]{(inertial) mass}.
\end{quotation}
Images have neither inertia nor mass.
Summing up, for the description of motion we must distinguish bodies, which
can be touched and are impenetrable, from images, which cannot and are not.
Everything visible is either an object or an image; there is no third
possibility. (Do you agree?)\challengenor{third} If the object is so far away
that it cannot be touched, such as a star or a comet, it can be difficult to
decide whether one is dealing with an image or an object; we will encounter
this difficulty repeatedly. For example, how would you show that comets 
such as the beautiful example of \figureref{icometmcnaught}  are objects
and not images, as Galileo (falsely) claimed?\challengenor{comets}
In the same way that objects are made of \ii{matter}, images are made of
\ii{radiation}. Images are the domain of shadow theatre, cinema, television,
computer\cite{cinefex} graphics, \iin{belief systems} and drug experts.
%halos, Kirlian
Photographs, motion pictures, \iin{ghosts}, \iin{angels}, dreams and many
hallucinations are images (sometimes coupled with brain malfunction). To
understand images, we need to study radiation (plus the eye and the brain).
However, due to the importance of objects  after all we are objects
ourselves  we study the latter first.
\cssmallepsf{isteer}{scale=1}{In which direction does the bicycle turn?}
%
% Impr. Jul 2016
\subsection{Motion and contact}
% Index OK
\begin{quote}
Democritus affirms that there is only one type of movement:
That resulting from collision.\\
\iname{Aetius}, \btsim Opinions/.\cite{presocr567}
\end{quote}
\np When a child rides a \iin{unicycle}, she or he makes use of a general rule
in our world: one body acting on another puts it in motion. Indeed, in about
six hours, anybody can learn to ride and enjoy a unicycle. As in all of
life's pleasures, such as toys, animals, women, machines, children, men, the
sea, wind, cinema, juggling, rambling and loving, something pushes something
else. Thus our first challenge is to describe the transfer of motion due to
contact\index{contact!and motion}\index{collision!and motion}\index{motion!and
contact}\index{motion!and collision}  and to collisions  in more precise
terms.
Contact is not the only way to put something into motion; a counterexample is
an \iin[apple!and fall]{apple} falling from a tree or one magnet pulling
another. %\label{contact}
Noncontact influences are more fascinating: nothing is hidden, but
nevertheless something mysterious happens. Contact motion seems easier to
grasp, and that is why one usually starts with it. However, despite this
choice, noncontact interactions cannot be avoided. Our choice to start with
contact will lead us to a similar experience to that of riding a
\iin[bicycle!riding]{bicycle}. (See \figureref{isteer}.) If we ride a
bicycle at a {sustained} speed and try to turn left by pushing the righthand
steering bar, we will turn \emph{right}. By the way, this surprising effect,
also known to motor bike riders,\index{motor bike} obviously works only above
a certain minimal speed. Can you determine what this speed
is?\challengenor{bikespeed} Be careful! Too strong a push will make you fall.
Something similar will happen to us as well; despite our choice for contact
motion, the rest of our walk will rapidly force us to study noncontact
interactions.
%
% Reread Jul 2016
\subsection{What is mass?}
% Index OK
\label{mass1}%
%
%
\begin{quote}
% Improved Jan 2012
\csgreekok{D'oc mo'i (fhsi) po\~u st\~w ka`i kin\~w t`hn g\~hn.}
Da ubi consistam, et terram %[caelumque]
movebo.\footnote{`Give me a place to stand, and I'll move the Earth.'
Archimedes \lived(\circa 283212), Greek scientist and engineer. This phrase
is attributed to him by \iname{Pappus}.\cite{chilhadetto2} % frase 319
Already Archimedes knew that the distinction used by \iin{lawyers} between
\iin[object!movable]{movable} and \iin[object!immovable]{immovable}
objects made no sense.}\\
\inames{Archimedes}
\end{quote}
% EMAILED FEB 2008  webmaster@bipm.org
\cstftlepsfpsfragone{ibilliard}{scale=1}{Collisions define
mass.}[28mm]{fkilogram}{scale=1}{The standard kilogram
({\textcopyright}~\protect\iname{BIPM}).}%
[%
\psfrag{v1}{\small $v_{1}$}%
\psfrag{v2}{\small $v_{2}$}%
\psfrag{v1d}{\small $v_{1}+\Delta v_{1}$}%
\psfrag{v2d}{\small $v_{2}+\Delta v_{2}$}%
]
% (OK) check typesetting, spurious ``[]'' (April 2013)
\np When we push something we are unfamiliar with, such as when we kick an
object on the street, we automatically pay attention to the same aspect that
children explore when they stand in front of a mirror for the first time, or
when they see a red laser spot for the first time. They check whether the
unknown entity can be pushed or caught, and they pay attention to how the
unknown object moves under their influence. All these are collision
experiments. The high precision version of any collision experiment is
illustrated in \figureref{ibilliard}. Repeating such experiments with
various pairs of objects, we find:
\begin{quotation}
% as in everyday life  that a
\npcsrhd A \emph{fixed} quantity
$m_{i}$ can be ascribed to every object $i$, determined by the relation
\begin{equation}
\frac{ m_{2}}{ m_{1}}
= \frac{\Delta v_{1} }{ \Delta v_{2}}
\label{eq:masdef1}
\end{equation}
where $\Delta v$ is the velocity change produced by the collision. The number
$m_{i}$ is called the \ii{mass} of the object $i$.
\end{quotation}
\np The more difficult it is to move an object, the higher the number. In
order to have mass values that are common to everybody, the mass value for one
particular, selected object has to be fixed in advance. This special object,
shown in \figureref{fkilogram}, is called the \ii{standard kilogram} and is
kept with great care in a glass container in Sèvres near Paris.
%
The standard kilogram is touched only once every few years because otherwise
dust, humidity, or scratches would change its mass.
% Jan 2011
By the way, the standard kilogram is \emph{not} kept under vacuum, because
this would lead to outgassing and thus to changes in its mass.
%
The standard kilogram determines the value of the mass of every other object
in the world.\label{massdef}
\cssmallepsfnb{ilavoisier}{scale=0.45}{Antoine Lavoisier
\livedfig(17431794) and his wife.}
The \ii[mass!concept of]{mass} thus measures \emph{the difficulty of getting
something moving.} High\index{mass!measures motion difficulty} masses are
harder to move than low masses. Obviously, only objects have mass; images
don't. (By the way, the word `mass' is derived, via Latin, from the Greek
\csgreekok{maza}  bread  or the\cite{ety} Hebrew `mazza'  unleavened
bread. That is quite a change in meaning.)
% May 2004
Experiments with everyday life objects also show that throughout any
collision, the sum of all masses is \emph{conserved}:\index{mass!is conserved}
\begin{equation}
\sum_{i} m_{i}={\rm const} \cp
\label{mdef1b}
\end{equation}
The principle of conservation of mass was first stated by
\iinns{AntoineLaurent Lavoisier}.%
%
\footnote{AntoineLaurent Lavoisier \livedplace(1743 Paris 1794 Paris),
chemist and genius. Lavoisier was the first to understand that combustion
is a reaction with oxygen; he discovered the components of water and
introduced mass measurements into chemistry. A famous story about his
character: When he was (unjustly) sentenced to the guillotine during the
French revolution, he decided to use the situations for a scientific
experiment. He announced that he would try to blink his eyes as frequently
as possible after his head was cut off, in order to show others how long it
takes to lose consciousness.\index{eye!blinking after guillotine} Lavoisier
managed to blink eleven times. It is unclear whether the story is true or
not. It is known, however, that it could be true. Indeed, after a
decapitation without pain or shock, a person can remain conscious for up to
half a minute.\cite{headdead}} %
%
Conservation of mass also implies that the mass of a composite system is the
sum of the mass of the components. In short, \emph{mass is also a measure for
the \iin[matter!quantity of]{quantity of matter}.}\index{mass!measures
quantity of matter}
In a famous experiment in the sixteenth century, for several weeks
\inames{Santorio Santorio} (\inames{Sanctorius}) \lived(15611636), friend of
Galileo, lived with all his food and drink supply, and also his toilet, on a
large balance. He wanted to test mass conservation. How did the measured
weight change with time?\challengenor{weightman}
% Nov 2016
Various cult leaders pretended and still pretend that they can produce matter
out of nothing. This would be an example of nonconservation of mass. How can
you show that all such leaders are crooks?\challengenor{cultcrooks}
\cssmallepsfnb{ihuygens}{scale=0.13}{Christiaan Huygens
\livedfig(16291695).}
%
% Impr. July 2016
\subsection{Momentum and mass}
% Index OK
The definition of mass can also be given in another way.
We\index{mass!definition} can ascribe a number $m_{i}$ to every object $i$
such that for collisions free of outside interference the following sum is
unchanged \emph{throughout} the collision:
\begin{equation}
\sum_{i} m_{i} {\bm v_{i}} = {\rm const} \cp
\label{mc}
\end{equation}
The product of the velocity ${\bm v_{i}}$ and the mass $m_{i}$ is called the
(linear) \ii{momentum} of the body. The sum, or \ii[momentum!total]{total
momentum} of the system, is the same before and after the collision;
momentum is a \emph{conserved} quantity.
\begin{quotation}
\noindent \csrhd {Momentum conservation defines mass.}
\end{quotation}
\np The two conservation principles (\ref{mdef1b}) and (\ref{mc}) were first
stated in this way by the important %Dutch
physicist\label{huygensvita} \iinns{Christiaan Huygens}:%
%
\footnote{Christiaan Huygens \livedplace(1629 's Gravenhage1695 Hofwyck) was
one of the main physicists and mathematicians of his time. Huygens
clarified the concepts of mechanics; he also was one of the first to show
that light is a wave. He wrote influential books on probability theory,
clock mechanisms, optics and astronomy. Among other achievements, Huygens
showed that the Orion Nebula consists of stars, discovered Titan, the moon
of Saturn, and showed that the rings of Saturn consist of rock. (This is in
contrast to Saturn itself, whose density is lower than that of water.)} %
%
% Apr 2006
\emph{Momentum and mass are conserved in everyday motion of objects.} In
particular, neither quantity can be defined for the motion of images. Some
typical momentum values are given in \tableref{momentumtab}.
% \cssmallepsf[11]{fconcrete}{scale=1}{Is this dangerous?}
\cssmallepsf{fconcrete}{scale=1}{Is this dangerous?}
Momentum conservation implies that when a moving sphere hits a resting one of
the same mass and without loss of energy, a simple rule determines the angle
between the directions the two spheres take after the collision. Can you find
this rule?\challengenor{poolrule} It is particularly useful when playing
\iin{billiards}. We will find out later that the rule is \emph{not} valid for
speeds near that\seepagetwo{ipool} of light.
%
% {Table of momenta}
%
{\small
\begin{table}[t]
\sbox{\cshelpbox}{\small\dirrtabular%
\begin{tabular}{@{\hspace{0em}} >{\PBS\raggedright\hspace{0.0em}%
\columncolor{hks152}[0pt][1cm]}
p{75mm} @{\hspace{1em}} p{25mm} @{\hspace{0em}}}
%
\toprule
%
\tabheadf{Observation} & \tabhead{Momentum} \\[0.5mm]
%
\midrule
%
Images & 0 \\
%
Momentum of a green photon & $\csd{1.2\cdot 10^{27}}{Ns}$ \\
%
Average momentum of oxygen molecule in air & $\csd{10^{26}}{Ns}$ \\
%
Xray photon momentum & \csd{10^{23}}{Ns} \\
%
$\gamma$ photon momentum & \csd{10^{17}}{Ns} \\
%
Highest particle momentum in accelerators & \csd{1}{fNs} \\
%
Highest possible momentum of a single elementary particle  the
Planck momentum & $\csd{6.5}{Ns}$ \\
%
Fast billiard ball & $\csd{3}{Ns}$ \\
% % %
% % Highest possible momentum of a single elementary particle  the
% % corrected
% % Planck momentum & $\csd{3.2}{Ns}$ \\
%
Flying rifle bullet & $\csd{10}{Ns}$ \\
%
Box punch & 15 to \csd{50}{Ns} \\
%
Comfortably walking human & $\csd{80}{Ns}$ \\
%
Lion paw strike & \def\circa\csd{0.2}{kNs} \\ % my own estimate
%
Whale tail blow & \def\circa\csd{3}{kNs} \\ % my own estimate
%
Car on highway & $\csd{40}{kNs}$ \\
%
Impact of meteorite with \csd{2}{km} diameter & $\csd{100}{TNs}$ \\
%
Momentum of a galaxy in\index{galaxy!collision} galaxy collision& up to $\csd{10^{46}}{Ns}$ \\
%
%
\bottomrule
\end{tabular}%
}% end of sbox
%
\centering
\begin{minipage}{\wd\cshelpbox}%
\captionsetup{width=\wd\cshelpbox} % needed since Dec 2015
\caption{Some measured\protect\index{momentum!values, table} momentum
values.}%
\label{momentumtab}\noindent\usebox{\cshelpbox}
\end{minipage}
\end{table}
}
Another consequence of momentum conservation is shown in
\figureref{fconcrete}:
%
% was shown on the cover photograph of the \iin{CERN Courier}
% %(before January 1995)
% in 1994.
a man is lying on a bed of nails with a large block of concrete on his
stomach. Another man is hitting the concrete with a heavy sledgehammer.
%Sometimes the concrete is even full of nails at its bottom.
As the impact is mostly absorbed by the concrete, there is no pain and no
danger  unless the concrete is missed. Why?\challengenor{nails}
The above definition (\ref{eq:masdef1}) of mass has been generalized by the
physicist and philosopher \iinns{Ernst Mach}%
%
\footnote{Ernst Mach \lived(1838 Chrlice1916 Vaterstetten), Austrian
physicist and philosopher. The \ii{mach} unit for \iin[aeroplane!speed
unit]{aeroplane} speed as a multiple of the \iin{speed of sound} in air
(about \csd{0.3}{km/s}) is named after him.
% His SON developed the socalled MachZehnder \iin{interferometer}.
He also studied the basis of mechanics.
His thoughts about mass and inertia influenced the development of general
relativity, and led to \iin{Mach's principle}, which will appear later on.
He was also proud to be the last scientist denying  humorously, and
against all evidence  the existence of atoms.} %
%
in such a way that it is valid even if the two objects interact without
contact, as long as they do so along the line connecting their positions.
\begin{quotation}
\npcsrhd The
mass ratio between two bodies is defined as a negative inverse acceleration
ratio, thus as\label{machmassdef}
\begin{equation}
\frac{ m_{2}}{ m_{1}} = \frac{a_{1} }{ a_{2}} \cvend
\label{mdef2}
\end{equation}
where $a$ is the acceleration of each body during the interaction.
\end{quotation}
This definition of mass has been explored in much detail in the physics
community, mainly in the nineteenth century. A few points sum up the results:
\begin{Strich}
\item The definition of mass \emph{implies} the conservation of total momentum
$\sum mv$. Momentum conservation is \emph{not} a separate principle.
Conservation of momentum {cannot} be checked
experimentally,\index{mass!definition implies momentum conservation} because
mass is defined in such a way that the momentum conservation
holds.\index{momentum!conservation follows from mass definition}
\item The definition of mass \emph{implies} the equality of the products
$m_{1}a_{1}$ and $m_{2}a_{2}$. Such products are called
\ii[force!definition]{forces}. The equality of acting and reacting forces
is not a separate principle; mass is defined in such a way that the
principle holds.
\item The definition of mass is \emph{independent} of whether contact is
involved or not, and whether the accelerations are due to electricity,
gravitation, or other interactions.%
%
\footnote{As mentioned above, only \ii[force!central]{central} forces obey
the relation (\ref{mdef2}) used to define mass. Central forces act
between the centre of mass of bodies. We give a precise definition
later.\seepageone{masscentre} However, since all fundamental forces are
central, this is not a restriction. There seems to be one notable
exception: \iin{magnetism}. Is the definition of mass
valid\challengenor{yesmass} in this case?} %
%
Since the interaction does not enter the definition of mass, mass values
defined with the help of the electric, nuclear or gravitational interaction
all agree, as long as momentum is conserved. All known interactions
conserve momentum. For some unfortunate historical reasons, the mass value
measured with the electric or nuclear interactions is called the `inertial'
mass and the mass measured using gravity is called the `gravitational' mass.
As it turns out, this artificial distinction makes no sense; this becomes
especially clear when we take an observation point that is \emph{far away}
from all the bodies concerned.
\item The definition of mass requires observers at rest or in inertial
motion. % More about this issue later. (NO)
\end{Strich}
%
% {Table of masses}
{\small
\begin{table}[t]
\small
\centering
\caption{Some measured\protect\index{mass!values, table} mass values.}
\label{massmetab}
\dirrtabularstar
\begin{tabular*}{\textwidth}{@{}>{\PBS\raggedright\hspace{0.0em}%
\columncolor{hks152}[0pt][1cm]}p{78mm}
@{\extracolsep{\fill}} p{55mm}@{}}
%
\toprule
%
\tabheadf{Observation} & \tabhead{Mass} \\[0.5mm]
%
\midrule
%
Probably lightest known object: neutrino & \circa$\csd{2\cdot 10^{36}}{kg}$ \\
%
Mass increase due to absorption of one green photon & $\csd{4.1\cdot
10^{36}}{kg}$ \\
%
Lightest known charged object: electron & $\csd{9.109\,381\,88(72)\cdot
10^{31}}{kg}$ \\
%
Atom of argon & $\csd{39.962\; 383\; 123(3)}{u}=\csd{66.359\,1(1)}{yg}$ \\
%
Lightest object ever weighed (a gold particle) & $\csd{0.39}{ag}$\\
%
Human at early age (fertilized egg) & $\csd{10^{8}}{g}$\\
%
Water adsorbed on to a kilogram metal weight & $\csd{10^{5}}{g}$\\
%
Planck mass & $\csd{2.2\cdot 10^{5}}{g}$\\
%
Fingerprint & $\csd{10^{4}}{g}$\\
%
Typical ant & $\csd{10^{4}}{g}$ \\
%
Water droplet & $\csd{1}{mg}$\\
%
{Honey bee},\index{honey bees} \iie{Apis mellifera} & $\csd{0.1}{g}$\\
%
Euro coin & $\csd{7.5}{g}$\\
%
{Blue whale}, \iie{Balaenoptera musculus} & $\csd{180}{Mg}$\\
%
{Heaviest living things},\index{living thing, heaviest} such as
the\indexe{Armillaria ostoyae} fungus \emph{Armillaria ostoyae} or a
large\indexe{Sequoiadendron giganteum} Sequoia \emph{Sequoiadendron
giganteum}&
$\csd{10^{6}}{kg}$\\
%
Heaviest \iin{train} ever & $\csd{99.7\cdot 10^{6}}{kg}$\\
%
Largest oceangoing \iin{ship} & $\csd{400\cdot 10^{6}}{kg}$\\
%
Largest object moved by man (Troll gas rig) & $\csd{687.5\cdot
10^{6}}{kg}$\\
%
Large antarctic \iin{iceberg} & $\csd{10^{15}}{kg}$\\
%
Water on Earth & $\csd{10^{21}}{kg}$\\
%
Earth's mass & $\csd{5.98 \cdot 10^{24}}{kg}$\\
%
Solar mass\index{Sun} & $\csd{2.0 \cdot 10^{30}}{kg}$\\
%
Our galaxy's visible mass & $\csd{3 \cdot 10^{41}}{kg}$\\ % from stellar
% numbers
%
Our galaxy's estimated total mass & $\csd{2 \cdot 10^{42}}{kg}$\\ % wikipedia
%
virgo supercluster & $\csd{2 \cdot 10^{46}}{kg}$\\ % wikipedia
%
Total mass visible in the universe & $\csd{10^{54}}{kg}$\\
% wikipedia says 10^60 in one place ; that seems wrong
%
\bottomrule
\end{tabular*}
\end{table}
}
\np By measuring the masses of bodies around us we can explore the science and
art of experiments. An overview of mass measurement devices is given in
\tableref{masssensors} and \figureref{imassmeasurement}. Some
measurement results are listed in \tableref{massmetab}.
Measuring mass vales around us we confirm the main properties of mass. First
of all, mass is \emph{additive} in everyday life, as the mass of two bodies
combined is equal to the sum of the two separate masses. Furthermore, mass is
\emph{continuous}; it can seemingly take any positive value. Finally, mass is
\emph{conserved} in everyday life; the mass of a system, defined as the sum of
the mass of all constituents, does not change over time if the system is kept
isolated from the rest of the world. Mass is not only conserved in collisions
but also during melting, evaporation, digestion and all other everyday
processes.
%
% {Properties of mass}
%
\begin{table}[t]
\small
\caption{Properties\protect\index{mass!properties, table} of % Galilean
mass in everyday life.}
\label{massprop}
\centering
\dirrtabularstarnolines
\begin{tabular*}{\textwidth}{@{} p{32mm}
@{\extracolsep{\fill}} p{32mm}
@{\extracolsep{\fill}} p{42mm}
@{\extracolsep{\fill}} p{22mm} @{}}
%
\toprule
%
\tabheadf{Masses} % \break does not help
& \tabhead{Physical property} & \tabhead{Mathematical
name} & \tabhead{Definition}\\[0.5mm]
% & \tabhead{property} & \tabhead{name} & \\[0.5mm]
%
\midrule
%
Can be distinguished &\iin{distinguishability} & \iin{element of set} &
\seepagethree{setdefi} \\
Can be ordered &\iin{sequence} & \iin{order} & \seepagefour{orderdefi} \\
Can be compared & \iin{measurability}& \iin{metricity}&
\seepagefour{mespde4} \\
Can change gradually &\iin{continuity} & \iin{completeness} &
\seepagefive{topocont} \\
Can be added & \iin[matter!quantity]{quantity of matter} &\iin{additivity}
& \seepageone{eulc} \\
Beat any limit & \iin{infinity}& \iin{unboundedness}, \iin{openness}&
\seepagethree{settab} \\
Do not change & \iin{conservation} & \iin{invariance} & $m = \hbox{const}$\\
Do not disappear & \iin{impenetrability} & \iin{positivity}& $m \geq 0$ \\
%
\bottomrule
\end{tabular*}
\end{table}
All the properties of everyday mass are summarized in \tableref{massprop}.
% We also speak of \ii[mass!Galilean, definition]{Galilean mass}.
Later we will find that several of the properties are only approximate.
Highprecision experiments show deviations.%
%
\footnote{For example, in order to define mass we must be able to
\emph{distinguish} bodies. This seems a trivial requirement, but we
discover that this is not always possible in nature.} %
%
% For the moment we continue with the present, Galilean concept of
% mass,\indexs{mass!Galilean} as we have not yet a better one at our disposal.
%
% Dec 2016
However, the definition of mass remains unchanged throughout our adventure.
The definition of mass through momentum conservation implies that when an
object falls, the Earth is accelerated upwards by a tiny amount. If we could
measure this tiny amount, we could determine the mass of the Earth.
Unfortunately, this measurement is impossible. Can you find a better way to
determine the mass of the Earth?\challengenor{eaw}
% Jul 2016
\cssmallepsf{iklotzraetsel}{scale=1}{Depending on the way you pull, either
the upper of the lower thread snaps. What are the options?}
% Jul 2016
The definition of mass and momentum allows to answer the question of
\figureref{iklotzraetsel}. A brick hangs from the ceiling; a second thread
hangs down from the brick, and you can pull it. How can you tune your pulling
method to make the upper thread break?\challengn The lower one?
\label{negmass} %
%
Summarizing \tableref{massprop}, the mass\index{mass} of a body is thus most
precisely described by a \emph{positive real number,} often abbreviated $m$ or
$M$. This is a direct consequence of the impenetrability of matter. Indeed,
a \emph{negative} (inertial) mass would mean that such a body would move in
the opposite direction of any applied force or
acceleration.\index{mass!negative} % force already defined? YES
Such a body could not be kept in a box; it would break through any wall trying
to stop it. Strangely enough, negative mass bodies would still fall downwards
in the field of a large positive mass (though more slowly than an equivalent
positive mass). Are you able to confirm this?\challengn However, a small
positive mass object would float away from a large negativemass body, as you
can easily deduce by comparing the various accelerations involved. A positive
and a negative mass of the same value would stay at constant distance and
spontaneously accelerate away along the line connecting the two
masses.\challengn Note that both energy and momentum are conserved in all
these situations.%
%
\footnote{For more curiosities, see \asi R.H. Price/ \ti Negative mass can be
positively amusing/ \jo American Journal of Physics/ \vo 61/ \pp 216217/
\yrend 1993/ Negative mass particles in a box would heat up a box made of
positive mass while traversing its walls, and accelerating, i.e.,{} losing
energy, at the same time. They would\seepageone{perpdef} allow one to build
a \ii{perpetuum mobile} of the second kind, i.e.,{} a device circumventing
the second principle of thermodynamics.\challengn Moreover, such a system
would have no thermodynamic equilibrium, because its energy could decrease
forever. The more one thinks about negative mass,\index{mass!negative} the
more one finds strange properties contradicting observations. By the way,
what is the range of possible mass values for\challengenor{tamas}
tachyons?} % \cite{a37}}
%
% ref a37 is commented out
%
Negativemass bodies have never\seepagetwo{fitachy} been observed.
Antimatter,\index{antimatter} which will be discussed later, also has positive
mass.\seepagefour{antimatter}
%
% {Table of mass sensors}
{\small
\begin{table}[t]
\small
\caption{Some mass\protect\index{mass!sensors, table} sensors.}
\label{masssensors} % (OK) table must be improved
\centering
\dirrtabularstar
\begin{tabular*}{\textwidth}{%
@{}>{\PBS\raggedright\hspace{0.0em}\columncolor{hks152}[0pt][1cm]} p{50mm}
@{\extracolsep{\fill}}
>{\PBS\raggedright\hspace{0.0em}\columncolor{hks152}[0pt][1cm]} p{47mm}
@{\extracolsep{\fill}} p{28mm} @{}}
%
\toprule
%
\tabheadf{Measurement} & \tabhead{Sensor} & \tabhead{Range} \\[0.5mm]
%
\midrule
%
Precision scales & balance, pendulum, or spring & \csd{1}{pg} to \csd{10^3}{kg}
\\
%
Particle collision & speed & below \csd{1}{mg} \\
%
Sense of touch & pressure sensitive cells & \csd{1}{mg} to \csd{500}{kg} \\
%
Doppler effect on light reflected off the object & interferometer &
\csd{1}{mg} to \csd{100}{g} \\
%
Cosmonaut\index{cosmonaut} body mass measurement device & spring frequency &
around \csd{70}{kg} \\
%
Truck scales & hydraulic balance & $10^3$ to \csd{60\cdot 10^3}{kg} \\
%
Ship weight & water volume measurement & up to \csd{500\cdot 10^6}{kg} \\
%
%
\bottomrule
%
\end{tabular*}
\end{table}
}
\csepsfnb{imassmeasurement}{scale=0.8}{Mass measurement devices: a vacuum
balance used in 1890 by \protect\iinn{Dmitriy~Ivanovich Mendeleyev}, a modern
laboratory balance, a device to measure the mass of a
cosmonaut\protect\index{cosmonaut} in space and a truck scales
({\textcopyright}~\protect\iname{Thinktank Trust},
% EMAILED FEB 2008  Jack.Kirby@thinktank.ac  OK for one print run only!
\protect\iname{MettlerToledo},
\protect\iname{NASA},
\protect\iname{Anonymous}).}
% EMAILED FEB 2008  mary.finnegan@mt.com
%
% find a mass measurement device in living systems
%
% Impr. Jul 2016
\subsection{Is motion eternal?  Conservation of momentum}
% Index OK
\begin{quote}
Every body continues in the state of rest or of uniform motion
in a straight line except in so far as it doesn't.\\
\iinns{Arthur Eddington}\footnote{Arthur Eddington \lived(18821944),
British astrophysicist.}
\end{quote}
\np The product ${\bm p}= m{\bm v}$ of mass and velocity is called the
\ii{momentum} of a particle; it describes the tendency of an object to keep
moving during collisions. The larger it is, the harder it is to stop the
object. Like velocity, momentum has a direction and a magnitude: it is a
vector. In French, momentum is called `quantity of motion', a more
appropriate term. In the old days, the term `motion' was used instead of
`momentum', for example by {Newton}. The conservation of momentum, relation
(\ref{mc}), therefore expresses the conservation of\index{disappearance!of
motion}\index{conservation!of momentum} motion during
interactions.\index{creation!of
motion}\index{motion!creation}\index{motion!disappearance}\index{momentum!conservation}
% Feb 2010, added solutions of drawn puzzles
\csepsf{iselfprop}{scale=1}{What happens in these
four\protect\challengenor{foursit} situations?}
Momentum is an \emph{extensive quantity}.\index{quantity!extensive} That means
that it can be said that it \emph{flows} from one body to the
other,\index{momentum!flows}\index{energy!flows} and that it can be
\emph{accumulated} in bodies, in the same way that water flows and can be
accumulated in containers. Imagining momentum as something that can be
\emph{exchanged} between bodies\cite{disessa} in collisions is always useful
when thinking about the description of moving
objects.\index{bottle}\index{wine!bottle}\index{cork}\index{sailing}\index{boat}
Momentum is conserved. That explains the limitations you might experience
when being on a perfectly frictionless surface, such as ice or a polished,
\iin{oil} covered \iin[marble!oil covered]{marble}: you cannot propel yourself
forward by patting your own back. (Have you ever tried to put a \iin{cat} on
such a marble surface? It is not even able to stand on its four legs.
Neither are humans. Can you imagine why?)\challengenor{legs} Momentum
conservation also answers the puzzles of \figureref{iselfprop}.
The conservation of momentum and mass also means that
\iin[teleportation!impossibility of]{teleportation} (`beam me up') is
impossible in nature. Can you explain this to a
nonphysicist?\challengenor{telepuno}
% May 2005
Momentum conservation implies that momentum can be imagined to be like an
invisible \emph{fluid}.\index{momentum!as fluid} In an interaction, the
invisible fluid is transferred from one object to another. In such transfers,
the amount of fluid is always constant.
Momentum conservation implies that motion never stops; it is only
\emph{exchanged}. On the other hand, motion often `disappears' in our
environment, as in the case of a \iin[stones]{stone} dropped to the ground, or
of a ball left rolling on grass. Moreover, in daily life we often observe the
creation of motion, such as every time we open a hand. How do these examples
fit with the conservation of momentum?
It turns out that apparent momentum disappearance is due to the microscopic
aspects of the involved systems. A muscle only \ii[transformation!of motion
in engines]{transforms} one type of motion, namely that of the electrons in
certain chemical compounds%
%
\footnote{The fuel of most processes in animals usually
is\cite{ATPbook} adenosine triphosphate (\csaciin{ATP}).} %
%
into another, the motion of the fingers. The working of muscles is similar to
that of a car engine transforming the motion of electrons in the fuel into
motion of the wheels. Both systems need fuel and get warm in the process.
We must also study the microscopic behaviour when a ball rolls on grass until
it stops. The apparent disappearance of motion is called \ii{friction}.
Studying the situation carefully, we find that the grass and the ball heat up
a little during this process. \emph{During friction, visible motion is
transformed into heat.} A striking observation of this effect for a bicycle
is shown below, in \figureref{ibiketire}.\seepageone{ibiketire} Later, when
we discover the structure of matter, it will become clear that heat is the
disorganized motion of the microscopic constituents of every material. When
the microscopic constituents all move in the same direction, the object as a
whole moves; when they oscillate randomly, the object is at rest, but is warm.
Heat is a form of motion. Friction thus only seems to be disappearance of
motion; in fact it is a transformation of ordered into unordered motion.
Despite momentum\label{perpdef} conservation, \emph{macroscopic} perpetual
motion does not\seepageone{entruzsadi}
exist, since friction cannot be completely eliminated.%
%
\footnote{Some funny examples of past attempts to built a \ii[perpetuum
mobile]{perpetual motion machine} are described in \asi Stanislav Michel/
\bt Perpetuum mobile/ VDI Verlag, \yrend 1976/ Interestingly, the idea of
eternal motion came to Europe from India, via the Islamic world, around the
year 1200, and became popular as it opposed the then standard view that all
motion on Earth disappears over time. See also the
\url{web.archive.org/web/20040812085618/http://www.geocities.com/mercutio78_99/pmm.html}
and the \url{www.lhup.edu/~dsimanek/museum/unwork.htm} websites. The
conceptual mistake made by eccentrics and used by crooks is always the same:
the hope of overcoming friction. (In fact, this applied only to the
perpetual motion machines of the {second} kind; those of the first kind 
which are even more in contrast with observation\index{perpetuum mobile!
first and second kind}  even try to generate energy from nothing.)
If the machine is well constructed, i.e.,{} with little friction, it can
take the little energy it needs for the sustenance of its motion from very
subtle environmental effects.\index{clock!air pressure powered} For example,
in the Victoria and Albert Museum in London one can admire a beautiful
\iin{clock} powered by the variations of \iin[air!pressure]{air pressure}
over time.\cite{vict}
Low friction means that motion takes a long time to stop. One immediately
thinks of the motion of the
planets.\index{Sun}\index{planet}\index{Sunplanet
friction}\index{planetSun friction}\index{friction!between planets and
the Sun} In fact, there \emph{is} friction between the Earth and the Sun.
(Can you guess one of the mechanisms?)\challengenor{easunfri} But the value
is so small that the Earth has already circled around the Sun for thousands
of millions of years, and will do so for quite some time more.} %
%
Motion is eternal only at the microscopic scale. In other words, the
disappearance and also the spontaneous appearance of motion in everyday life
is an illusion due to the limitations of our senses. For example, the motion
proper of every living being exists before its birth, and stays after its
\iin[death!conservation and]{death}. The same happens with its energy. This
result is probably the closest one can get to the idea of {everlasting life}
from\index{life!everlasting} evidence collected by observation. It is perhaps
less than a coincidence that energy used to be called \ii{vis viva}, or
`living force', by \iname[Leibniz, Gottfried Wilhelm]{Leibniz} and many
others.
Since motion is conserved, it has no origin. Therefore, at this stage of our
walk we cannot answer the fundamental questions: Why does motion exist? What
is its origin? The end of our adventure is nowhere near.
%
% Impr. Jul 2016
\subsection{More conservation  energy}
% Index OK
When\label{enconszz} collisions are studied in detail, a second conserved
quantity turns up.
%
%
% But the example of the ball rolling on grass also shows that motion cannot
% be
% described adequately by momentum alone. Momentum is never lost, only
% exchanged. One also needs a quantity which distinguishes situations with
% friction from situations without friction. In daily life, one feels that
% in a
% collision without friction, an \ii{elastic} collision, in which the bodies
% bounce well, little is lost to friction, whereas in a collision where the
% bodies stick to each other, the losses are greater. What is the quantity
% we
% are looking for? Obviously, the quantity must depend on the mass of a
% body,
% and on its velocity.
%
Experiments show that in the case of perfect, or elastic \iin[collision!and
momentum]{collisions}  collisions without \iin{friction}  the following
quantity, called the \ii[energy!kinetic]{kinetic energy} $T$ of the
system,\indexs{energy} is also conserved:
\begin{equation}
T = \sum_{i}
{\te \frac{1}{2}}
m_{i} {\bm
v}_{i}^{2} = \sum_{i}
{\te \frac{1}{2}}
m_{i} {v}_{i}^{2} ={\rm const} \cp
%\label{eq:energydef}
\end{equation}
Kinetic energy is the ability that a body has to induce change in bodies it
hits. Kinetic energy thus depends on the mass and on the square of the speed
$v$ of a body. The full name `kinetic energy' was introduced by
\iinns{GustaveGaspard Coriolis}.%
%
\footnote{GustaveGaspard Coriolis \livedplace(1792 Paris1843 Paris) % French
was engineer and mathematician. He introduced the modern concepts of
`\iin{work}' and of `\iin{kinetic energy}', and explored the Coriolis effect
discovered by Laplace.\seepageone{lcoriolis} Coriolis also introduced the
factor 1/2 in the kinetic energy $T$, in order that the relation $\diffd T/
\diffd v=p$
would be obeyed.\challengenor{kinco} (Why?)} %
%
Some measured energy values are given in \tableref{enemetab}.
% He also introduced the term `work' says the Britannica, already said below.
% %
% %
% In nonelastic collisions, part or all of the kinetic
% energy is lost. In these and in other cases one finds thus the general
% rule:
% \ii{friction} leads to the loss of kinetic energy.
The experiments and ideas mentioned so far can be summarized in the following
definition:
\begin{quotation}
\noindent \csrhd {(Physical) \ii{energy} is the measure of the ability to
generate motion.}
\end{quotation}
\np A body has a lot of energy if it has the\index{energy!definition} ability
to move many other bodies. Energy is a number; energy, in contrast to
momentum, has no direction. The total momentum of two equal masses moving
with opposite velocities is zero; but their total energy is not, and it
increases with velocity. Energy thus also measures motion, but in a different
way than momentum. Energy measures motion in a more global way.
% Dec 2005
An equivalent definition is the following:
\begin{quotation}
\noindent \csrhd {Energy is the ability to perform work.}
\end{quotation}
\np %Energy is the ability to perform work.
Here, the physical\index{energy!definition} concept of work is just the
precise version of what is meant by work in everyday life. As usual,
(physical) \ii[work!definition in physics]{work} is the product of force and
distance in direction of the force. In other words, work is the \emph{scalar
product} of force and distance. Physical work is a quantity that describes
the effort of pushing of an object along a distance. As a result, in physics,
work is a form of energy.
Another, equivalent definition of energy will become clear shortly:
\begin{quotation}
\noindent \csrhd {Energy is what can be transformed into heat.}
\end{quotation}
\np Energy is a word taken from ancient Greek; originally it was used to
describe character, and meant `intellectual or moral vigour'. It was taken
into physics by \iinn{Thomas Young} \lived(17731829) in 1807
% Cited by A. Pais.
%
% OR by \iinn{William Thomson} and \iinn{William Rankine} around 1860
%
%
because its literal meaning is `force within'. (The letters $E$, $W$, $A$ and
several others are also used to denote energy.)
Both energy and momentum measure how systems change.\label{endefqq}
\iin[momentum!as change per distance]{Momentum} tells how systems change
\emph{over distance}: momentum is action (or change) divided by distance.
Momentum is needed to compare motion here and there.
\iin[energy!as change per time]{Energy} measures how systems change \emph{over
time}: energy is action (or change) divided by time. Energy is needed to
compare motion now and later.
% Improved Nov 2012
Do not be surprised if you do not grasp the difference between momentum and
energy straight away: physicists took about a century to figure it out! So
you are allowed to take some time to get used to it. Indeed, for many
decades, English physicists insisted on using the same term for both concepts;
this was due to Newton's insistence that  no joke  the existence of
god\index{gods!and energy} implied that energy was the same as momentum.
Leibniz, instead, knew that energy increases with the square of the speed and
proved Newton wrong.\index{Newton!his energy mistake} In 1722, Willem Jacob
's~Gravesande\indname{Gravesande@'s Gravesande, Willem Jacob} even showed the
difference between energy and momentum experimentally.\cite{gravesande} He let
metal balls of different masses fall into mud from different heights. By
comparing the size of the imprints he confirmed that Newton was wrong both
with his physical statements and his theological ones.
%
% {Table of energies}
% !.!1 more enegy values about animals and plants, and sports
%
%
{\small
\begin{table}[t]
\sbox{\cshelpbox}{\small\dirrtabular%
\begin{tabular}{@{\hspace{0em}} p{75mm} p{20mm} @{\hspace{0em}}}
%
\toprule
%
\tabheadf{Observation} & \tabhead{Energy} \\[0.5mm]
%
\midrule
%
Average kinetic energy of oxygen molecule in air & $\csd{6}{zJ}$
\\
%
Green photon energy & $\csd{0.37}{aJ}$ \\
%
Xray photon energy & \csd{1}{fJ} \\
%
$\gamma$ photon energy & \csd{1}{pJ} \\
%
Highest particle energy in accelerators & \csd{0.1}{\muunit J} \\
%
Kinetic energy of a flying mosquito & \csd{0.2}{\muunit J} \\
%
Comfortably walking human & $\csd{20}{J}$ \\
%
Flying arrow & $\csd{50}{J}$ \\
%
Right hook in boxing & \csd{50}{J} \\
%
Energy in torch battery & \csd{1}{kJ} \\
%
Energy in explosion of \csd{1}{g} TNT & \csd{4.1}{kJ} \\
%
Energy of \csd{1}{kcal} & \csd{4.18}{kJ} \\
%
Flying rifle bullet & $\csd{10}{kJ}$ \\
%
One gram of fat & $\csd{38}{kJ}$ \\
%
One gram of gasoline & $\csd{44}{kJ}$ \\
%
Apple digestion & $\csd{0.2}{MJ}$ \\
%
Car on highway & 0.3 to $\csd{1}{MJ}$ \\
%
Highest laser pulse energy & $\csd{1.8}{MJ}$ \\
%
Lightning flash & up to $\csd{1}{GJ}$ \\
%
Planck energy & $\csd{2.0}{GJ}$ \\
%
Small nuclear\index{kilotonne} bomb (\csd{20}{ktonne}) & $\csd{84}{TJ}$ \\
%
Earthquake of magnitude 7 & $\csd{2}{PJ}$ \\
%
Largest\index{megatonne} nuclear bomb (\csd{50}{Mtonne}) & $\csd{210}{PJ}$ \\
%
Impact of meteorite with \csd{2}{km} diameter & $\csd{1}{EJ}$ \\
%
Yearly machine energy use & $\csd{420}{EJ}$ \\
%
Rotation energy of Earth & $\csd{2\cdot 10^{29}}{J}$ \\
%
Supernova explosion & $\csd{10^{44}}{J}$ \\
%
Gammaray burst & up to $\csd{10^{47}}{J}$ \\
%
Energy content $E=c^2m$ of Sun's mass & $\csd{1.8\cdot 10^{47}}{J}$ \\
%
Energy content of Galaxy's central black hole & $\csd{4\cdot 10^{53}}{J}$ \\
%
\bottomrule
\end{tabular}%
}% end of sbox
%
\centering
\begin{minipage}{\wd\cshelpbox}%
\captionsetup{width=\wd\cshelpbox} % needed since Dec 2015
\caption{Some measured\protect\index{energy!values, table} energy values.}%
\label{enemetab}\noindent\usebox{\cshelpbox}
\end{minipage}
\end{table}
}
\cssmallepsfnb{irobertmayer}{scale=0.14}{Robert Mayer \livedfig(18141878).}
% (NO) add table of energy measurement devices and methods,
% e.g., photos of energy meters
% !!!2 add energy scaling picture in biology
One way to explore the difference between energy and momentum is to think
about the following challenges.
% note that if a body is accelerated by
% a constant force, momentum is what increases with time, and energy is what
% increases with distance.
Is it more difficult to stop a running man with mass $m$ and speed $v$, or one
with mass $m/2$ and speed $2v$, or one with mass $m/2$ and speed
$\sqrt{2}v$?\challengn You may want to ask a \iin{rugby}playing friend for
confirmation.
% !!!3 find ref  did not succeed in 2016
Another distinction between energy and momentum is illustrated by athletics:
the \emph{real} \iin[long jump!record]{long jump} world record, almost
\csd{10}{m}, is still kept by an athlete who in the early twentieth century
ran with two weights in his hands, and then threw the weights behind him at
the moment he took off. Can you explain the feat?\challengenor{longjump}
When a car travelling at \csd{100}{m/s} runs headon into a parked car of the
same kind and make, which car receives the greatest\challengenor{cardamage}
damage? What changes if the parked car has its brakes on?
To get a better feeling for energy, here is an additional aspect. The world
consumption of energy by human machines (coming from solar, geothermal,
biomass, wind, nuclear, hydro, gas, oil, coal, or animal sources) in the year
2000 was about \csd{420}{EJ},%
%
\footnote{For the explanation of the abbreviation E,
see \appendixref{units1}.\seepageone{units1}} %
%
for a world population of about 6000 million people.\cite{enchall}
%
% a Shell study says 500 EJ in 2000
%
To see what this energy consumption means, we translate it into a personal
power consumption; we get about \csd{2.2}{kW}.
% I checked the number in May 2002
%%\csd{2.6}{kW}. % I checked the number in jan 2001 for 500 EJ
The watt W is the unit of power, and is simply defined as
$\csd{1}{W}=\csd{1}{J/s}$, reflecting the definition of
\ii[power!physical]{(physical) power} as energy used per unit time. The
precise wording is: power is energy flowing per time through a defined closed
surface. See \tableref{powmetab} for some power values found in nature, and
\tableref{powersensors} for some measurement devices.
As a working person can produce mechanical work of about \csd{100}{W}, the
average human energy consumption corresponds to about 22 humans working 24
hours a day. In particular, if we look at the \iin[energy!consumption in
First World]{energy consumption in First World countries}, the average
inhabitant there has machines working for him or her that are equivalent to
several hundred `servants'.\index{servant!machines}\index{machine!servants}
Machines do a lot of good. Can you point out some of these\challengenor{mach}
machines?
Kinetic energy is thus not conserved in everyday life. For example, in
nonelastic collisions, such as that of a piece of chewing gum hitting a wall,
kinetic energy is lost. \emph{Friction} destroys kinetic energy.
% as it destroys momentum. % WRONG! external forces do
At the same time, friction produces heat. It was one of the important
conceptual discoveries of physics that \emph{total} energy {is} conserved if
one includes the discovery that heat is a form of energy. Friction is thus a
process transforming kinetic energy, i.e.,{} the energy connected with the
motion of a body, into heat. On a microscopic scale, \emph{energy is always
conserved}.
Any example of nonconservation of energy is only apparent.
%
\footnote{In fact, the conservation of energy was stated in its full
generality in public only in 1842, by \iinn{Julius~Robert Mayer}. He was a
medical doctor by training, and the journal \emph{Annalen der Physik} refused
to publish his paper, as it supposedly contained `fundamental errors'. What
the editors called errors were in fact mostly  but not only 
contradictions of their prejudices. Later on, \iname[Helmholtz,
Hermann~von]{Helmholtz}, \iname[ThomsonKelvin]{ThomsonKelvin}, \iname[Joule,
James P.]{Joule} and many others acknowledged Mayer's genius. However, the
first to have stated energy conservation in its modern form was the French
physicist \iinns{Sadi Carnot} \lived(17961832) in 1820. To him the issue was
so clear that he did not publish the result. In fact he went on and
discovered the \iin[thermodynamics!second law]{second `law' of
thermodynamics}. Today, energy conservation, also called the
\iin[thermodynamics!first law]{first `law' of thermodynamics}, is
one of the pillars of physics, as it is valid in all its domains.} %
%
Indeed, without \iin[energy!conservation and time]{energy conservation}, the
concept of time would not be definable! We will show this important connection
shortly.
In summary, in addition to mass and momentum, everyday linear motion also
conserves energy. To discover the last conserved quantity, we explore another
type of motion: rotation.
%
% {Table of powers}
%
{\small
\begin{table}[p]
\small
\caption{Some measured\protect\index{power!values, table} power values.}
\label{powmetab}
\centering
\dirrtabularstar
\begin{tabular*}{\textwidth}{@{}>{\columncolor{hks152}[0pt][1cm]} p{97mm}
@{\extracolsep{\fill}} p{32mm} @{}}
%
\toprule
\tabheadf{Observation} & \tabhead{Power} \\[0.5mm]
%
\midrule
%
Radio signal from the Galileo space probe sending from Jupiter
& \csd{10}{zW}\\
%
Power of \iin{flagellar motor} in \iin{bacterium} & \csd{0.1}{pW}\\
%
Power consumption of a typical cell & \csd{1}{pW}\\
%
sound power at the ear at hearing threshold & \csd{2.5}{pW}\\
% from wikipedia
%
CRR laser, at \csd{780}{nm} & 40\csd{80}{mW}\\
% from wikipedia
%
Sound output from a piano playing fortissimo & \csd{0.4}{W}\\
%
Dove (\csd{0.16}{kg}) \iin{basal metabolic rate} & \csd{0.97}{W}\\
%
Rat (\csd{0.26}{kg}) basal metabolic rate & \csd{1.45}{W}\\
%
Pigeon (\csd{0.30}{kg}) basal metabolic rate & \csd{1.55}{W}\\
%
Hen (\csd{2.0}{kg}) basal metabolic rate & \csd{4.8}{W}\\
%
Incandescent light bulb light output & 1 to \csd{5}{W}\\
%
Dog (\csd{16}{kg}) basal metabolic rate & \csd{20}{W}\\
%
Sheep (\csd{45}{kg}) basal metabolic rate & \csd{50}{W}\\
%
Woman (\csd{60}{kg}) basal metabolic rate & \csd{68}{W}\\
%
Man (\csd{70}{kg}) basal metabolic rate & \csd{87}{W}\\
%
Incandescent light bulb electricity consumption& 25 to \csd{100}{W}\\
%
A human, during one work shift of eight hours & \csd{100}{W}\\
%
Cow (\csd{400}{kg}) basal metabolic rate & \csd{266}{W}\\
%
One \iin{horse}, for one shift of eight hours & \csd{300}{W}\\
%
Steer (\csd{680}{kg}) basal metabolic rate & \csd{411}{W}\\
%
Eddy Merckx,\indname{Merckx, Eddy} the great bicycle athlete, during one hour
& \csd{500}{W}\\
% see also http://jap.physiology.org/cgi/content/full/89/4/1522
% with the same result
%
Metric \iin[horse!power]{horse power} power unit
($\csd{75}{kg}\cdot\csd{9.81}{m/s^2}\cdot\csd{1}{m/s}$)& \csd{735.5}{W}\\
%
% Jan 2008
British \iin{horse power} power unit & \csd{745.7}{W}\\
%
Large motorbike & $\csd{100}{kW}$\\
%
Electrical power station output &$\hbox{0.1 to $\csd{6}{GW}$}$\\
%
World's electrical power production in 2000 \cite{enchall} &$\csd{450}{GW}$\\
%
%
Power used by the geodynamo & 200 to $\csd{500}{GW}$\\
%
% Nov 2012
Limit on wind energy production \cite{winden} & 18 to $\csd{68}{TW}$\\
%
Input on Earth surface: Sun's irradiation of Earth \cite{hehet}
&$\csd{0.17}{EW}$\\
%
Input on Earth surface: thermal energy from inside of the Earth
&$\csd{32}{TW}$\\
%
Input on Earth surface: power from tides (i.e.,{} from Earth's rotation) &
$\csd{3}{TW}$\\
%
Input on Earth surface: power generated by man from fossil fuels &
8~to~$\csd{11}{TW}$\\ % 11 from cite enchall, 8 from the AMJPhy
%
Lost from Earth surface: power stored by plants' photosynthesis &
$\csd{40}{TW}$\\
%
World's record laser power & $\csd{1}{PW}$\\
%
Output of Earth surface: sunlight reflected into space & $\csd{0.06}{EW}$\\
%
Output of Earth surface: power radiated into space at \csd{287}{K} &
$\csd{0.11}{EW}$\\
%
Peak power of the largest nuclear bomb & $\csd{5}{YW}$\\
%
Sun's output &$\csd{384.6}{YW}$\\
%
Maximum power in nature, $c^5/4G$ &$\csd{9.1\cdot 10^{51}}{W}$\\
%
\bottomrule
\end{tabular*}
\end{table}
}
%
% {Table of power sensors}
%
%
{\small
\begin{table}[t]
\small
\caption{Some power\protect\index{power!sensors, table} sensors.}
\label{powersensors} %
\centering
\dirrtabularstar
\begin{tabular*}{\textwidth}{%
@{}>{\PBS\raggedright\hspace{0.0em}\columncolor{hks152}[0pt][1cm]} p{50mm}
@{\extracolsep{\fill}}
>{\PBS\raggedright\hspace{0.0em}\columncolor{hks152}[0pt][1cm]} p{50mm}
@{\extracolsep{\fill}} p{25mm} @{}}
%
\toprule
%
\tabheadf{Measurement} & \tabhead{Sensor} & \tabhead{Range} \\[0.5mm]
%
\midrule
%
Heart beat as power meter & deformation sensor and clock & 75 to
\csd{2\,000}{W}\\
%
Fitness power meter & piezoelectric sensor & 75 to \csd{2\,000}{W}\\
%
Electricity meter at home & rotating aluminium disc & 20 to \csd{10\,000}{W}\\
%
Power meter for car engine & electromagnetic brake & up to \csd{1}{MW}\\
%
Laser power meter & photoelectric effect in semiconductor & up to
\csd{10}{GW} \\
%
Calorimeter for chemical reactions & temperature sensor & up to \csd{1}{MW}\\
%
Calorimeter for particles & light detector & up to a few \csd{}{\muunit J/ns}\\
% ns added in June 2011 after reader complaint, my estimate
%
%
\bottomrule
%
\end{tabular*}
\end{table}
}
% Dec 2012, Nov 2013
\csepsfnb{ipowermeasurement}{scale=1}{%
Some power measurement devices: a bicycle power meter, a laser power meter,
and an electrical power meter ({\textcopyright}~\protect\iname{SRAM},
\protect\iname{Laser Components}, \protect\iname{Wikimedia}).}
%
% Impr. July 2016
\subsection{The cross product, or vector product}
% Index OK
The discussion\label{crossproduct} of rotation is easiest if we introduce an
additional way to multiply vectors. This new product between two vectors
${\bm a}$ and ${\bm b}$ is called the \ii{cross product} or
\ii[vector!product]{vector product}
${\bm a}\times{\bm b}$.\index{product!vector}
The result of the vector product is another \emph{vector}; thus it differs
from the \emph{scalar} product, whose result is a scalar, i.e., a number. The
result of the vector product is that vector
\begin{Strich}
\item that is orthogonal to both vectors to be multiplied,
\item whose orientation is given by the \ii{righthand rule}, and
\item whose length is given by the surface area of the parallelogram spanned
by the two vectors, i.e., by $a b \sin \sphericalangle ({\bm a}, {\bm b})$.
\end{Strich}
The definition implies that the cross product vanishes if and only if the
vectors are parallel. From the definition you can also show that the vector
product has the\challengn properties
%
\begin{align}
%
&{\bm a} \times {\bm b} =  {\bm b} \times {\bm a} \;,\quad {\bm a} \times
({\bm b} + {\bm c}) = {\bm a} \times {\bm b} + {\bm a} \times {\bm c} \;,
%
\non
%
&\lambda {\bm a} \times {\bm b} = \lambda ({\bm a} \times {\bm b}) =
{\bm a} \times \lambda {\bm b} \;,\quad {\bm a} \times {\bm a} = {\bm 0}
\;,
%
\non
%
&{\bm a} ({\bm b}\times {\bm c}) = {\bm b} ({\bm c}\times {\bm a}) = {\bm c}
({\bm a}\times {\bm b}) \;,\quad
%
{\bm a} \times ({\bm b}\times {\bm c}) = ({\bm a} {\bm c}) {\bm b} 
({\bm a} {\bm b}) {\bm c} \;,
%
\non
%
&({\bm a} \times {\bm b})({\bm c}\times {\bm d}) = {\bm a} ({\bm b} \times
({\bm c} \times {\bm d} )) = ({\bm a} {\bm c}) ({\bm b} {\bm d})  ({\bm b}
{\bm c})({\bm a} {\bm d}) \;,
%
\non
%
%
& ({\bm a} \times {\bm b})\times ({\bm c}\times {\bm d}) = (({\bm a} \times
{\bm b} ) {\bm d} ) {\bm c}  (({\bm a} \times {\bm b} ) {\bm c} ) {\bm d} \;,
%
\non
%
&{\bm a} \times ({\bm b}\times {\bm c}) + {\bm b} \times ({\bm c}\times {\bm
a}) + {\bm c} \times ({\bm a}\times {\bm b}) = {\bm 0} \cp
%
\end{align}
%
The vector product exists only in vector spaces with \emph{three} dimensions.
We will explore more details on this connection later on.\seepagefour{vecipo}
% Feb 2010, Jul 2016
The vector product is useful to describe systems that \emph{rotate}  and
(thus) also systems with magnetic forces. The motion of an orbiting body is
always perpendicular both to the axis and to the line that connects the body
with the axis. In rotation, axis, radius and velocity form a righthanded set
of mutually orthogonal vectors. This connection lies at the origin of the
vector product.
% Feb 2010, July 2010
Confirm\challengn that the best way to calculate the vector product
${\bm a}\times {\bm b}$ component by component is given by the symbolic
determinant
\begin{equation}
{\bm a}\times {\bm b} =
\begin{vmatrix}
{\bm e}_{x} & a_{x} & b_{x}\; \\
{\bm e}_{y} & a_{y} & b_{y}\; \\
{\bm e}_{z} & a_{z} & b_{z}\; \\
\end{vmatrix}
\quad\hbox{or,\ sloppily}\quad
{\bm a}\times {\bm b} =
\begin{vmatrix}
+ &  & + \\
a_{x} & a_{y} & a_{z}\; \\
b_{x} & b_{y} & b_{z}\; \\
\end{vmatrix}
\cp
\end{equation}
These symbolic determinants are easy to remember and easy to perform, both
with letters and with numerical values. (Here, ${\bm e}_{x}$ is the unit
basis vector in the $x$ direction.) Written out, the symbolic determinants
are equivalent to the relation
\begin{equation}
{\bm a}\times {\bm b} = (a_{y}b_{z}b_{y}a_{z}, b_{x}a_{z}a_{x}b_{z},
a_{x}b_{y}b_{x}a_{y})
\end{equation}
which is harder to remember, though.
Show that the \ii{parallelepiped} spanned\challengn by three arbitrary vectors
${\bm a}$, ${\bm b}$ and ${\bm c}$ has the volume
$V= {\bm c}\,({\bm a}\times {\bm b})$. Show that the \ii{pyramid} or
\ii{tetrahedron} formed by the same three\challengn vectors has one sixth of
that volume.
%
% Impr. Jul 2016
\subsection{Rotation and angular momentum}
% Index OK
\np Rotation keeps us alive. Without the change of day and night, we would be
either fried or frozen to \iin[death!rotation and]{death}, depending on our
location on our planet. But rotation appears in many other settings, as
\tableref{rotmetab} shows. A short exploration of rotation is thus
appropriate.
All objects have the ability to rotate. We saw before that a body is
described by its reluctance to move, which we called mass; similarly, a body
also has a \iin[reluctance!to rotation]{reluctance to
turn}.\index{rotation!reluctance} This quantity is called its \ii[moment!of
inertia]{moment of inertia} and is often abbreviated $\Theta$  pronounced
`theta'. The speed or rate of
rotation\index{rotation!rate}\index{rotation!speed} is described by
\ii[velocity!angular]{angular velocity}, usually abbreviated $\omega$ 
pronounced `omega'. A few values found in nature are given in
\tableref{rotmetab}.
%
% {Table of rotation velocities}
%
%
{\small
\begin{table}[t]
\sbox{\cshelpbox}{\small\dirrtabular%
\begin{tabular}{@{\hspace{0em}} p{61mm} p{51mm} @{\hspace{0em}}}
%
\toprule
%
\tabheadf{Observation} & \tabhead{Angular velocity $\omega=2\pi/T$}\\[0.5mm]
%
\midrule
%
Galactic rotation & $2 \pi\; \cdot$ \csd{0.14 \cdot 10^{15}/}{s} \csd{=2
\pi\;/(220\cdot 10^6}{a})\\
%
Average Sun rotation around its axis& $2 \pi\;\cdot $\csd{3.8 \cdot
10^{7}/}{s}
\csd{=\;2 \pi\;/}{30\,d}\\
%
Typical lighthouse & $2 \pi\; \cdot$ \csd{0.08/}{s}\\
%
Pirouetting ballet dancer & $2 \pi\;\cdot$ \csd{3/}{s}\\
%
Ship's diesel engine & $2 \pi\;\cdot$ \csd{5/}{s}\\
%
Helicopter rotor & $2 \pi\;\cdot$ \csd{5.3/}{s}\\
%
Washing machine & up to $2 \pi\;\cdot$ \csd{20/}{s}\\
%
Bacterial flagella & $2 \pi\;\cdot$ \csd{100/}{s}\\
%
Fast CD recorder & up to $2 \pi\;\cdot$ \csd{458/}{s}\\
%
Racing car engine & up to $2 \pi\;\cdot$ \csd{600/}{s}\\
%
Fastest turbine built & $2 \pi\;\cdot$ \csd{10^{3}/}{s}\\
%
Fastest pulsars (rotating stars) & up to at least $2 \pi\;\cdot$
\csd{716/}{s}\\
%
% Checked Feb 2014
Ultracentrifuge & $>2 \pi\;\cdot$ \csd{3 \cdot 10^{3}/}{s}\\
%
Dental drill & up to $2 \pi\;\cdot$ \csd{13 \cdot 10^{3}/}{s}\\
%
Technical record & $2 \pi\;\cdot$ \csd{333 \cdot 10^{3}/}{s}\\
%
Proton rotation & $2 \pi\;\cdot$ \csd{10^{20}/}{s}\\
%
Highest possible, Planck angular velocity & $2 \pi\cdot$
\csd{10^{35}/}{s}\\
%
\bottomrule
\end{tabular}%
}% end of sbox
%
\centering
\begin{minipage}{\wd\cshelpbox}%
\captionsetup{width=\wd\cshelpbox} % needed since Dec 2015
\caption{Some measured\protect\index{rotation!frequency values, table}
rotation frequencies.}%
\label{rotmetab}\noindent\usebox{\cshelpbox} %
\end{minipage}
\end{table}
}
The observables that describe rotation are similar to those describing linear
motion, as shown in \tableref{rotlintab}. Like mass, the moment of inertia
is defined in such a way that the sum of \ii[angular momentum]{angular
momenta} $L$  the product of moment of inertia and angular velocity  is
conserved in systems that do not interact with the outside world:
\begin{equation}
\sum_{i} \Theta_{i}\bm\omega_{i} = \sum_{i} \bm L_{i}=
{\rm const} \cp
% \label{eq:angcons}
\end{equation}
In the same way that the conservation of linear momentum defines mass, the
conservation of angular momentum defines the moment of inertia.
% Dec 2016
Angular momentum is a concept introduced in the 1730s and 1740s by
\iinn{Leonhard Euler} and \iinn{Daniel Bernoulli}.
The moment of inertia can be related to the mass and shape of a body. If the
body is imagined to consist of small parts or mass elements, the resulting
expression is
%
\begin{equation}
\Theta = \sum_{{n}} m_{n} r_{ n}^{2} \cvend
\end{equation}
where $r_{ n}$ is the distance from the mass element $m_{ n}$ to the axis of
rotation. Can you confirm the expression?\challengn Therefore,
the\label{angmomtendef} moment of inertia of a body depends on the chosen
axis of rotation. Can you confirm that this is\challengenor{brick} so for a
brick?
%
% \subsubsubsubsubsubsubsubsection{Table of angular momenta}
% New in Oct 2011
{\small
\begin{table}[t] % t added in Dec 2016
\small
\centering
\caption{Some measured\protect\index{angular momentum!values, table} angular
momentum values.}
\label{angmomvaltab}
\dirrtabularstar
\begin{tabular*}{\textwidth}{@{}>{\columncolor{hks152}[0pt][2cm]} p{80mm}
@{\extracolsep{\fill}} p{45mm} @{}}
%
\toprule
%
\tabheadf{Observation} & \tabhead{Angular momentum} \\[0.5mm]
%
\midrule
%
{Smallest observed} value in nature, $\hbar/2$, in
% applies to the zcomponent of
elementary matter particles (fermions) & \csd{0.53\cstimes10^{34}}{Js} \\
%
\iin[spinning top!angular momentum]{Spinning top} & \csd{5\cstimes10^{6}}{Js} \\
% my estimate
%
\iin[CD!angular momentum]{CD} (compact disc) playing& \circa\csd{0.029}{Js} \\
%
\iin[walking!angular momentum]{Walking man} (around body axis) &
\circa\csd{4}{Js} \\
%
\iin[dancer!angular momentum]{Dancer} in a pirouette &
\csd{5}{Js} \\ % my estimate
%
\iin[car!wheel angular momentum]{Typical car wheel} at \csd{30}{m/s}&
\csd{10}{Js} \\ % my estimate
%
\iin[wind generator!angular momentum]{Typical wind generator} at \csd{12}{m/s}
(6 Beaufort) & \csd{10^4}{Js} \\ % my estimate
%
\iin[atmosphere!angular momentum]{Earth's atmosphere} & 1 to \csd{2\cstimes
10^{26}}{Js}\\ % from an el Nino paper, impr. Aug 2013
%
\iin[ocean!angular momentum]{Earth's oceans} & \csd{5\cstimes
10^{24}}{Js}\\ % from a paper on the issue, Aug 2013
%
\iin[Earth!angular momentum]{Earth around its axis} & \csd{7.1 \cstimes
10^{33}}{Js}\\
%
\iin[Moon!angular momentum]{Moon around Earth} & \csd{2.9 \cstimes
10^{34}}{Js}\\
%
\iin[Earth!angular momentum]{Earth around Sun} & \csd{2.7 \cstimes
10^{40}}{Js}\\ % two sources
%
\iin[Sun!angular momentum]{Sun} around its axis
& \csd{1.1 \cstimes 10^{42}}{Js} \\
%
\iin[Jupiter!angular momentum]{Jupiter around Sun} & \csd{1.9 \cstimes
10^{43}}{Js} \\
%
\iin[Solar System!angular momentum]{Solar System around Sun} & \csd{3.2
\cstimes 10^{43}}{Js} \\
%
\iin[Milky Way!angular momentum]{Milky Way} & \csd{10^{68}}{Js} \\
%
\iin[universe!angular momentum]{All masses in the universe} & 0 (within
measurement error) \\
%
\bottomrule
%
\end{tabular*}
\end{table}
}
%
% \subsubsubsubsubsubsubsubsection{Linear  Rotation correspondence}
%
% fill in more, if possible
%
{\small
\begin{table}[t]
\small \caption{Correspondence\protect\index{motion!linearrotational
correspondence table} between linear and rotational motion.}
\label{rotlintab} %
\centering
\dirrtabularstarnolines
\begin{tabular*}{\textwidth}{@{} p{29mm}
@{\hspace{10mm}}@{\extracolsep{\fill}} p{19mm} p{15mm}
p{30mm} p{15mm} @{}}
%
\toprule
%
\tabheadf{Quantity} & \multicolumn{2}{@{}l}{\tabhead{Linear motion}} &
\tabhead{Rotational motion} & \\[0.5mm]
%
\midrule
%
%
State & time & ${t}$ & time & $t$ \\
& position & $\bm {x}$ & {angle}\index{angle} & $\bm\phi$ \\
& momentum & ${p}=m \bm v$ &
{angular momentum}\index{angular momentum!aspect of state} & $\bm {L}=\Theta \bm \omega$
\\
& energy & $mv^2/2$ & energy & $\Theta \omega^2/2$ \\
%
%
Motion & velocity & $\bm v$ & {angular velocity}\index{velocity!angular} &
$\bm \omega$ \\
%
& acceleration & $\bm a$ &
{angular acceleration}\index{acceleration!angular} & $\bm \alpha$ \\
%
Reluctance to move & mass & $m$ &
{moment of inertia}\index{moment!of inertia} & $\Theta$ \\
%
%
Motion change & {force}\index{force} & $m \bm a$ &
torque\index{torque} & $\Theta \bm \alpha$ \\
%
\bottomrule
\end{tabular*}
\end{table}
}
In contrast to the case of mass, there is \emph{no} conservation of the moment
of inertia. In fact, the value of the moment of inertia depends both on the
direction and on the location of the axis used for its definition. For each
axis direction, one distinguishes an \ii[moment!of inertia,
intrinsic]{intrinsic} moment of inertia, when the axis passes through the
centre of mass of the body, from an \ii[moment!of inertia,
extrinsic]{extrinsic} moment of inertia, when it does not.%
%
\footnote{Extrinsic and intrinsic moment of inertia are related by
\begin{equation}
\Theta_{\rm ext}=\Theta_{\rm int} + m d^{2} \cvend
\end{equation}
where $d$ is the distance between the centre of mass and the axis of extrinsic
rotation. This relation is called \ii{Steiner's parallel axis theorem}. Are
you able to deduce it?\challengenor{steiner}} %
%
In the same way, we distinguish intrinsic and extrinsic angular
momenta.\indexs{momentum!angular, intrinsic}\indexs{momentum!angular,
extrinsic}\indexs{momentum, angular} (By the way, the\label{masscentre}
\ii[mass!centre of]{centre of mass} of a body is that imaginary point which
moves straight during vertical fall, even if the body is rotating. Can you
find a way to\challengenor{loccm} determine its location for a specific body?)
\csepsf{iangmomentum}{scale=1}{Angular momentum and other quantities for a
point particle in circular motion, and the two versions of the righthand
rule.}[%
\psfrag{L}{\small ${\bm L} = {\bm r} \times {\bm p} = \Theta {\bm \omega} = m r^2
\omega$}%
\psfrag{p}{\small ${\bm p} = m {\bm v} = m {\bm \omega} \times {\bm r}$}%
\psfrag{r}{\small ${\bm r}$}%
\psfrag{A}{\small ${A}$}%
]
\cstftlepsf{iaperot}{scale=1}{Can the ape reach the banana?}[25mm]
{isnake}{scale=0.8}{How a snake turns itself around its axis.}
We now define the \ii[energy!rotational, definition]{rotational energy} as
\begin{equation}
E_{\rm rot} =
{\te\frac{1}{2}}\,
\Theta \; \omega^{2}=
\frac{L^{2}}{2 \Theta} \cp
% \label{rotengcons}
\end{equation}
The expression is similar to the expression for the kinetic energy of a
particle. For rotating objects with fixed shape, rotational energy is
conserved.
Can you guess how much larger the rotational energy of the Earth is compared
with the yearly electricity usage of humanity?\challengenor{encoroz} In fact,
if you could find a way to harness the Earth's rotational energy, you would
become famous.
% Improved Aug 2013
Every object that has an orientation also has an intrinsic angular momentum.
(What about a sphere?)\challengenor{sphereori} Therefore, \emph{point}
particles do \emph{not} have intrinsic angular momenta  at least in
classical physics. (This statement will change in quantum theory.) The
\ii[momentum!angular, extrinsic]{extrinsic} angular momentum $\bm L$ of a
point particle is defined as
\begin{equation}
{\bm L} = {\bm r} \times {\bm p} % = \frac{2 {\bm a}(T) m }{ T }
% \qhbox{implying} L= r\, p = \frac{2 A(T) m }{ T } \cv
\end{equation}
where $\bm p$ is the momentum of the particle and $\bm r$ the position vector.
%
The angular momentum thus points along the rotation axis, following the
righthand rule, as shown in \figureref{iangmomentum}.
% Oct 2011
A few values observed in nature are given in \tableref{angmomvaltab}.
%
The definition implies that the angular momentum can also be determined using
the expression\challengn
\begin{equation}
L= \frac{2 A(t) m }{ t } \cv
\end{equation}
where $A(t)$ is the area \emph{swept} by the position vector $\bm r$ of the
particle during time $t$. For example, by determining the swept area with the
help of his telescope, Johannes Kepler discovered in the year 1609 that each
planet orbiting the Sun has an angular momentum value that is \emph{constant}
over time.
% Feb 2012
A physical body can rotate simultaneously about \emph{several} axes. The film
of \figureref{gyroprec} shows an example:\seepageone{gyroprec} The top rotates
around its body axis and around the vertical at the same time. A detailed
exploration shows that the exact rotation of the top is given by the
\emph{vector sum} of these two rotations. To find out, `freeze' the\challengn
changing rotation axis at a specific time. Rotations\index{rotation!as
vector} thus are a type of vectors.\seepageone{rotpseuvec}
% corrected mistake in Apr 2006, impr. 2013
As in the case of linear motion, rotational energy and angular momentum are
not always conserved in the macroscopic world: rotational energy can change
due to friction, and angular momentum can change due to external forces
(torques). But for \emph{closed} (undisturbed) systems, both angular momentum
and rotational energy are always conserved. In particular, on a microscopic
scale, most objects are undisturbed, so that conservation of rotational energy
and angular momentum usually holds on microscopic scales.
% We will study the consequences in more detail in quantum
% theory.\seepage{...}
\emph{Angular momentum is conserved.} This statement is valid for any axis of
a physical system, \emph{provided} that
% friction plays no role. % WRONG!
external forces (torques) play no role. To make the point, \iinn{JeanMarc
LévyLeblond} poses the problem\cite{hewex} of \figureref{iaperot}. Can
the ape reach the banana without leaving the plate,\index{banana!catching
puzzle} assuming that the plate on which the ape rests can turn around the
axis without any friction?\challengenor{aperot}
% May 2007, Jul 2016
We note that many effects of rotation are the same as for acceleration: both
acceleration and rotation of a car pushed us in our seat. Therefore, many
sensors for rotation are the same as the acceleration sensors we explored
above.\seepageone{accelsensors} But a few sensors for rotation are
fundamentally new. In particular, we will meet the \iin{gyroscope}
shortly.\seepageone{gyroscope}
On a frictionless surface, as approximated by smooth ice or by a marble floor
covered by a layer of oil, it is impossible to move forward. In order to move,
we need to push \emph{against} something. Is this also the case for rotation?
Surprisingly, it is\label{twodimoric} possible to %\provoc
{turn} even \emph{without} pushing against something.\index{orientation!change
needs no background} You can check this on a welloiled rotating office
chair: simply rotate an arm above the head. After each turn of the hand, the
orientation of the chair has changed by a small amount. Indeed, conservation
of angular momentum and of rotational energy do \emph{not} prevent bodies from
changing their orientation. Cats\index{cat!falling}\index{rotation!cat} learn
this in their youth. After they have learned the trick, if cats are dropped
legs up, they can turn themselves in such a way that they always land feet
first.\cite{catfall} Snakes\index{snake!rotation}\index{rotation!snake} also
know how to rotate themselves, as \figureref{isnake} shows. Also humans have
the ability: during the Olympic Games you can watch \iin{board divers} and
\iin{gymnasts} perform similar tricks. Rotation thus differs from translation
in this important aspect. (Why?)\challengedif{whyrotdif}
%
% Reread Jul 2016
\subsection{Rolling wheels}
% Index OK
Rotation is an interesting phenomenon in many ways. A rolling wheel does
\emph{not} turn around its axis, but around its point of contact. Let us show
this.\index{rolling!wheels}\index{wheel!rolling}
\cstftlepsf{iwheel2}{scale=1}{The velocities and unit vectors for a rolling
wheel.}[20mm]{isimwheel}{scale=1}{A simulated photograph of a rolling wheel
with spokes.}
A wheel of radius $R$ is \ii{rolling} if the speed of the axis $v_{\rm axis}$
is related to the angular velocity $\omega$ by
\begin{equation}
\omega= \frac{v_{\rm axis}}{R} \cp
\end{equation}
For any point $\rm P$ on the wheel, with distance $r$ from the axis, the
velocity $v_{\rm P}$ is the sum of the motion of the axis and the motion
around the axis. \figureref{iwheel2} shows that $v_{\rm P}$ is orthogonal to
$d$, the distance between the point $\rm P$ and the contact point of the
wheel. The figure also shows\challengn that the length ratio between
$v_{\rm P}$ and $d$ is the same as between $v_{\rm axis}$ and $R$. As a
result, we can write
%
% \begin{equation}
% {\bm v}_{\rm P}= \omega \, R \, {\bm e}_{\rm x}  \omega \, r \, {\bm
% e}_{\theta} \cvend
% \end{equation}
% where ${\bm e}_{\theta}$ in the second term is a unit vector orthogonal to
% the line connecting the point $\rm P$ and the axis. Now take ${\bm e}_{\rm z}$
% as the unit vector along the axis; then one can transform the previous
% expression into
%
\begin{equation}
{\bm v}_{\rm P}=
% (\omega \, {\bm e}_{\rm z}) \times (R \, {\bm e}_{\rm y}
% + {\bm r}) =
{\bm \omega} \times {\bm d} \cvend
\end{equation}
which shows that a rolling wheel does indeed rotate about its point of contact
with the ground.
Surprisingly, when a wheel rolls, some points on it move \emph{towards} the
wheel's axis, some stay at a \emph{fixed} distance and others move \emph{away}
from it. Can you determine where these various points are
located?\challengenor{cirwheel} Together, they lead to an interesting pattern
when a rolling wheel with spokes, such as a bicycle wheel, is photographed, as
show in \figureref{isimwheel}.\cite{rollwheel}
With these results you can tackle the following beautiful
challenge.\cite{intfails} When a turning bicycle wheel is deposed on a
slippery surface, it will slip for a while, then slip and roll, and finally
roll only. How does the final speed depend on the initial speed and on the
friction?\challengedif{intfailt}
%
% Reread Jul 2016
\subsection{How do we walk and run?}
% Index OK
\begin{quote}
Golf is a good walk spoiled.\\
The Allens
%Mark Twain\indname{Twain, Mark}
\end{quote}
\np Why do we move our arms when walking or running?\index{walking!human} To
save energy or to be graceful? In fact, whenever a body movement is performed
with as little energy as possible, it is both natural and graceful. This
correspondence can indeed be taken as the actual definition of
grace.\index{grace} The connection is common knowledge in the world of dance;
it is also a central aspect\cite{a8} of the methods used by actors to learn
how to move their bodies as beautifully as possible.
% \csepsfnb{fwalking}{scale=0.964}{The measured motion of a walking human
\cssmallepsfnb{fwalking}{scale=0.75}{The measured motion of a walking human
({\textcopyright}~\protect\iinn{Ray McCoy}).}
To convince yourself about the energy savings, try walking or running with
your arms fixed or moving in the opposite direction to usual: the effort
required is considerably higher. In fact, when a leg is moved, it produces a
torque around the body axis which has to be counterbalanced. The method using
the least energy is the swinging of arms,\index{arm!swinging} as depicted in
\figureref{fwalking}. Since the arms are lighter than the legs, they must
move further from the axis of the body, to compensate for the momentum;
evolution has therefore moved the attachment of the arms, the \iin{shoulders},
farther apart than those of the legs, the \iin{hips}. Animals on two legs but
without arms, such as \iin{penguins} or \iin{pigeons}, have more difficulty
walking; they have to move their whole torso with every step.
% Feb 2014
Measurements show that\index{pendulum!and walking}\index{walking!and pendulum}
all walking animals follow\cite{biomech}
\begin{equation}
v_{\hbox{max walking}}= (\csd{2.2 \pm 0.2}{m/s}) \, \sqrt{l/{\rm m}} \cp
\end{equation}
Indeed, walking, the moving of one leg after the other, can be described as a
concatenation of (inverted) pendulum swings. The pendulum length is given by
the leg length $l$. The typical time scale of a pendulum is
$t \sim \sqrt{l/g}$. The maximum speed of walking then becomes
$v \sim l/t \sim \sqrt{gl}$, which is, up to a constant factor, the measured
result.
Which muscles do most of the work when walking, the motion that experts call
\ii[gait!human]{gait}? In 1980, \iinn{Serge Gracovetsky}\cite{Gracovet} found
that in human gait a large fraction of the power comes from the muscles along
the \emph{spine}, not from those of the legs. (Indeed, people without legs
are also able to walk. However, a number of muscles in the legs must work in
order to walk normally.) When you take a step, the lumbar muscles straighten
the spine; this automatically makes it turn a bit to one side, so that the
knee of the leg on that side automatically comes forward. When the foot is
moved, the lumbar muscles can relax, and then straighten again for the next
step.
% The arm swing helps to reduce the
% necessary energy.
In fact, one can experience the increase in tension in the \emph{back} muscles
when walking without moving the arms,\challengn thus confirming where the
human engine, the socalled \ii{spinal engine} is located. %
%
% I took this starting info from a letter in New Scientist
%
% Found on the net:
%
% rentsv1.uokhsc.edu/dthompson/: walking expert, assistant professor.
%
% rentsv1.uokhsc.edu/dthompson/gait/sked.htm: a lecture course on gait
%
% Graco is cited in
% rentsv1.uokhsc.edu/dthompson/gait/kinetics/mmactsum.htm
%
% A colloquium by him is:
% cug.concordia.ca/~scol/publect/gracov_lec.html
%
% He has written a book in the Springer Verlag: The spinal engine.
% Dec 2004
Human legs differ from those of apes in a fundamental aspect: humans are able
to \ii[running!human]{run}. In\index{evolution!and running} fact the whole
human body has been optimized for running, an ability that no other primate
has. The human body has shed most of its hair to achieve better cooling, has
evolved the ability to run while keeping the head stable, has evolved the
right length of arms for proper balance when running, and even has a special
ligament in the back that works as a shock absorber while running. In other
words, running is the most human of all forms of motion.
%
% Reread 2016
\subsection{Curiosities and fun challenges about mass, conservation and
rotation}
% Index ok
\begin{quote}
It is a mathematical fact that the casting of this pebble from my hand
alters the centre of gravity of the universe.\\
\iinns{Thomas Carlyle},\footnote{Thomas Carlyle \lived(17971881), Scottish
essayist. Do you agree with the\challengenor{uniqu} quotation?} %
\emph{Sartor Resartus III}.
\end{quote}
% \np Here are a few facts to ponder about motion.
% April 2016
\csepsf{iscalepuzzle}{scale=1}{How does the displayed weight value
change when an object hangs into the water?}
\cssmallepsf{irotpuzzle}{scale=1}{How many rotations does the tenth coin
perform in one round?}
\begin{curiosity}
% April 2016
\item[] A cup with water is placed on a weighing scale, as shown in
\figureref{iscalepuzzle}. How does the mass result change if you let a
piece of metal attached to a string hang into the water?\challengn
% The weight increases
% Jan 2014, impr. Oct 2017
\item Take ten coins of the same denomination. Put nine of them on a table
and form a closed loop with them of any shape you like, a shown in
\figureref{irotpuzzle}. (The nine coins thus look like a section of pearl
necklace where the pearls touch each other.) Now take then tenth coin and
let it roll around the loop, thus without ever sliding it. How many turns
does this last coin make during one
round?\challengn % Ten times, if I remember correctly.
% Mar 2014
\item Conservation of momentum is best studied playing and exploring
billiards, snooker or pool. The best introduction are the trickshot films
found across the internet. Are you able to use momentum conservation to
deduce ways for improving your billiards game?\challengn
\cssmallepsfnb{imariotte}{scale=0.7}{%
The ballchain\protect\index{cradle!Mariotte}\protect\index{cradle!Newton}
or cradle invented by Mariotte allows to explore momentum conservation,
energy conservation, and the difficulties of precision manufacturing
({\textcopyright}~\protect\url{www.questacon.edu.au}).}
% October 2017
Another way to explore momentum conservation is to explore the ballchain,
or ball collision pendulum, that was invented by \iinn{Edme
Mariotte}. Decades later, Newton claimed it as his, as he often did with
other people's results. Playing with the toy is fun  and explaining its
behaviour even more. Indeed, if you lift and let go three balls on one
side, you will see three balls departing on the other side; for the
explanation of this behaviour the conservation of momentum and energy
conservation are \emph{not} sufficient, as you should be able to find
out.\challengedif{ballchain} Are you able to build a highprecision
ballchain?
% Feb 2015
\item There is a wellknown way to experience 81 sunrises in just 80
days. How?\challengenor{days80}
% May 2010
\item Walking is a source of many physics problems. When climbing a mountain,
the most energyeffective way is not always to follow the steepest
ascent;\cite{zigzaglit} indeed, for steep slopes, zigzagging is more energy
efficient. Why? And can you estimate the slope angle at which this will
happen?\challengenor{zigzag}
% July 2016
\item \iin{Asterix} and his friends from the homonymous comic strip, fear only
one thing: that the sky might fall down.\index{sky!nature of} Is the sky an
object? An image?\challengn
% Sep 2011
\item Death\label{deatmetabolism} is a physical process and thus can be
explored.\index{death!energy and} In general, animals have a
\ii[lifespan!animal]{lifespan} $T$ that scales with fourth root of their
mass $M$. In other terms, $T=M^{1/4}$. This is valid from bacteria to
insects to blue whales. Animals also have a power consumption per mass, or
\ii{metabolic rate} per mass, that scales with the \emph{inverse} fourth
root. We conclude that death occurs for all animals when a certain fixed
energy consumption per mass has been achieved. This is indeed the case;
death occurs for most animals when they have consumed around
\csd{1}{GJ/kg}.\cite{metabdeath} (But quite a bit later for humans.) This
surprisingly simple result is valid, \emph{on average}, for all known
animals.\index{death!and energy consumption}
Note that the argument is only valid when \emph{different} species are
compared. The dependence on mass is \emph{not} valid when specimen of the
same species are compared. (You cannot live longer by eating less.)
In short, animals die after they metabolized \csd{1}{GJ/kg}. In other
words, once we ate all the calories we were designed for, we die.
% Dec 2005
\item A car at a certain speed uses 7 litres of gasoline per \csd{100}{km}.
What is the combined air and rolling resistance?\challengenor{airfule}
(Assume that the engine has an efficiency of 25\,\%.)
% Apr 2005
\cssmallepsf{iwineglass}{scale=1}{Is it safe to let the cork go?}
% Apr 2005
\item A cork is attached to a thin string a metre long.\index{cork} The string
is passed over a long rod held horizontally, and a wine glass is attached at
the other end. If you let go the cork in \figureref{iwineglass}, nothing
breaks. Why not? And what happens exactly?\challengenor{wineglass}
% Sep 2005
\item In 1907, \iinn{Duncan MacDougalls}, a medical doctor, measured the
weight of dying people, in the hope to see whether death leads to a mass
change.\cite{ddd} He found a sudden decrease between 10 and \csd{20}{g} at
the moment of death.\index{death!mass change with} He attributed it to the
\iin{soul} exiting the body. Can you find a more satisfying
explanation?\challengenor{soulmawe}
% March 2010
\item It is well known that the weight of a oneyear old child depends on
whether it wants to be carried or whether it wants to reach the
floor.\index{child's mass}\index{mass!of children} Does this contradict mass
conservation?\challengn
% Impr. May 2014
\csepsfnb{imountcont}{scale=1}{A simple model for continents and
mountains.}
% May 2005
\item The Earth's crust\label{contswim} is less dense (\csd{2.7}{kg/l}) than
the Earth's mantle\index{Earth!crust} (\csd{3.1}{kg/l}) and floats on it.
As a result, the lighter crust below a mountain ridge must be much deeper
than below a plain. If a mountain rises \csd{1}{km} above the plain, how
much deeper must the crust be below it?\challengenor{moiudep}
% Jul 2005
The simple block model shown in \figureref{imountcont} works fairly well;
first, it explains why, near mountains, measurements of the deviation of
free fall from the vertical line lead to so much lower values than those
expected without a deep crust. Later, sound measurements have confirmed
directly that the continental crust is indeed thicker beneath mountains.
% Oct 2009
\item All homogeneous cylinders roll down an inclined plane in the same way.
True or false?\challengn And what about spheres? Can you show that spheres
roll faster than cylinders?
% Apr 2013
% http://blogs.scienceforums.net/swansont/archives/2066
\item Which one rolls faster: a soda can filled with liquid or a soda can
filled with ice?\challengenor{sodaroll} (And how do you make a can filled
with ice?)
\item Take two \iin[cans of peas]{cans} of\index{cans of ravioli} the same
size and weight, one full of \iin{ravioli} and one full of \iin[pea!in
can]{peas}. Which one rolls faster on an inclined plane?\challengn
% Sep 2011
\item Another difference between matter and images: matter smells. In fact,
the nose is a matter sensor. The same can be said of the tongue and its
sense of taste.
% June 2005
\item Take a pile of coins. You can push out the coins, starting with the one
at the bottom, by shooting another coin over the table surface. The method
also helps to visualize twodimensional momentum conservation.\challengn
\item In early 2004, two\label{roulettemoney} men and a woman earned
{\textsterling}\,1.2\,million
% how to typeset ? !.!4
in a single evening in a London casino. They did so by applying the
formulae of Galilean mechanics. They used the method pioneered by various
physicists in the 1950s who built various small computers that could predict
the outcome of a roulette ball from the initial velocity imparted by the
croupier.\cite{roulettebook} In the case in Britain, the group added
a\index{roulette and Galilean mechanics} laser scanner to a smart phone that
measured the path of a roulette ball and predicted the numbers where it
would arrive. In this way, they increased the odds from 1 in 37 to about 1
in 6. After six months of investigations, Scotland Yard ruled that they
could keep the money they won.
% Nov 2007
In fact around the same time, a few people earned around 400\,000 euro over
a few weeks by using the same method in Germany, but with no computer at
all.
% Told to me by the CTO of a casino in southern Germany
In certain casinos, machines were throwing the roulette ball. By measuring
the position of the zero to the incoming ball with the naked eye, these
gamblers were able to increase the odds of the bets they placed during the
last allowed seconds and thus win a considerable sum purely through fast
reactions.
% Dec 2016
\item Does the universe rotate?\challengenor{unirotdoesxy}
\item The toy of \figureref{icctoy} shows interesting behaviour: when a
number of spheres are lifted and dropped to hit the resting ones, the same
number of spheres detach on the other side, whereas the previously dropped
spheres remain motionless. At first sight, all this seems to follow from
energy and momentum conservation. However, energy and momentum conservation
provide only two equations, which are insufficient to explain or determine
the behaviour of five spheres. Why then do the spheres behave in this way?
And why do they all swing in phase when a longer time has
passed?\challengedif{cctoy}
\cssmallepsf{icctoy}{scale=1}{A wellknown toy.}
\cssmallepsf{imomnot}{scale=1}{An elastic collision that seems not to obey
energy conservation.}
\item A surprising effect is used in home tools such as \iin{hammer drills}.
We remember that when a small ball elastically hits a large one at rest,
both balls move after the hit, and the small one obviously moves faster than
the\cite{zweck} large one. Despite this result, when a short cylinder hits
a long one of the same diameter and material, but with a length that is some
\emph{integer} multiple of that of the short one, something strange happens.
After the hit, the small cylinder remains almost at rest, whereas the large
one moves, as shown in \figureref{imomnot}. Even though the collision is
elastic, conservation of energy seems not to hold in this case. (In fact
this is the reason that demonstrations of elastic collisions in schools are
always performed with spheres.) What happens to the
energy?\challengedif{hammerdrill}
\item Is the structure shown in \figureref{isoup} possible?
\cstftlepsf{isoup}{scale=1}{Is this possible?}[30mm]{iladder}{scale=1}{How
does the ladder fall?}
\item Does a wall get a stronger jolt when it is hit by a ball rebounding from
it or when it is hit by a ball that remains stuck to it?\challengenor{wall}
% Impr. Oct 2015
\item Housewives know how to extract a \iin{cork} of a \iin[wine!bottle]{wine}
\iin{bottle} using a cloth or a shoe. Can you imagine
how?\challengenor{winecloth} They also know how to extract the cork with the
cloth if the cork has fallen inside the bottle. How?
\item The \iin[ladder!sliding]{sliding ladder} problem, shown schematically in
\figureref{iladder}, asks for the detailed motion of the ladder over time.
The problem is more difficult than it looks, even if friction is not taken
into account. Can you say whether the lower end always touches the floor,
or if is lifted into the air for a short time
interval?\challengenor{slidladd}
\item A homogeneous \iin{ladder} of length \csd{5}{m} and mass \csd{30}{kg}
leans on a wall. The angle is \csd{30}{\csdegrees}; the static friction
coefficient on the wall is negligible, and on the floor it is 0.3. A person
of mass \csd{60}{kg} climbs the ladder. What is the maximum height the
person can climb before the ladder starts sliding? This and many puzzles
about ladders can be found on
\url{www.mathematischebasteleien.de/leiter.htm}.
% Lösung: Kräftebetrachtung: ...... Bei einer Leiter treten die
% Gewichtskräfte der Person (FP=Mg) und der Leiter (FL=mg) auf. Durch den
% Boden und die Wand entstehen die Normalkräfte FN und FW. Durch die
% Reibungskraft FR am Boden wird die Leiter gehalten.
%
% Kräftebilanz: Es gilt im abgeschlossenen System FW=FR und FN=FP+FL. Bilanz
% der Drehmomente bezüglich des Drehpunktes A: (#) FW*c cos(phi)=FP*s
% sin(phi)+FL*(c/2)*sin(phi)
%
% Weitere Rechnung: Die Leiter beginnt zu rutschen, wenn gerade FR=f FN ist.
% Daraus folgt, dass in (#) FW durch f(FP+FL) ersetzt werden muss: f (FP+FL)*c
% cos(phi)=FP*s sin(phi)+FL*(c/2)*sin(phi) Die Kräfte ersetzt man durch die
% Massen über FP=Mg und FL=mg: f(M+m)c*cos(phi)=Ms sin(phi)+m*(c/2)*sin(phi)
% oder (##) cf(M+m)=Ms tan(phi)+m(c/2)tan(phi) Nach s aufgelöst: Zahlenlösung:
% Setzt man c=5m, f=1/3, m=30kg, M=60kg und phi=30° ein, ergibt sich s=3,1m.
% Ergebnis: Die Person kann 3,1m hoch steigen.
\item A common fly on the stern of a \csd{30\,000}{ton} ship\cite{a9} of
\csd{100}{m} length tilts it by %\csd{..}{nm},
less than the diameter of an atom. Today, distances that
small %\label{cheapsmalldis}
are easily measured. Can you think of at least two methods, one of which
should not cost more than 2000 euro?\challengenor{afmch}
% Sep 2007
\cssmallepsfnb{igyrosstacked}{scale=1}{Is this a possible situation or is it
a fake photograph? ({\textcopyright}~\protect\iname{Wikimedia})}
% Sep 2007
\item Is the image of three stacked spinning tops shown in
\figureref{igyrosstacked} a true photograph, showing a real observation,
or is it the result of digital composition, showing an impossible
situation?\challenge %nor{stackedgyros} % !!!5
\item How does the kinetic energy of a rifle bullet compare to that of a
running man?\challengenor{rifle}
\item What happens to the size of an egg when one places it in a jar of
vinegar for a few days?\challengenor{eggvin}
%\item highest electric acceleration with a field: \csd{30}{MeV} in \csd{6}{mm},
%UCLA, 1994
\item What is the amplitude of a pendulum oscillating in such a way that the
absolute value of its acceleration at the lowest point and at the return
point are equal?\challengenor{oscaccpend}
\item Can you confirm that the value of the acceleration of a drop of water
falling through mist is $g/7$?\challengedif{fallingdrop}
\item You have two hollow spheres: they have the same weight, the same size
and are painted in the same colour. One is made of copper, the other of
aluminium. Obviously, they fall with the same speed and acceleration. What
happens if they both roll down\challengenor{hollsphe} a tilted plane?
\item What is the shape of a rope when rope jumping?\challengenor{jumpro}
\item How can you determine the speed of a rifle bullet with only a scale and
a metre stick?\challengenor{bullet}
\item Why does a gun make a hole in a door but cannot push it open, in exact
contrast to what a finger can do?\challengn
% Dec 2006
\item What is the curve described by the mid point of a ladder sliding down a
wall?\challengenor{laddercircle}
% Aug 2007
\item A hightech company, see \url{www.enocean.com}, sells electric switches
for room lights that have no cables and no power cell (battery). You can
glue such a switch to the centre of a window pane. How is this
possible?\challengenor{npbatswitch}
\csepsfnb{iatmosclock}{scale=1}{A commercial clock that needs no special
energy source, because it takes its energy from the environment
({\textcopyright}~\protect\iname{JaegerLeCoultre}).}
% Aug 2007
\item For over 50 years now, a famous Swiss clock maker is selling table
clocks with a rotating pendulum that need no battery and no manual
rewinding, as they take up energy from the environment. A specimen is shown
in \figureref{iatmosclock}. Can you imagine how this clock
works?\challengenor{bimetal}
% Feb 2012
\csepsfnb[p]{ishiplift}{scale=1}{The spectacular ship lift at StrépyThieux in
Belgium. What engine power is needed to lift a ship, if the right and left
lifts were connected by ropes or by a hydraulic system?
({\textcopyright}~\protect\iinn{JeanMarie Hoornaert})}
% from wikimedia commons
% Feb 2012
\item Ship lifts, such as the one shown in \figureref{ishiplift}, are
impressive machines.\index{lift!for ships}\index{ship!lift} How does the
weight of the lift change when the ship enters?\challengenor{shiplift}
% Feb 2012
\item How do you\index{ship!mass of} measure the mass of a ship?\challengn
% Aug 2009, Apr 2013
\item All masses are measured by comparing them, directly or indirectly, to
the \ii{standard kilogram} in Sèvres near Paris. Since a few years, there
is the serious doubt that the standard kilogram is losing weight, possibly
through outgassing, with an estimated rate of around \csd{0.5}{\muunit g/a}.
This is an awkward situation, and there is a vast, worldwide effort to find
a better definition of the kilogram. Such an improved definition must be
simple, precise, and make trips to Sèvres unnecessary. No such alternative
has been defined yet.{\present}
% Nov 2007
\item Which engine is more efficient: a \iin{moped} or a human on a bicycle?
% April 2011
\item Both mass and moment of inertia can be defined and measured both with
and without contact. Can you do so?\challengn
% Photo Mar 2012
\csepsfnb{iceltic}{scale=1}{The famous Celtic wobble stone  above and
right  and a version made by bending a spoon  bottom left
({\textcopyright}~\protect\iinn{Ed Keath}).} % from wikimedia commons
\item \figureref{iceltic} shows the socalled \ii{Celtic wobble stone}, also
called \ii{anagyre} or \ii{rattleback}, a \iin[stones]{stone} that starts
rotating on a plane surface when it is put into upanddown
oscillation.\cite{zweck} The size can vary between a few centimetres and a
few metres. By simply bending a \iin{spoon} one can realize a primitive
form of this strange device, if the bend is not completely symmetrical. The
rotation is always in the same direction. If the stone is put into rotation
in the wrong direction, after a while it stops and starts rotating in the
other sense! Can you explain the effect that seems to contradict the
conservation of angular momentum?\challengedif{celticstone}
% Nov 2014
\item A beautiful effect, the \ii[chain!fountain]{chain
fountain},\index{fountain!chain} was discovered in 2013 by \iinn{Steve
Mould}. Certain chains, when flowing out of a container, first shoot up
in the air. See the video at \url{www.youtube.com/embed/_dQJBBklpQQ} and
the story of the discovery at \url{stevemould.com}. Can you explain the
effect to your grandmother?\challenge % !!!5
% see also
% http://www.physikdidaktik.unikarlsruhe.de/publication/Chain_fountain.pdf
\end{curiosity}
%
% Aug 2007, Jul 2016
\subsection{Summary on conservation in motion}
% Index OK
% Aug 2007
\begin{quote}
The gods\index{gods!and conservation} are not as rich as one might think:
what they give to one, they take away %\index{blasphemy}
from the other.\\
Antiquity
\end{quote}
% Aug 2007
\np We have encountered four \iin[conservation!principles]{conservation
principles} that are valid for the motion of all closed systems in everyday
life:\index{principle!conservation}
\begin{Strich}
\item conservation of total linear momentum,
\item conservation of total angular momentum,
\item conservation of total energy,
\item conservation of total mass.
\end{Strich}
\np None of these conservation principles applies to the motion of
images. These principles thus allow us to distinguish objects from images.
% Aug 2007
\np The conservation principles are among the great results in science. They
limit the surprises that nature can offer: conservation means that linear
momentum, angular momentum, and massenergy can neither be created from
nothing, nor can they disappear into nothing. Conservation limits creation.
The quote below the section title %, almost blasphemous,
expresses this idea.
% Aug 2007
Later on\seepageone{noenoe} we will find out that these results could have
been deduced from three simple observations: closed systems behave the same
independently of where they are, in what direction they are oriented and of
the time at which they are set up. % Motion is universal.
In more abstract % and somewhat more general
terms, physicists like to say that all conservation principles are
consequences of the \emph{invariances}, or \emph{symmetries}, of nature.
% Aug 2007, Feb 2012
Later on, the theory of special relativity will show that energy and mass are
conserved only when taken together. Many adventures still await us.
\vignette{classical}
%
%
%
%
\newpage
% Reread Jul 2016, Jan 2018
\chapter{From the rotation of the earth to the relativity of motion}
% Index OK
\markboth{\thesmallchapter\ from the rotation of the earth}%
{to the relativity of motion}
\begin{quote}
\selectlanguage{italian}Eppur si muove!\selectlanguage{british}\\
Anonymous\indname{Galileo}%
\footnote{`And yet she moves' is the sentence about the Earth attributed,
most probably incorrectly, to Galileo since the
1640s. % see english wikipedia
It is true, however, that at his trial he was forced to publicly retract
the statement of a moving Earth to save his life. For more details of
this famous story, see the section on
\cspageref{redondipage}.} % this vol I
\end{quote}
% Impr. May 2014
\cssmallepsfnb{iparallaxis}{scale=1}{The parallax  not drawn to scale.}
\csini{I}{s} the Earth rotating?\index{rotation!of the
Earth(}\index{Earth!rotation(} The search for definite answers to this
question gives an\linebreak nteresting cross section of the history of
classical physics. Around the year 265 {\bce},\linebreak n Samos, %
the Greek thinker \iname[Aristarchus of Samos]{Aristarchus} was the first to
maintain that the \iin{Earth} rotates.\cite{aristar} He had measured the
parallax of the Moon (today known to be up to \csd{0.95}{\csdegrees}) and of
the Sun (today known to be \csd{8.8}{\csminutes}).%
%
% !.!1 improve typesetting of csminutes!
%
\footnote{For the definition of the concept of angle, see
\cspageref{iangles}, % this vol I
and for the definition of the measurement units for %
angle see \appendixref{units1}.} %
%
The \ii{parallax} is an interesting effect; it is the angle describing the
difference between the directions of a body in the sky when seen by an
observer on the surface of the Earth and when seen by a hypothetical observer
at the Earth's centre. (See \figureref{iparallaxis}.) Aristarchus noticed
that the Moon and the Sun \emph{wobble} across the sky, and this wobble has a
period of 24 hours. He concluded that the Earth rotates.
% Feb 2012  !.!1 check
It seems that Aristarchus received death threats for his result.
% Did this measurement provide Aristarchus with enough arguments for
% his conclusion?
% Oct 2012
\csepsfnb{ipolestartrails}{scale=1}{The motion of the stars during the
night, observed on 1 May 2012 from the South Pole, together with the green
light of an aurora australis ({\textcopyright}~\protect\iinn{Robert
Schwartz}).} % I have his permission!
% http://apod.nasa.gov/apod/ap120802.html
% % Feb 2012
% Aristarchus' observation yields a more powerful argument than the trails of
% the stars shown in \figureref{istartrails}. Can you explain why?\challengn
% Oct 2012
Aristarchus' observation yields an even more powerful argument than
the trails of the stars shown in \figureref{ipolestartrails}. Can
you explain why?\challengn (And how do the trails look at the most
populated places on Earth?)\challengenor{ayopic}
Experiencing \figureref{ipolestartrails} might be one reason that people
dreamt and still dream about reaching the poles. Because the rotation and the
motion of the Earth makes the poles extremely cold places, the adventure of
reaching them is not easy. Many tried unsuccessfully. A famous
crook,\cite{pearylit} \iinn{Robert Peary}, claimed to have reached the North
Pole in 1909. (In fact, \iinn{Roald Amundsen} reached both the South and the
North Pole first.) Among others, Peary claimed to have taken a picture there,
but that picture, which went round the world, turned out to be one of the
proofs that he had not been there.\challengenor{peary} Can you imagine how?
\cssmallepsf{iflattening}{scale=1}{Earth's deviation from spherical shape due
to its rotation (exaggerated).}
% Jan 2015
If the Earth rotates instead of being at rest, said the unconvinced, the speed
at the equator has the substantial value of \csd{0.46}{km/s}. How did Galileo
explain why we do not feel\challengn or notice this speed?
Measurements of the aberration\seepagetwo{aberrrr} of light also show the
rotation of the Earth; it can be detected with a telescope while looking at
the stars. The \ii{aberration} is a change of the expected light direction,
which we will discuss shortly.
% , discovered in 1728 by \iinn{James Bradley}, the
% astronomer royal, shows the rotation of the Earth.
At the Equator, Earth rotation adds an angular deviation of
\csd{0.32}{\csminutes}, changing sign every 12 hours, to the aberration due to
the motion of the Earth around the Sun, about \csd{20.5}{\csminutes}. In
modern times, astronomers have found a number of additional proofs
for the rotation of the Earth, but none
is accessible to the man on the street.
% Furthermore,
Also the measurements\index{Earth!flattened}\index{flattening!of the
Earth} showing that the Earth is not a sphere, but is \emph{flattened} at
the poles, confirmed the rotation of the Earth. \figureref{iflattening}
illustrates\label{maupertlab} the situation. Again, however, this eighteenth
century measurement by \inames[Maupertuis, Pierre Louis Moreau
de]{Maupertuis}%
%
\footnote{\iinns{Pierre~Louis Moreau~de Maupertuis}
\lived(16981759), %French
physicist and mathematician, was one of the key figures in the quest
for the \iin[principle!of least action]{principle of least action},
which he named in this way. He was also founding president of the
Berlin Academy of Sciences. Maupertuis thought that the principle
reflected the maximization of goodness in the universe. This idea
was thoroughly ridiculed by Voltaire in his \bt Histoire du Docteur
Akakia et du natif de SaintMalo/ \yrend 1753/ (Read it at
%\url{www.voltaireintegral.com/Html/23/08DIAL.htm}.) % Mar 2021
\url{gallica.bnf.fr/ark:/12148/bpt6k6548988f.texteImage}.)
Maupertuis
performed his measurement of the Earth to distinguish between the
theory of gravitation of Newton and that of Descartes, who had
predicted that the
Earth is elongated at the poles, instead of flattened.} %
%
is not accessible to everyday observation.
% OK PSFRAG APR 2014
\cssmallepsf{ifallhit}{scale=1}{The deviations of free fall towards the east
and towards the Equator due to the rotation of the Earth.}[%
\psfrag{vh}{\small $v_{h}=\omega (R+h)$}%
\psfrag{vwr}{\small $v=\omega R$}%
\psfrag{h}{\small $h$}%
\psfrag{j}{\small $\phi$}%
]
Then, in the years 1790 to 1792 in Bologna, \iinn{Giovanni~Battista
Guglielmini} \lived(17631817) finally succeeded in measuring what
\iname[Galilei, Galileo]{Galileo} and \iname[Newton, Isaac]{Newton} had
predicted to be the simplest proof for the Earth's rotation. On the rotating
Earth, \emph{objects do not fall vertically},\index{fall!is not vertical} but
are slightly deviated to the east. This deviation appears because an object
keeps the larger horizontal velocity it had at the height from which it
started falling, as shown in \figureref{ifallhit}. Guglielmini's result was
the first nonastronomical proof of the Earth's rotation.
%
%1802, Johann Friedrich Benzenberg \lived(17771846), a not hamburger
%
The experiments were repeated in 1802 by \iinn{Johann~Friedrich Benzenberg}
\lived(17771846). Using metal balls % my guess
which he dropped from the Michaelis tower
% from the church
in Hamburg  a height of \csd{76}{m}~ Benzenberg found that the deviation
to the east was \csd{9.6}{mm}. Can you confirm that the value measured by
Benzenberg almost agrees with the assumption that the Earth turns once every
24 hours?\challengedif{benzen} There is also a much smaller deviation towards
the Equator, not measured by Guglielmini, Benzenberg or anybody after them up
to this day; however, it completes the list of effects on free fall by the
rotation of the Earth.
Both deviations from vertical fall are easily understood if we use the result
(described below)\seepageone{ugorbits} that falling objects describe an
ellipse around the centre of the rotating Earth. The elliptical shape shows
that the path of a thrown \iin[stones]{stone} does not lie on a plane for an
observer standing on Earth; for such an observer, the exact path of a stone
thus cannot be drawn on a flat piece of paper!
%
% , the point below it on the ground does not
% follow a straight line; the motion of the stone is \emph{not} in a plane,
% as
% the \iin{moving pictures in the lower left corner} seem to suggest. The
% rotation of the Earth makes the real path a curve in all three dimensions.
% Added fig in Feb 2010
\csepsf{icorioliseffect}{scale=1}{A typical carousel allows observing the
Coriolis effect in its most striking appearance: if a person lets a ball roll
with the proper speed and direction, the ball is deflected so strongly that it
comes back to her.}
% Nov 2006, Nov 2012
In 1798,\label{lcoriolis} \iinns{Pierre~Simon Laplace}%
%
\footnote{Pierre Simon Laplace \livedplace(1749 \hbox{BeaumontenAuge}1827
Paris), important % French
mathematician. His famous treatise \btsim Traité de mécanique céleste/
appeared in five volumes between 1798 and 1825. He was the first to propose
that the Solar System was formed from a rotating gas cloud, and one of the
first people to imagine and explore black holes.} %
%
% meine enzy sagt 1799 bis 1825
%
explained how bodies move on the rotating Earth and showed that they feel an
apparent force.\cite{gerkema} In 1835, \iinn{GustaveGaspard Coriolis} then
reformulated and simplified the description. Imagine a ball that rolls over a
table. For a person on the floor, the ball rolls in a straight line. Now
imagine that the table rotates. For the person on the floor, the ball still
rolls in a straight line. But for a person on the rotating table, the ball
traces a \emph{curved} path. In short, any object that travels in a rotating
background is subject to a transversal acceleration. The acceleration,
discovered by Laplace, is nowadays called \ii{Coriolis acceleration} or
\ii{Coriolis effect}.
% !.! reader asked: explain this much better in words! done in Nov 2006
% Added Feb 2010:
On a rotating background, travelling objects deviate from the straight line.
The best way to understand the Coriolis effect is to experience it yourself;
this can be done on a carousel, as shown in \figureref{icorioliseffect}.
Watching films on the internet on the topic is also
helpful.\cite{coriolisvideo} You will notice that on a rotating carousel it is
not easy to hit a target by throwing or rolling a ball.
% Nov 2006
Also the Earth is a rotating background. On the northern hemisphere, the
rotation is anticlockwise. As the result, any moving object is slightly
deviated to the right (while the magnitude of its velocity stays constant).
On Earth, like on all rotating backgrounds, the \ii{Coriolis acceleration}
${\bm a}_{\rm C}$ results from the change of distance to the rotation axis.
Can you deduce the analytical expression for the Coriolis effect, namely
${\bm a}_{\rm C} =  2 {\bm \omega} \times {\bm v}$?\challengenor{Coriolis}
% Sep 2016
\csepsfnb{icyclones}{scale=1}{Cyclones, with their low pressure centre,
differ in rotation sense between the southern hemisphere, here cyclone Larry
in 2006, and the northern hemisphere, here hurricane Katrina in
2005. (Courtesy NOAA)}
% Nov 2006, Sep 2016
On Earth, the \iin{Coriolis acceleration} generally has a small value.
Therefore it is best observed either in largescale or highspeed
phenomena. Indeed, the Coriolis acceleration determines the
handedness of many largescale phenomena with a spiral shape, such as
the directions of cyclones and anticyclones in meteorology  as shown
in \figureref{icyclones}  the general wind patterns on Earth and
the deflection of ocean currents and \iin[tide!and Coriolis
effect]{tides}. These phenomena have opposite handedness on the
northern and the southern hemisphere. Most beautifully, the Coriolis
acceleration explains why icebergs do not follow the direction of the
wind as they drift away from the polar caps.\cite{ekman} The {Coriolis
acceleration} also plays a role in the flight of cannon balls (that
was the original interest of Coriolis), in satellite launches, in the
motion of sunspots and even in the motion of electrons in
molecules.\cite{molcori} All these Coriolis accelerations are of
opposite sign on the northern and southern hemispheres and thus prove
the rotation of the Earth. For example, in the First World War, many
naval \iin[guns and the Coriolis effect]{guns} missed their targets in
the southern hemisphere because the engineers had compensated them for
the Coriolis effect in the northern hemisphere.
Only in 1962, after several earlier attempts by other researchers, \iinn{Asher
Shapiro}\cite{a53} was the first to verify that the Coriolis effect has a
tiny influence on the direction of the vortex formed by the water flowing out
of a bathtub.\index{bathtub vortex} Instead of a normal bathtub, he had to
use a carefully designed experimental setup because, contrary to an
oftenheard assertion, no such effect can be seen in a real bathtub. He
succeeded only by carefully eliminating all disturbances from the system; for
example, he waited 24 hours after the filling of the reservoir (and never
actually stepped in or out of it!) in order to avoid any leftover motion of
water that would disturb the effect, and built a carefully designed,
completely rotationallysymmetric opening mechanism. Others have repeated the
experiment in the southern hemisphere,\cite{a53} finding opposite rotation
direction and thus confirming the result. In other words, the handedness of
usual bathtub vortices is \emph{not} caused by the rotation of the Earth, but
results from the way the water starts to flow out.
% Sep 2011
(A number of crooks in \iin{Quito}, a city located on the Equator,
show gullible tourists that the vortex in a sink changes when crossing
the Equator line drawn on the road.)
%
But let us go on with the story about the Earth's rotation.
\cssmallepsf{ifoucault}{scale=1}{The turning motion of a pendulum showing the
rotation of the Earth.}[\psfrag{psi0}{\small $\psi_{0}$}\psfrag{psi1}{\small
$\psi_{1}$}\psfrag{phi}{\small $\phi$}]
In 1851,\label{foucpend} the %French
physicianturnedphysicist \iinns{{Jean Bernard Léon} Foucault}
\livedplace(1819 Paris1868 Paris) performed an experiment that removed all
doubts and rendered him worldfamous practically overnight. He suspended a
\csd{67}{m} long pendulum%
%
\footnote{Why was such a long pendulum necessary?\challengedif{written}
Understanding the reasons allows one to repeat the experiment at home, using
a pendulum as short as \csd{70}{cm}, with\cite{rotpend2} the help of a few
tricks.
% Dec 2006
To observe Foucault's effect with a simple setup, attach a pendulum to your
office chair and rotate the chair slowly. Several pendulum animations, with
exaggerated deviation, can be found at
\url{commons.wikimedia.org/wiki/Foucault_pendulum}.} %
%
% May 2004, corrected text and sign (!) in March 2007
in the Panthéon in Paris and showed the astonished public that the direction
of its swing changed over time, rotating slowly. To anybody with a few
minutes of patience to watch the change of direction, the experiment proved
that the Earth rotates. If the Earth did not rotate, the swing of the
pendulum would always continue in the same direction. On a rotating Earth, in
Paris, the direction changes to the right, in clockwise sense, as shown in
\figureref{ifoucault}. The swing direction does not change if the pendulum
is located at the Equator, and it changes to the left in the southern
hemisphere.%
\footnote{The discovery also shows how precision and genius go together. In
fact, the first person to observe the effect was \iinn{Vincenzo Viviani}, a
student of Galileo, as early as 1661! Indeed, Foucault had read about
Viviani's work in the publications of the Academia dei Lincei. But it took
Foucault's genius to connect the effect to the rotation of the Earth; nobody
had done so before him.} %
%
% Mar 2007
%
A modern version of the pendulum can be observed via the web cam\index{web
cam!Foucault's pendulum}\index{Foucault's pendulum!web cam} at
\url{pendelcam.kip.uniheidelberg.de}; high speed films of the pendulum's
motion during day and night can also be downloaded at
\url{www.kip.uniheidelberg.de/oeffwiss/pendel/zeitraffer/}.
% OK in Jan 2014
% Old, not ok any more:
% \url{www.kip.uniheidelberg.de/OeffWiss/PendelInternetauftritt/zeitraffer.php}.
The time over which the orientation of the pendulum's swing performs a full
turn  the \emph{precession time}\indexs{precession!of a pendulum}  can be
calculated. Study a pendulum starting to swing in the NorthSouth direction
and you will find that the precession time $T_{\rm Foucault}$ is given
by\challengedif{foucault}
\begin{equation}
T_{\rm Foucault}=
\frac{\csd{23}{h}\;\csd{56}{min}}{\sin \phi}
\end{equation}
where $\phi$ is the latitude of the location of the pendulum,
e.g.~\csd{0}{\csdegrees} at the Equator and \csd{90}{\csdegrees} at the North
Pole. This formula is one of the most beautiful results of Galilean
kinematics.%
%
\footnote{The calculation of the period of Foucault's pendulum assumes that
the precession rate is constant during a rotation. This is only an
approximation (though usually a good one).}
% May 2007
\csepsfnb{igyro4}{scale=1}{The gyroscope: the original system by Foucault
with its freely movable spinning top, the mechanical device to bring it to
speed, the optical device to detect its motion, the general construction
principle, and a modern (triangular) ring laser gyroscope, based on colour
change of rotating laser light instead of angular changes of a rotating mass
({\textcopyright}~\protect\iname{CNAM}, \protect\iname{JAXA}).}
% EMAILED FEB 2008  fukuda.kyoko@jaxa.jp
% Mar 2014, Jan 2018 % checked that Zach's u3d works correctly
% % add figure of Zach espiritu (email 23. Mai 2013, in colour)
% % and u3d (9 December 2013)
\csuuudfile{SubmittedGyro3D2}{scale=0.75}% new number in June 2014  works
{A threedimensional model of Foucault's original gyroscope: in the
pdf verion of this text, the model can be rotated and zoomed by moving
the cursor over it ({\textcopyright}~\protect\iinn{Zach~Joseph
Espiritu}).}
Foucault\label{gyroscope} was also the inventor and namer of the
\ii{gyroscope}. He built the device, shown in \figureref{igyro4} and
\figureref{SubmittedGyro3D2}, in 1852, one year after his pendulum.
With it, he again demonstrated the rotation of the Earth. Once a
gyroscope rotates, the axis stays fixed in space  but only when seen
from distant stars or galaxies. (By the way, this is not the same as
talking about \iin[space!absolute]{absolute space}.
Why?)\challengenor{absgyro} For an observer on Earth, the axis
direction changes regularly with a period of 24 hours. Gyroscopes are
now routinely used in ships and in aeroplanes to give the direction of
north, because they are more precise and more reliable than magnetic
compasses. The most modern versions use laser light running in
circles instead of rotating masses.\footnote{Can you guess how
rotation is detected in this\challengenor{rotldet} case?}
In 1909, \iinn{Roland~von Eötvös} measured a small but surprising
effect: due to the rotation of the Earth, the weight of an object
depends on the direction in which it moves. As a result, a balance in
rotation around the vertical axis does not stay perfectly horizontal:
the balance starts to oscillate slightly. Can you explain the origin
of the effect?\challengenor{eoeffect}
In 1910, \iinn{John Hagen}
% I first found only E. Hagen  internet searches gave no results
published the results of an even simpler experiment,\cite{Hagenbet} proposed
by \iinn{Louis Poinsot} in 1851. % \lived(17771859)
Two masses are put on a horizontal bar that can turn around a vertical axis, a
socalled \ii{isotomeograph}. Its total mass was \csd{260}{kg}. If the two
masses are slowly moved towards the support, as shown in \figureref{ihagen},
and if the friction is kept low enough, the bar rotates. Obviously, this
would not happen if the Earth were not rotating. Can you explain the
observation?\challengenor{coribet} This littleknown effect is also useful for
winning \iin[bets!how to win]{bets} between physicists.
\cstftlepsfpsfragboth{ihagen}{scale=1}{Showing the rotation of the Earth
through the rotation of an axis.}[\psfrag{m}{$m$}]
{icomptonwheelnew}{scale=1}{Demonstrating the rotation of the Earth with
water.}[\psfrag{r}{$r$}] % OK psfrag of APR 2014
% An important consequence of the rotational motion of the Earth is its
% magnetic field. We will come back to the topic later on.
% Feb 2004, Dec 2004
In 1913, \iinn{Arthur Compton} showed that a closed tube filled with water and
some small floating particles (or bubbles) can be used to show the rotation of
the Earth.\cite{comptwheel} The device is called a \ii{Compton tube} or
\ii{Compton wheel}. {Compton} showed that when a horizontal tube filled with
water is rotated by \csd{180}{\csdegrees}, something happens that allows one
to prove that the Earth rotates. The experiment, shown in
\figureref{icomptonwheelnew}, even allows measuring the latitude of the point
where the experiment is made. Can you guess what
happens?\challengedif{comptonwheel}
\csepsfnb{ilasergyrointerf}{scale=1}{%One of the early interferometers built
% !.!1 is not the same  correct the error!
% by Michelson and Morley, and a
A modern precision ring laser interferometer
({\textcopyright}~\protect\iname{Bundesamt für Kartographie und Geodäsie},
% EMAILED FEB 2008  kluegel@fs.wettzell.de
\protect\iname{Carl Zeiss}).}
% % In 1925, \iinns{Albert Michelson}\footnote{Albert~Abraham
% % Michelson \livedplace(1852 Strelno1931 Pasadena) %Jewish
% % % taken out after comments about such details in the press
% % PrussianPolishUSAmerican physicist,
% % obsessed by the precise measurement of the speed of light, received
% % the Nobel Prize in Physics in 1907.} %
% % %
% % and his collaborators in Illinois built several \iin{interferometer}s for
% % light and detected the rotation of the Earth.
% % ; the largest interferometer they constructed
% % had an arm length of \csd{32}{m}
% Rewritten Feb 2010
Another method to detect the rotation of the Earth using light was first
realized in 1913 %by the French physicist
\iinns{Georges Sagnac}:%
%
\footnote{Georges Sagnac \livedplace(1869 Périgeux1928 MeudonBellevue) was a
physicist in Lille and Paris, friend of the Curies, Langevin, Perrin, and
Borel. Sagnac also deduced from his experiment that the speed of light was
independent from the speed of its source, and thus confirmed a prediction of
special relativity.} %
%
he used an \ii{interferometer} to produce bright and dark fringes of
light\seepagefour{lightnotbend} with two light beams, one circulating in
clockwise direction, and the second circulating in anticlockwise direction.
The interference fringes are \emph{shifted} when the whole system rotates; the
faster it rotates, the larger is the shift. A modern, highprecision
version\cite{wettzellref} of the experiment, which uses lasers instead of
lamps, is shown in \figureref{ilasergyrointerf}.
%
(More details on interference and fringes are found
in\seepagethree{ilibosphotovol3} volume III.)
%
Sagnac also determined the relation between the fringe shift and the
details of the experiment. The rotation of a complete ring
interferometer with angular frequency (vector) $\bm{\Omega}$ produces
a fringe shift of angular phase $\Delta \phi$ given
by\challengenor{sagnac}
\begin{equation}
\Delta \phi =
\frac{8 \pi \; \bm{\Omega} \, {\bm a} }{ c \,\lambda}
\label{eq:sagnac}
\end{equation}
where ${\bm a}$ is the area (vector) enclosed by the two interfering light
rays, $\lambda$ their wavelength and $c$ the speed of light. The effect
is now called the \ii{Sagnac effect} after its discoverer. It had already
been predicted 20 years earlier by\cite{a55} \iinns{Oliver Lodge}.%
%
\footnote{Oliver Lodge \livedplace(1851, StokeonTrent1940,
Wiltshire) was a physicist and spiritualist who studied
electromagnetic waves and tried to communicate with the dead. A
strange but influential figure, his ideas are often cited when fun
needs to be made of physicists; for example, he was one of those
(rare) physicists who believed
that at the end of the nineteenth century physics was complete.} %
%
% This is partly in c2d:
%
%\footnote{At the end of the nineteenth century, both Michelson and Lodge 
% mainly
%experimental physicists  claimed that electrodynamics and Galilean physics
%meant that the major laws of physics were well known. This contrasts nicely
%with Kelvin, who had spotted two major problems which later would lead to
%relativity and quantum theory. Ironically, the results of Lodge and
% Michelson
%were important} %
%
% % Also for a fixed interferometer, Michelson and his team found a fringe shift
% % with a period of 24 hours and of exactly the magnitude predicted by the
% % rotation of the Earth with equation (\ref{eq:sagnac}).
Today, Sagnac interferometers are the central part of \iin{laser gyroscopes}
 shown in \figureref{igyro4}  and\index{gyroscope!laser} are found in
every passenger aeroplane, missile and submarine, in order to measure the
changes of their motion and thus to determine their actual position.
% Improved Feb 2010
A part of the fringe shift is due to the rotation of the Earth. Modern
highprecision Sagnac interferometers use ring lasers with areas of a few
square metres, as shown in \figureref{ilasergyrointerf}. Such a ring
interferometer is able to measure variations of the rotation rates of the
Earth of less than one part per million. Indeed, over the course of a year,
the length of a day varies irregularly by a few milliseconds, mostly due to
influences from the Sun or the Moon, due to weather changes and due to hot
magma flows deep inside the Earth.%
%
\footnote{% Jun 2005
The growth of \iin[tree!leaves and Earth rotation]{leaves on trees} and the
consequent change in the Earth's moment of inertia, already thought of in
1916 by \iinn{Harold Jeffreys}, is way too small to be seen, as it is hidden
by larger effects.}\cite{dayle}
%
% Added Apr 2005
But also earthquakes, the El Ni\~no effect in the climate and the filling of
large water dams have effects on the rotation of the Earth.
%
All these effects can be studied with such highprecision interferometers;
they can also be used for research into the motion of the soil due to lunar
tides or earthquakes, and for checks on the theory of special relativity.
% May 2007
\cssmallepsf{ibuckarot}{scale=1}{Observing the rotation of the Earth in two
seconds.}
% May 2007
Finally, in 1948, \iinn{Hans Bucka} developed the simplest experiment so far
to show the rotation of the Earth.\cite{buckarot} A metal rod allows anybody
to detect the rotation of the Earth after only a few seconds of observation,
using the setup of \figureref{ibuckarot}. The experiment can be easily be
performed in class. Can you guess how it works?\challengenor{buckaexp}
In summary: all observations\index{Earth!rotation speed} show that the
Earth surface rotates at \csd{464}{m/s} at the Equator, a larger value
than that of the \iin{speed of sound} in air, which is about
\csd{340}{m/s} at usual conditions. The rotation of the Earth also
implies an acceleration, at the Equator, of \csd{0.034}{m/s^2}. We
are in fact \emph{whirling} through the universe.
%
%
% Reread Jul 2016
\subsection{How does the Earth rotate?}
% Index OK
Is the rotation of the Earth, the length of the day,\index{day!length
of} \emph{constant} over geological time scales? That is a hard
question. If you find a method leading\index{rotation!change of
Earth}\index{Earth!rotation!change of} to an answer, publish it! (The
same is true for the question whether the length of the year is
constant.)\cite{dayyear} Only a few methods are known, as we will find
out shortly.
The rotation of the Earth is not even constant during a human lifespan. It
varies by a few parts in $10^8$. In particular, on a `secular' time scale,
the length of the day increases by about 1 to \csd{2}{ms} per century, mainly
because of the friction by the Moon and the melting of the polar ice caps.
This was deduced by studying historical astronomical observations of the
ancient Babylonian and Arab astronomers.\cite{babi}
% others say: 2500 years ago, 50 ms longer. But I think they made the
% standard mistake of assuming linear relationship
Additional `decadic' changes have an amplitude of 4 or \csd{5}{ms} and are due
to the motion of the liquid part of the Earth's core.
%
(The centre of the Earth's core is solid; this was discovered in 1936
by the Danish
% female
seismologist \iinn{Inge Lehmann} \lived(18881993); her discovery was
confirmed most impressively by two
% male % for sure
British seismologists in 2008, who detected shear waves of the inner
core, thus confirming Lehmann's conclusion. There is a liquid core
around the solid core.)
The seasonal and biannual changes of the length of the day  with an
amplitude of \csd{0.4}{ms} over six months, another \csd{0.5}{ms} over the
year, and \csd{0.08}{ms} over 24 to 26 months  are mainly due to the effects
of the \emph{atmosphere}. In the 1950s the availability of precision
measurements showed that there is even a 14 and 28 day period with an
amplitude of \csd{0.2}{ms}, due to the Moon. In the 1970s, when \emph{wind
oscillations} with a length scale of about 50 days were discovered, they
were also found to alter the length of the day, with an amplitude of about
\csd{0.25}{ms}. However, these last variations are quite irregular.
% Aug 2009
Also the oceans influence the rotation of the Earth, due to the tides, the
ocean currents, wind forcing, and atmospheric pressure forcing.\cite{bchao}
Further effects are due to the ice sheet variations and due to water
evaporation and rain falls. Last but not least, flows in the interior of the
Earth, both in the mantle and in the core, change the rotation. For example,
earthquakes, plate motion, postglacial rebound and volcanic eruptions all
influence the rotation.
% Aug 2009, Oct 2013
But why does the Earth rotate at all? The rotation originated in the rotating
gas cloud at the origin of the Solar System.\index{Solar System!formation}
This connection explains that the Sun and all planets, except two, turn around
their axes in the same direction, and that they also all orbit the Sun in that
same direction. But the complete story\cite{ssrot} is outside the scope of
this text.
\csepsfnb{iprecession}{scale=1}{The precession and the nutation of the
Earth's axis.}
The rotation\label{moonprecsn} around its axis is not the only motion of the
Earth; it performs other motions as well. This was already known long ago.
In 128 {\bce}, the Greek astronomer \iname{Hipparchos} discovered what is
today called the \ii[precession!equinoctial]{(equinoctial) precession}. He
compared a measurement he made himself with another made 169 years before.
Hipparchos found that the Earth's axis points to different stars at different
times. He concluded that the sky was moving.\index{Vega!at the North
pole}\index{sky!moving}
%
% around the ecliptic pole, says the web
%
Today we prefer to say that the axis of the Earth is moving. (Why?)\challengn
During a period of 25\,800 years % says the wikipedia
% insolation changes differently,
% 23,700 and 22,400 and 19,000 years (see below)
the axis draws a cone with an opening angle of \csd{23.5}{\csdegrees}. This
motion, shown in \figureref{iprecession}, is generated by the tidal forces of
the Moon and the Sun on the equatorial bulge of the Earth that results form
its flattening. The Sun and the Moon try to align the axis of the Earth at
right angles to the Earth's path; this torque leads to the precession of the
Earth's axis.
\csmpgfilmrepeat{gyroprec}{scale=1}{Precession of a suspended spinning top
(mpg film {\textcopyright}~\protect\iinn{Lucas %~V.
Barbosa})}
% EMAILED FEB 2008  dnukem@gmail.com
% Aug 2007
Precession is a motion common to all rotating systems: it appears in planets,
spinning tops and atoms. (Precession is also at the basis of the surprise
related to the suspended wheel shown on \cspageref{ihanged}.) % this vol I
Precession is most easily seen in spinning tops, be they suspended or not. An
example is shown in \figureref{gyroprec}; for atomic nuclei or planets, just
imagine that the suspending wire is missing and the rotating body less
flattened.
% Nov 2013 (Nature, 12 Sept 2013)
On the Earth, precession leads to upwelling of deep water in the equatorial
Atlantic Ocean and regularly changes the ecology of algae.
% Sep 2007, Jul 2016
\csepsfnb{ipolarmotion}{scale=1}{The motion of the North Pole  roughly
speaking, the Earth's effective \protect\ii{polhode}  from 2003 to 2007,
including the prediction until 2008 (left) and the average position since
1900 (right)  with 0.1\,arcsecond being around 3.1\,m on the surface of
the Earth  not showing the diurnal and semidiurnal variations of a
fraction of a millisecond of arc due to the tides (from
\protect\url{hpiers.obspm.fr/eoppc}).}
In addition, the \iin[axis!Earth's, motion of]{axis of the Earth} is not even
fixed relative to the Earth's surface. In 1884, by measuring the exact angle
above the horizon of the celestial North Pole, \iinns{Friedrich Küstner}
\lived(18561936) found that the axis of the Earth \emph{moves} with respect
to the Earth's crust, as \iname{Bessel} had suggested 40 years earlier. As a
consequence of Küstner's discovery, the \iin{International Latitude Service}
was created. The \ii[motion!polar]{polar motion} Küstner discovered turned
out to consist of three components: a small linear drift  not yet understood
 a yearly elliptical motion due to seasonal changes of the air and water
masses, and a circular motion%
%
% impr. Jan 2015
\footnote{The circular motion, a wobble, was predicted by the great
Swiss mathematician \iinn{Leonhard Euler}
\lived(17071783).\index{wobble!Euler's}\index{Euler's
wobble}\index{wobble!falsely claimed by Chandler} In a
disgusting story, using Euler's and Bessel's predictions and
Küstner's data, in 1891 \iinn{Seth Chandler}
%\iinn{Seth~Carlo Chandler} %\lived(18461913)
claimed to be the discoverer of the circular component.} %
%
with a period of about 1.2 years due to fluctuations in the pressure at the
bottom of the oceans. In practice, the \iin{North Pole} moves with an
amplitude of about \csd{15}{m} around an average central position, as shown in
\figureref{ipolarmotion}.\cite{axismotion} Short term variations of the
North Pole position, due to local variations in atmospheric pressure, to
weather change and to the tides, have also been measured.\cite{poleshortterm}
%
% it is 435 days and 0.5 arc seconds
%
% Aug 2009, Dec 2014
The high precision of the \csac{GPS} system is possible only with the help of
the exact position of the Earth's axis; and only with this knowledge can
artificial satellites be guided to Mars or other planets.
% Mar 2012
The details of the motion of the Earth have been studied in great detail.
\tableref{Eamotafasciul} gives an overview of the knowledge and the precision
that is available today.
% Source added in Oct 2014
\cssmallepsfnb{iplate}{scale=0.4}{The continental plates are the objects of
tectonic\protect\index{tectonics} motion (HoloGlobe project, NASA).}
% http://www.animatedearth.com/et/archivalHoloGlobe/HoloGlobeSourceInformation.html
%
% See also http://svs.gsfc.nasa.gov/cgibin/details.cgi?aid=1288
%
% There is even a film of a rotating Earth
In 1912, the % German
meteorologist and geophysicist \iinn{Alfred Wegener} \lived(18801930)
discovered an even larger effect. After studying the shapes of the
continental shelves and the geological layers on both sides of the Atlantic,
he conjectured that the \iin[continent!motion of]{continents} \emph{move}, and
that\index{motion!of continents} they are all fragments of a single continent
that broke up 200 million years ago.%
%
% Jun 2005, impr. Jan 2015
\footnote{In this old continent, called \iin{Gondwanaland}, there was a huge
river that flowed westwards from the Chad to Guayaquil in Ecuador. After the
continent split up, this river still flowed to the west. When the Andes
appeared, the water was blocked, and many millions of years later, it
reversed its flow. Today, the river still flows eastwards: it is called the
\iin{Amazon} River.}
% Sep 2016
\csepsfnb{itectonicplates}{scale=1}{The tectonic plates of the Earth, with
the relative speeds at the
boundaries. ({\textcopyright}~\protect\iname{NASA})}
Even though at first derided across the world, Wegener's discoveries were
correct. Modern satellite measurements, shown in \figureref{iplate}, confirm
this model. For example, the American continent moves away from the European
continent by about \csd{23}{mm} every year, as shown in
\figureref{itectonicplates}. There are also speculations that this velocity
may have been much higher at certain periods in the past. The way to check
this is to look at the magnetization of sedimental
rocks.\index{rock!magnetization}\index{magnetization!of rocks} At present,
this is still a hot topic of research. Following the modern version of the
model, called \ii{plate tectonics}, the continents (with a density of
\csd{2.7\cdot 10^3}{kg/m^3}) float\seepageone{contswim} on the fluid mantle of
the Earth (with a density of \csd{3.1\cdot 10^3}{kg/m^3}) like pieces of cork
on water,\seepagethree{earthstru} and the convection inside the mantle
provides the driving mechanism for the\index{rotation!of the
Earth)}\index{Earth!rotation)} motion.\cite{platetect}
%
% Sep 2007
% \subsubsubsubsubsubsubsubsection{Earth data}
%
{\small
\begin{table}[tp]
\small \caption{Modern measurement\protect\index{Earth!rotation data, table}
data about the motion of the Earth (from
\protect\url{hpiers.obspm.fr/eoppc}).}
\label{Eamotafasciul} %
\dirrtabularstarnolines
\begin{tabular*}{\textwidth}{@{}
>{\PBS\raggedright} p{67 mm} @{\extracolsep{\fill}}
>{\PBS\raggedright} p{21.4mm} @{\extracolsep{\fill}}
% 21 is minimum
% In Jan 2015 I changed H to h (dyn falttening) to gain space
>{\PBS\raggedright} p{45.8mm} @{}}
% 45 was too small by .9 pt
% In Jan 2015, 45.5. was too small by 1 pt
%
\toprule
%
\tabheadf{Observable} & \tabhead{Sym\bol} & \tabhead{Value} \\
%
\midrule
%
Mean angular velocity of Earth & $\Omega$ & \csd{72.921\,150(1)}{\muunit
rad/s} \\
%
Nominal angular velocity of Earth (epoch 1820) & $\Omega_{\rm N}$ &
\csd{72.921\,151\, 467\,064}{\muunit rad/s} \\
%
Conventional mean solar day\index{day!mean solar} (epoch 1820) & d
& \csd{86\,400}{s} \\
%
Conventional sidereal day\index{day!sidereal} & d${}_{\rm si}$ &
\csd{86\,164.090\,530\,832\,88}{s} \\
%
Ratio conv.~mean solar day to conv.~sidereal day& $k={\rm d}/{\rm d}_{\rm
si}$ &
1.002\,737\,909\,350\,795 \\
%
Conventional duration of the stellar day &
d${}_{\rm st}$ & \csd{86\,164.098\,903\,691}{s} \\
%
Ratio conv.~mean solar day to conv.~stellar day& $k^\prime={\rm d}/{\rm
d}_{\rm st}$ & 1.002\,737\,811\,911\,354\,48 \\
%
General precession in longitude& $p$ &
\csd{5.028\,792(2)}{\csseconds/a} \\
%
Obliquity of the ecliptic (epoch 2000) & $\e_{0}$ &
$23\,\csdegrees\, 26\,\csminutes\, 21.4119\,\csseconds$ \\
%
KüstnerChandler period in terrestrial frame& $T_{\rm KC}$ &
\csd{433.1(1.7)}{d} \\
%
Quality factor of the KüstnerChandler peak& $Q_{\rm KC}$ &
170 \\
%
Free core nutation period in celestial frame& $T_{\rm F}$ &
\csd{430.2(3)}{d} \\
%
Quality factor of the free core nutation & $Q_{\rm F}$ &
$2\cdot 10^4$ \\
%
Astronomical unit & AU & \csd{149\,597\,870.691(6)}{km} \\
%
Sidereal year (epoch 2000) & $a_{\rm si}$ & \csd{365.256\,363\,004}{d}
$=365\,{\rm d}\,
6\,{\rm h}\, 9\,{\rm min}\,
9.76\,{\rm s}$ \\
%
Tropical year & $a_{\rm tr}$ & \csd{365.242\,190\,402}{d} $=365\,{\rm d}\,
5\,{\rm h}\, 48\,{\rm min}\,
45.25\,{\rm s}$ \\
%
Mean Moon period & $T_{\rm M}$ & \csd{27.321\,661\,55(1)}{d} \\
%
Earth's equatorial radius & $a$ & \csd{6\, 378\, 136.6(1)}{m} \\
%
First equatorial moment of inertia & $A$ & \csd{8.0101(2)\cdot
10^{37}}{kg\,m^2} \\
%
Longitude of principal inertia axis $A$ & $\lambda_{A}$ &
\csd{14.9291(10)}{\csdegrees} \\
%
Second equatorial moment of inertia & $B$ & \csd{8.0103(2)\cdot
10^{37}}{kg\,m^2} \\
%
Axial moment of inertia & $C$ & \csd{8.0365(2)\cdot 10^{37}}{kg\,m^2} \\
%
Equatorial moment of inertia of mantle & $A_{\rm m}$ &
\csd{7.0165\cdot 10^{37}}{kg\,m^2} \\
%
Axial moment of inertia of mantle & $C_{\rm m}$ &
\csd{7.0400\cdot 10^{37}}{kg\,m^2} \\
%
Earth's flattening & $f$ & $1/298.25642(1)$ \\
%
Astronomical Earth's dynamical flattening & $h=(CA)/C$ &
$0.003\,273\,794\,9(1)$\\
%
Geophysical Earth's dynamical flattening & $e=(CA)/A$ &
$0.003\,284\,547\,9(1)$\\
%
Earth's core dynamical flattening & $e_{\rm f}$ & $0.002\,646(2)$\\
%
Second degree term in Earth's gravity potential
& $J_{\rm 2}=(A+B2C)/(2MR^2)$ &
$1.082\,635\,9 (1) \cdot 10^{3}$ \\
%
Secular rate of $J_{\rm 2}$
& $\diffd J_{\rm 2}/ \diffd t$ &
\csd{2.6(3) \cdot 10^{11}}{/a} \\
%
Love number (measures shape distortion by tides) & $k_{\rm 2}$ & 0.3 \\
%
Secular \iin{Love number} & $k_{\rm s}$ & 0.9383 \\
%
Mean equatorial gravity & $g_{\rm eq}$ & \csd{9.780\,3278 (10)}{m/s^2} \\
%
Geocentric constant of gravitation & $GM$ &
\csd{3.986\,004\,418(8)\cdot10^{14}}{m^3/s^2} \\
%
Heliocentric constant of gravitation & $GM_{\cssunsymbol}$ &
\csd{1.327\,124\,420\,76(50)\cdot10^{20}}{m^3/s^2} \\
%
MoontoEarth mass ratio & $\mu$ &
0.012\,300\,038\,3(5) \\
%
\bottomrule
\end{tabular*}
\end{table}
}
% Oct 2008
\cssmallepsfnb{isunsizechange}{scale=1.35}{The angular size of the Sun
changes due to the elliptical motion of the Earth
({\textcopyright}~\protect\iinn{Anthony Ayiomamitis}).}
% EMAILED FEB 2008  OK for book , also his other ones
%
% Reread July 2016
\subsection{Does the Earth move?}
% May 2005
\cssmallepsfnb{ibessel}{scale=0.4}{Friedrich~Wilhelm Bessel
\livedfig(17841846).}
Also the centre of the Earth is not at rest in the universe. In the third
century {\bce} \iname{Aristarchus of Samos} maintained that the Earth turns
around the Sun. Experiments such as that of \figureref{isunsizechange}
confirm that the orbit is an ellipse. However, a fundamental difficulty of
the heliocentric system is that the stars look the same all year long. How
can this be, if the Earth travels around the Sun?\label{besselbio} The
distance between the Earth and the Sun has been known since the seventeenth
century, but it was only in 1837 that
%\comment{others say
%in 1838, others 1839  I remember vaguely that he published 2 years later}
\iinns{Friedrich~Wilhelm Bessel}%
%
\footnote{Friedrich~Wilhelm Bessel \lived(17841846), Westphalian astronomer
who left a successful business career to dedicate his life to the stars, and
became the foremost astronomer of his time.} %
%
became the first person to observe the \ii{parallax} of a star. This was a
result of extremely careful measurements and complex calculations: he
discovered the \ii{Bessel functions} in order to realize it. He was able to
find a star, 61 Cygni, whose apparent position changed with the month of the
year. Seen over the whole year, the star describes a small ellipse in the
sky, with an opening of
\csd{0.588}{\csseconds} % (OK) check typesetting, used to be
% {}^{\csprime\csprime}
(this is the modern value). After carefully
eliminating all other possible explanations, he deduced that the change of
position was due to the motion of the Earth around the Sun, when seen
from distant stars. From the size
of the ellipse he determined the distance to the star to be \csd{105}{Pm}, or
11.1 light years.\challengenor{bessol}
% Impr. Jan 2015
Bessel had thus managed, for the first time, to measure the distance
of a star. By doing so he also proved that the Earth is not fixed
with respect to the stars in the sky. The motion of the Earth was not
a surprise. It confirmed the result of the mentioned \iin{aberration}
of light, discovered in 1728 by\label{bradleybio} \iinns{James
Bradley}%
%
% Impr. Sep 2016
%
\footnote{James Bradley \livedplace(1693 Sherborne  1762 Chalford), was an
important astronomer. He was one of the first astronomers to understand the
value of precise measurement, and thoroughly modernized the Greenwich
observatory. He discovered, independently of \iinn{Eustachio Manfredi}, the
aberration of light, and showed with it that the Earth moves. In particular,
the discovery allowed him to measure the speed of light and confirm the
value of \csd{0.3}{Gm/s}. He later discovered the nutation of the Earth's
axis.}
%
and to be discussed below.\seepagetwo{aberrrr} When seen from the sky, the
Earth indeed revolves around the Sun.
\csepsf{ieaorbit}{scale=1}{Changes in the Earth's
motion\protect\index{Earth!motion around Sun} around the Sun, as seen from
different observers outside the orbital plane.}
With the improvement of telescopes, other motions of the Earth were
discovered. In 1748, \iinn{James Bradley} announced that there is a small
regular \emph{change} of the precession, which he called \ii{nutation}, with a
period of 18.6 years and an angular amplitude of \csd{19.2}{\csseconds}.
Nutation occurs because the plane of the Moon's orbit around the Earth is not
exactly the same as the plane of the Earth's orbit around the Sun. Are you
able to confirm that this situation produces nutation?\challengn
Astronomers also discovered that the \csd{23.5}{\csdegrees} tilt  or
\ii{obliquity}  of the \iin[Earth!axis tilt]{Earth's axis}, the angle
between its intrinsic and its orbital angular momentum, actually changes from
\csd{22.1}{\csdegrees} to
\csd{24.5}{\csdegrees} % a source says stabilised by the Moon
with a period of 41\,000 years.\index{axis!Earth's, precession} This motion is
due\index{precession!Earth's axis} to the attraction of the Sun and the
deviations of the Earth from a spherical shape.
%
In 1941, during the Second World War, the Serbian astronomer \iinns{Milutin
Milankovitch} \lived(18791958) retreated into solitude and explored the
consequences. In his studies he realized that this 41\,000 year period of the
obliquity, together with an average period of 22\,000 years due to
precession,%
%
\footnote{In fact, the 25\,800 year \iin{precession} leads to three insolation
periods, of 23\,700, 22\,400 and 19\,000 years,
due to the interaction between precession and perihelion shift.} %
% www.soc.soton.ac.uk/soes/staff/ejr/DarkMed/ch5.html
%
gives rise to the more than 20 \ii{ice ages} in the last 2 million years.
This happens through stronger or weaker irradiation of the poles by the Sun.
The changing amounts of melted ice then lead to changes in average
temperature. The last ice age had its peak about 20\,000 years ago and ended
around 11\,800 years ago; the next is still far away.
% Improved Nov 2008
A spectacular
confirmation of the relation between ice age cycles and astronomy
% in addition to the many geological proofs,
came through measurements of oxygen isotope ratios in ice cores and sea
sediments, which allow the average temperature over the past million years to
be tracked.\cite{icetrio}
% Nov 2008
\figureref{imilanko} shows how closely the temperature follows the changes in
irradiation due to changes in obliquity and precession.
% lasted from 120\,000 to 10\,000, says Physics update, jan 2002
% 18\,000 from a New Scientist article in February 1998
% 22\,000 elsewhere
% 60\,000 from a graph
% The web is full of pages on the ice ages!
% Nov 2008
\csepsf{imilanko}{scale=1}{Modern measurements showing how Earth's precession
parameter (black curve A) and obliquity (black curve D) influence the
average temperature (coloured curve B) and the irradiation of the Earth
(blue curve C) over the past 800\,000 years: the obliquity deduced by
Fourier analysis from the irradiation data RF (blue curve D) and the
obliquity deduced by Fourier analysis from the temperature (red curve D)
match the obliquity known from astronomical data (black curve D); sharp
cooling events took place whenever the obliquity rose while the precession
parameter was falling (marked red below the temperature curve)
({\textcopyright}~\protect\iinn{Jean Jouzel}{/}\protect\iname{Science} from
\protect\citen{icetrio}).}
The Earth's orbit also changes its \emph{eccentricity} with time, from
completely circular to slightly oval and back.\index{eccentricity!of Earth's
axis} However, this happens in very complex ways, not with periodic
regularity, and is due to the influence of the large planets of the solar
system on the Earth's orbit. The typical time scale is 100\,000 to 125\,000
years.
% www.hampsteadscience.ac.uk/HSS_Apr_2005.htm
In addition, the Earth's orbit changes in \emph{inclination} with respect to
the orbits of the other planets; this seems to happen regularly every 100\,000
years.\index{inclination!of Earth's axis} In this period the inclination
changes from \csd{+2.5}{\csdegrees} to \csd{2.5}{\csdegrees}
and back. %!.!3 Why? I have not found out yet (Nov 2008, June 2010)
\cssmallepsfnb{igalaxymotion}{scale=1}{The motion of the Sun around the
galaxy.}
% There is a problem with the size of the image here; now solved.
Even the direction in which the ellipse points changes with time. This
socalled \ii[perihelion!shift]{perihelion shift} is\index{shift!perihelion}
due in large part to the influence of the other planets; a small remaining
part will be important in the chapter on general relativity. The perihelion
shift of Mercury was the first piece of data confirming Einstein's theory.
Obviously, the length of the year also changes with time. The measured
variations are of the order of a few parts in $10^{11}$ or about \csd{1}{ms}
per year. However, knowledge of these changes and of their origins is much
less detailed than for the changes in the Earth's rotation.
% improved April 2005
The next step is to ask whether the Sun itself moves. Indeed it does.
Locally, it moves with a speed of \csd{19.4}{km/s} towards the constellation
of Hercules. This was shown by \iinn{William Herschel} in 1783. But
globally, the motion is even more interesting. The diameter of the galaxy is
at least 100\,000 light years,\index{galaxy!and Sun} and we are located
26\,000 light years from the centre. (This has been known since 1918; the
\iin[galaxy!centre]{centre of the galaxy} is located in the direction of
\iin{Sagittarius}.) At our position, the galaxy is 1\,300 light years thick;
presently, we are 68 light years `above' the centre plane.\cite{galwh} The
Sun, and with it the Solar System, takes about 225 million years to turn once
around the galactic centre,\index{Solar System!motion} its orbital velocity
being around \csd{220}{km/s}. It seems that the Sun will continue moving away
from the galaxy plane until it is about 250 light years above the plane, and
then move back, as shown in \figureref{igalaxymotion}. The oscillation
period is estimated to be around 62 million years, and has been suggested as
the mechanism for the mass extinctions of animal life on Earth, possibly
because some gas cloud or some cosmic radiation source may be periodically
encountered on the way. The issue is still a hot topic of research.
% June 2014
\csmovfilm{SolarVortex}{scale=0.398622}{The helical motion of the first four
planets around the path traced by the Sun during its travel around the centre
of the Milky Way. Brown: Mercury, white: Venus, blue: Earth, red: Mars.
(QuickTime film {\textcopyright}~\protect\iinn{Rhys Taylor} at
\protect\url{www.rhysy.net}).}
% June 2014
The motion of the Sun around the centre of the Milky Way implies that the
planets of the Solar System can be seen as forming helices around the
Sun.\index{Solar System!as helix}\index{helix!in Solar System}
\figureref{SolarVortex} illustrates their helical path.
We turn around the galaxy centre because the formation of galaxies, like that
of planetary systems, always happens in a \iin{whirl}. By the way, can you
confirm from your own observation that\index{galaxy!rotation} our galaxy
itself rotates?\challengenor{galflat}
Finally, we can ask whether the galaxy itself moves. Its motion can indeed be
observed because it is possible to give a value for the motion of the Sun
through the universe, defining it as the motion against the background
radiation. This value has been measured to be \csd{370}{km/s}.\cite{cobemeas}
(The velocity of the \emph{Earth} through the background radiation of course
depends on the season.)\index{Earth!speed through the universe} This value
is a combination of the motion of the Sun around the galaxy centre and of the
motion of the galaxy itself. This latter motion is due to the gravitational
attraction of the other, nearby galaxies in our local group of
galaxies.\footnote{This is roughly the end of the ladder. Note that the
\iin[expansion!of the universe]{expansion of the universe}, to be studied
later, produces no motion.}
% Oct 2016, have permission per email
\csepsfnb{isnowmotion}{scale=1}{Driving\protect\index{motion!relative,
through snow} through snowflakes shows the effects of relative motion in
three dimensions. Similar effects are seen when the Earth speeds through the
universe.\protect\seepageone{iperseids}
({\textcopyright}~\protect\iinn{Neil Provo} at
\protect\url{neilprovo.com}).}
In summary, the Earth really moves, and it does so in rather complex ways. As
\iinn{Henri Poincaré} would say, if we are in a given spot today, say the
Panthéon in Paris, and come back to the same spot tomorrow at the same time,
we are in fact 31 million kilometres away. This state of affairs would make
\iin[time!travel, difficulty of]{time travel} extremely difficult even if it
were possible (which it is not); whenever you went back to the past, you would
have to get to the old spot exactly!
% We stop this discussion at this point
% have a look at motion in everyday life.
%
% Jan 2005, July 2016
\subsection{Is velocity absolute?  The theory of everyday relativity}
% Index OK
% \csepsf{igalileobarca}{scale=1}{Galileo's discussion of the relativity of
% rest}  I THINK I CONFUSED THIS WITH THE HUYGENS DRAWING OF MOMENTUM
% CONSERVATION?
Why don't we feel\index{relativity!and ships} all the motions of the
Earth?\label{galprrel} The two parts of the answer were already given in 1632.
First of all, as \iname[Galilei, Galileo]{Galileo}
explained, % I hope he really did I am just guessing here!
we do not feel the accelerations of the Earth because the effects they produce
are too small to be detected by our senses. Indeed, many of the mentioned
accelerations\label{eamot} do induce measurable effects only in highprecision
experiments, e.g.~in atomic clocks.\seepagetwo{eamot2}
But the second point made by Galileo is equally important: it is impossible to
feel the high speed at which we are moving. We do not feel translational,
unaccelerated motions because this is impossible \emph{in principle}. Galileo
discussed the issue by comparing the observations of two observers: one on the
ground and another on the most modern means of unaccelerated transportation of
the time, a ship.
% The illustration
% from his discussion is shown in \figureref{igalileobarca}.
Galileo asked whether a man on the ground and a man in a ship\index{ship!and
relativity} moving at constant speed experience (or `feel') anything
different. Einstein used observers in trains. Later it became fashionable to
use travellers in rockets. (What will come next?)\challengn {Galileo}
explained that only \emph{relative} velocities between bodies produce effects,
not the absolute values of the velocities. For the senses and for all
measurements we find:
%
\begin{quotation}
\noindent \csrhd {There is no difference between constant, undisturbed
motion, however rapid it may be, and rest. This is called
\ii[relativity!Galileo's principle of]{Galileo's principle of
relativity}.}
\end{quotation}
%
Indeed, in everyday\index{relativity!Galilean} life we feel motion only if the
means of transportation trembles  thus if it accelerates  or if we move
against the air. Therefore Galileo concludes that two observers in straight
and undisturbed motion against each other cannot say who is `really' moving.
Whatever their relative speed, neither of them `feels' in motion.%
%
% Cited from the wikipedia: https://en.wikipedia.org/wiki/Galileo%27s_ship
%
\footnote{In 1632, in his \bt Dialogo/ Galileo writes: `Shut yourself up with
some friend in the main cabin below decks on some large ship, and have with
you there some flies, butterflies, and other small flying animals. Have a
large bowl of water with some fish in it; hang up a bottle that empties drop
by drop into a wide vessel beneath it. With the ship standing still, observe
carefully how the little animals fly with equal speed to all sides of the
cabin. The fish swim indifferently in all directions; the drops fall into the
vessel beneath; and, in throwing something to your friend, you need throw it
no more strongly in one direction than another, the distances being equal:
jumping with your feet together, you pass equal spaces in every direction.
When you have observed all these things carefully (though there is no doubt
that when the ship is standing still everything must happen in this way), have
the ship proceed with any speed you like, so long as the motion is uniform and
not fluctuating this way and that, you will discover not the least change in
all the effects named, nor could you tell from any of them whether the ship
was moving or standing still. In jumping, you will pass on the floor the same
spaces as before, nor will you make larger jumps toward the stern than toward
the prow even though the ship is moving quite rapidly, despite the fact that
during the time you are in the air the floor under you will be going in a
direction opposite to your jump. In throwing something to your companion, you
will need no more force to get it to him whether he is in the direction of the
bow or the stern, with yourself situated opposite. The droplets will fall as
before into the vessel beneath without dropping toward the stern, although
while the drops are in the air the ship runs many spans. The fish in their
water will swim toward the front of their bowl with no more effort than toward
the back, and will go with equal ease to bait placed anywhere around the edges
of the bowl. Finally the butterflies and flies will continue their flights
indifferently toward every side, nor will it ever happen that they are
concentrated toward the stern, as if tired out from keeping up with the course
of the ship, from which they will have been separated during long intervals by
keeping themselves in the air. And if smoke is made by burning some incense,
it will be seen going up in the form of a little cloud, remaining still and
moving no more toward one side than the other. The cause of all these
correspondences of effects is the fact that the ship's motion is common to all
the things contained in it, and to the air also. That is why I said you
should be below decks; for if this took place above in the open air, which
would not follow the course of the ship, more or less noticeable differences
would be seen in some of the effects noted.' (Translation by Stillman Drake)} %
%
\emph{Rest is relative.}\index{rest!is relative} Or more clearly: rest is an
observerdependent concept. This result of Galilean physics is so important
that Poincaré introduced the expression `theory of relativity' and Einstein
repeated the principle explicitly when he published his famous theory of
special relativity. However, these names are awkward. Galilean physics is
also a theory of relativity! The relativity of rest is common to \emph{all}
of physics; it is an essential aspect of motion.
In summary, undisturbed or uniform motion has no observable effect; only
\emph{change} of motion does. Velocity cannot be felt; acceleration can. As
a result,\label{wittgsonne} every physicist can deduce something simple about
the following statement by Wittgenstein:\indname{Wittgenstein, Ludwig}
\begin{quotation}
{\np Da\ss\ die Sonne morgen aufgehen wird, ist eine Hypothese; und das
hei{\ss}t: wir \emph{wissen} nicht,
ob sie aufgehen wird.%
%
\footnote{`That the Sun will rise tomorrow, is an hypothesis; and that means
that we do not \emph{know} whether it will rise.' This wellknown statement
is found in Ludwig Wittgenstein, \bt Tractatus/ 6.36311.} % Odgen translation
%
} \end{quotation}
\np The statement is \emph{wrong}. Can you explain why Wittgenstein erred
here, despite his strong desire\challengenor{wrongwitt} not to?
%
% Impr. Jul 2016
\subsection{Is rotation relative?}
% Index OK
% Jul 2005
When we turn\index{relativity!of rotation} rapidly, our arms lift. Why does
this happen? How can our body detect whether we are rotating or
not?\index{rotation!and arms} There are two possible answers. The first
approach, promoted by Newton, is to say that there is an absolute space;
whenever we rotate against this space, the system reacts. The other answer is
to note that whenever the arms lift, the stars also rotate, and in exactly the
same manner. In other words, our body detects rotation because we move
against the average mass distribution in space.
% Jul 2005, impr. Jan 2015
The most cited discussion of this question is due to Newton. Instead of arms,
he explored the water in a rotating bucket.\index{bucket!experiment, Newton's}
In a rotating bucket, the water surface forms a concave shape, whereas the
surface is flat for a nonrotating bucket. Newton asked why this is the case.
As usual for philosophical issues, Newton's answer was guided by the mysticism
triggered by his father's early death. Newton saw absolute space as a mystical
and religious concept and was not even able to conceive an alternative.
Newton thus saw rotation as an \emph{absolute} type of motion. Most modern
scientists have fewer personal problems and more common sense than Newton; as
a result, today's consensus is that rotation effects are due to the mass
distribution in the universe:
%
\begin{quotation}
\npcsrhd Rotation is relative.\index{rotation!absolute or relative}
\end{quotation}
%
\np A number of highprecision experiments confirm this conclusion; thus it is
also part of Einstein's theory of relativity. % !!!3 ref
%
% Impr. Mar 2015
\subsection{Curiosities and fun challenges about rotation and relativity}
% Index ok
\begin{curiosity}
% Dec 2005
\item[] When\index{relativity!challenges} travelling in the train, you can
test Galileo's statement about everyday relativity of
motion.\index{train!motion puzzle} Close your eyes and ask somebody to turn
you around several times: are you able to say in which direction the train
is running?\challengn
\item A good \iin{bathroom scales},\index{scales!bathroom} % (OK) scale or
% scales?
used to determine the weight of objects, does not show a constant weight when
you step on it and stay motionless. Why not?\challengenor{scalenotconst}
\item If a gun located at the Equator shoots\index{canon!puzzle} a bullet
vertically, where does the bullet fall?\challengenor{verticalcannon}
\item Why are most \iin[rocket!launch site puzzle]{rocket launch sites} as
near as possible to the Equator?\challengenor{rocketequator}
% Sep 2011
\item At the Equator, the speed of rotation of the Earth is \csd{464}{m/s}, or
about Mach 1.4; the\index{Mach number} latter number means that it is 1.4
times the speed of sound. This supersonic motion has two intriguing
consequences.
% Sep 2011, impr. Mar 2015
% http://en.wikipedia.org/wiki/Rossby_radius_of_deformation
%
First of all, the rotation speed determines the size of typical weather
phenomena. This size, the socalled \ii{Rossby radius}, is given by the
speed of sound (or some other typical speed) divided by twice the local
rotation speed, multiplied with the radius of the Earth. At moderate
latitudes, the Rossby radius is about \csd{2000}{km}.\index{weather} This is
a sizeable fraction of the Earth's radius, so that only a few large weather
systems are present on Earth at any specific time. If the Earth rotated
more slowly, the weather would be determined by shortlived, local flows and
have no general regularities. If the Earth rotated more rapidly, the
weather would be much more violent  as on Jupiter  but the small Rossby
radius implies that large weather structures have a huge lifetime, such as
the red spot on Jupiter, which lasted for several centuries. In a sense,
the rotation of the Earth has the speed that provides the most interesting
weather.
% Sep 2011
The other consequence of the value of the Earth's rotation speed concerns
the thickness of the \iin{atmosphere}. Mach 1 is also, roughly speaking,
the thermal speed of air molecules. This speed is sufficient for an air
molecule to reach the characteristic height of the atmosphere, about
\csd{6}{km}. On the other hand, the speed of rotation $\Omega$ of the Earth
determines its departure $h$ from sphericity: the Earth is
flattened,\index{Earth!flattened}\index{flattening!of the Earth} as we saw
above.\seepageone{iflattening} Roughly speaking, we have
$gh=\Omega^2R^2/2$, or about \csd{12}{km}. (This is correct to within
50\,\%, the actual value is \csd{21}{km}.) We thus find that the speed of
rotation of the Earth implies that its flattening is comparable to the
thickness of the atmosphere.
% Feb 2012
\item The Coriolis effect influences rivers\index{Coriolis effect!and rivers}
and their shores. This surprising connection was made in 1860 by
\iinn{Karl~Ernst~von Baer}
% from kr.cs.ait.ac.th/~radok/physics/b6.htm
who found that in Russia, many rivers flowing north in lowlands had right
shores that are steep and high, and left shores that are low and flat. (Can
you explain the details?)\challengn He also found that rivers in the southern
hemisphere show the opposite effect.
% Mar 2014, Feb 2015
\csepsfnb[p]{icoriolisapplications}{scale=1}{The use of the Coriolis effect
in insects\protect\index{insect!navigation}  here a crane fly and a
hovering fly  and in microelectromechanic systems (size about a few mm);
all provide navigation signals to the systems to which they are attached
({\textcopyright}~\protect\iname{Pinzo}, \protect\iname{Sean McCann},
\protect\iname{ST Microelectronics}).}
%
% Did not add Apple, Samsung or Sony Playstation because permissions are a
% mess. Bosch sensortec, the market leader, has no images on the internet
% Mar 2014
\item The Coriolis effect\index{Coriolis effect!and navigation(} saves lives
and helps people. Indeed, it has an important application for navigation
systems; the typical uses are shown in \figureref{icoriolisapplications}.
Insects use vibrating masses\index{haltere!and insect navigation} to
stabilize their orientation, to determine their direction of travel and to
find their way. Most twowinged insects, or diptera,\index{diptera} use
\emph{vibrating halteres} for navigation: in particular, bees, houseflies,
hoverflies
% % dragon flies : no, they have 4 wings
and crane flies use them. Other insects, such as moths, use \emph{vibrating
antennae} for navigation. Cars, satellites, mobile phones,
remotecontrolled helicopter models, and computer games also use tiny
vibrating masses as orientation and navigation sensors, in exactly the same
way as insects do.
% Mar 2014
In all these navigation applications, one or a few tiny masses are made to
vibrate; if the system to which they are attached turns, the change of
orientation leads to a Coriolis effect. The effect is measured by detecting
the ensuing change in geometry; the change, and thus the signal strength,
depends on the angular velocity and its direction. Such orientation sensors
are therefore called \emph{vibrating Coriolis
gyroscopes}.\index{gyroscope!vibrating Coriolis} Their development and
production is a sizeable part of hightech business  and of biological
evolution.\index{Coriolis effect!and navigation)}
% August 2013
\item A wealthy and quirky customer asked his architect to plan and build a
house whose four walls all faced south. How did the architect realize the
request?\challengn
\item Would travelling through interplanetary space be healthy?\index{space
travel}\index{travel!space, and health} People often fantasize about long
trips through the cosmos. Experiments have shown that on trips of long
duration, cosmic radiation, bone weakening, muscle degeneration and
psychological problems are the biggest dangers. Many medical experts
question the viability of space travel lasting longer than a couple of
years. Other dangers are rapid sunburn, at least near the Sun, and exposure
to the vacuum.\index{vacuum!human exposure to} So far only one man has
experienced vacuum without protection.\cite{vacman} He lost consciousness
after 14 seconds, but survived unharmed.
\item In which direction does a \iin[flame puzzle]{flame}
lean\index{puzzle!flame} if it burns inside a jar on a rotating
turntable?\challengenor{jar}
%Dec 2006
\item Galileo's principle of everyday relativity states that it is impossible
to determine an absolute velocity. It is equally impossible to determine an
absolute position, an absolute time and an absolute direction. Is this
correct?\challengenor{absoall}
\label{centrifug}
%
\item Does \ii{centrifugal acceleration}
exist?\indexs{acceleration!centrifugal} Most university students go through
the shock of meeting a teacher who says that it doesn't because it is a
`fictitious' quantity, in the face of what one experiences every day in a
car when driving around a bend. Simply ask the teacher who denies it to
define `existence'. (The definition physicists usually use is given later
on.)\seepagethree{exisdefi} Then check whether the definition applies to the
term and make up your own mind.\challengenor{centrifacc}
% Feb 2010, Nov 2013
Whether you like the term `centrifugal acceleration' or avoid it by using
its negative, the socalled \ii{centripetal
acceleration},\index{acceleration!centripetal} you should know how it is
calculated. We use a simple trick. For an object in circular motion of
radius $r$, the magnitude $v$ of the velocity
${\bm v}= \diffd {\bm x}/\diffd t $ is $v = 2 \pi r / T$. The vector
${\bm v}$ behaves over time exactly like the position of the object: it
rotates continuously. Therefore, the magnitude $a$ of the
centrifugal/centripetal acceleration ${\bm a}= \diffd {\bm v}/\diffd t $ is
given by the corresponding expression, namely $a=2\pi v /T$. Eliminating
$T$, we find that the centrifugal/centripetal acceleration $a$ of a body
rotating at speed $v$ at radius $r$ is given by
\begin{equation}
a= \frac{v^2}{r} = \omega^2 r\cp
\end{equation}
This is the acceleration we feel when sitting in a car that goes around a
bend.
\item Rotational motion\label{rotpseuvec} holds a little surprise for anybody
who studies it carefully. Angular momentum is a quantity with a magnitude
and a direction. However, it is \emph{not} a vector, as any mirror shows.
The angular momentum of a body circling in a plane parallel to a mirror
behaves in a different way from a usual arrow: its mirror image is not
reflected if it points towards the mirror! You can\challengn easily check
this for yourself. For this reason, \iin[angular
momentum!pseudovector]{angular momentum} is called a \ii{pseudovector}.
(Are rotations pseudovectors?)\challengn The fact has no important
consequences in classical physics;\seepageone{pseudov} but we have to keep
it in mind for later, when we explore nuclear\seepagefive{pviol} physics.
\item What is the best way to transport a number of full coffee or tea cups
while at the same time avoiding spilling any precious
liquid?\challengenor{tea}
\item A \iin{pingpong ball} is attached by a string to a stone, and the whole
is put under water in a jar. The setup is shown in \figureref{ipingpong}.
Now the jar is accelerated horizontally, for example in a car. In which
direction does the ball move?\challengenor{pingpong} What do you deduce for
a jar at rest?
\item The Moon recedes from the Earth by \csd{3.8}{cm} a year, due to
friction. Can you find the mechanism responsible for the
effect?\challengenor{moonfrit}
% (AIP news)  solution later on
\cssmallepsf{ipingpong}{scale=1}{How does the ball move when the jar is
accelerated in direction of the arrow?}
% Feb 2005
\item What are earthquakes? \ii[earthquake]{Earthquakes} are large examples
of the same process that make a door squeak. The continental plates
correspond to the metal surfaces in the joints of the door.
Earthquakes can be described as energy sources.\index{earthquake!energy} The
Richter scale is a direct measure of this energy. The \ii{Richter
magnitude} $M_{\rm s}$ of an earthquake, a pure number, is defined from
its energy $E$ in joule via
\begin{equation}
M_{\rm s}= \frac{{\rm log} (E/\csd{1}{J})  4.8} {1.5} \cp
\end{equation}
The strange numbers in the expression have been chosen to put the earthquake
values as near as possible to the older, qualitative \iin{Mercalli scale} (now
called \csaciin{EMS98}) that classifies the intensity of earthquakes.
However, this is not fully possible; the most sensitive instruments today
detect earthquakes with magnitudes of $3$. The highest value ever measured
was a Richter magnitude of 10, in Chile in 1960. Magnitudes above 12 are
probably impossible. Can you show why?\challengenor{richterscale}
\item What is the motion of the point on the surface of the Earth that has Sun
in its \iin{zenith}  i.e., vertically above it  when seen on a map of
the Earth during one day? And day after day?\challenge % !!!5
\item Can it happen that a satellite dish for geostationary TV
\iin[satellite!geostationary]{satellites} focuses the sunshine onto the
receiver?\challengenor{dish}
\item Why is it difficult to fire a rocket from an aeroplane in the direction
opposite to the motion of the plane?\challengenor{rocket}
\cssmallepsf{iape}{scale=1}{What happens when the ape climbs?}
\item An ape hangs on a rope. The rope hangs over a wheel
and\index{puzzle!ape}\index{ape!puzzle} is attached to a mass of equal
weight hanging down on the other side, as shown in \figureref{iape}. The
rope and the wheel are massless and frictionless. What happens when the ape
climbs the rope?\challengenor{ape}
\item Can a water skier move with a higher speed than the boat pulling
him?\challengenor{waterski}
% Added in Apr 2005
\item You might know the `Dynabee',\index{dynabee} a handheld gyroscopic
device that can be accelerated to high speed by proper movements of the
hand. How does it work?\challengedif{dynabee}
% Apr 2006
\item It is possible to make a spinning top with a metal paper clip. It is
even possible to make one of those tops that turn onto their head when
spinning. Can you find out how?\challengenor{burospintop}
\item The\label{angmomten} moment of inertia of a body depends on the shape of
the body; usually, the angular momentum and the angular velocity do not
point in the same direction.\index{inertia!moment of} Can you confirm this
with an example?\challengenor{angt}
\item What is the moment of inertia of a homogeneous
sphere?\challengenor{sphere}
\item The complete moment of inertia of a rigid body is determined by the
values along its three principal axes. These values are all equal for a
sphere and for a cube. Does it mean that it is impossible to distinguish a
sphere from a cube by their inertial behaviour?\challengenor{cubesphere}
% % New in July 2016 !!!4 not finished  IShowU does not work any more, Sep
% 2016, despite my emails
%
% \csmovfilm{polhodevideo}{scale=1}{The motion of the angular velocity of a
% tumbling brick. The animation confirms the wellknown nerd statement: the
% polhode\protect\index{polhode} rolls without slipping on the
% herpolhode\protect\index{herpolhode} lying in the invariable plane
% (respectively the yellow curve, the white curve, and the bright blue cross
% pattern). Any rotating, irregular, free rigid body follows the same
% motion. For the Earth, which is rotating and irregular, but neither fully
% free nor fully rigid, this is only an
% approximation. ({\textcopyright}~\protect\iinn{Svetoslav Zabunov} at
% \protect\url{www.ialms.net/sim/3drigidbodysimulation/})}
% New in Sep 2016 % ONLY FIXED IMAGE
\csepsfnb{ipolhode}{scale=1}{The motion of the angular velocity of a
tumbling brick. The tip of the angular velocity vector moves along the
yellow curve, the \emph{polhode}. It moves together with the tumbling
object, as does the elliptical mesh representing the energy ellipsoid; the
\emph{herpolhode} is the white curve and lies in the plane bright blue
cross pattern that represents the invariable plane (see text). The
animation behind the screenshot illustrates the wellknown nerd statement:
\emph{the polhode\protect\index{polhode} rolls without slipping on the
herpolhode\protect\index{herpolhode} lying in the invariable plane.} The
full animation is available online at
\protect\url{www.ialms.net/sim/3drigidbodysimulation/}. Any rotating,
irregular, free rigid body follows such a motion. For the Earth, which is
rotating and irregular, but neither fully free nor fully rigid, this
description is only an
approximation. ({\textcopyright}~\protect\iinn{Svetoslav Zabunov})}
% New in July 2016
\item Here is some mathematical fun about the rotation of a free rigid
body. Even for a free rotating rigid body, such as a brick rotating in free
space, the angular velocity is, in general, \emph{not} constant: the brick
\emph{tumbles} while rotating. In this motion, both energy and angular
momentum are constant, but the angular velocity is not. In particular, not
even the direction of angular velocity is constant; in other words, the
north pole changes with time. In fact, the north pole changes with time both
for an observer on the body and for an observer in free space. How does the
north pole, the end point of the angular velocity vector, move?
The moment of inertia is a tensor\seepageone{symola} and can thus
represented by an ellipsoid. In addition, the motion of a free rigid body is
described, in the angular velocity space, by the kinetic energy ellipsoid 
because its rotation energy is constant.
% see http://mysite.du.edu/~jcalvert/phys/ellipso.htm
%
% http://einstein.stanford.edu/highlights/hl_polhode_story.html
When a free rigid body moves, the energy ellipsoid  not the moment of
inertia ellipsoid  \emph{rolls} on the invariable plane that is
perpendicular to the initial (and constant) angular momentum of the
body.\challengn This is the mathematical description of the tumbling
motion. The curve traced by the angular velocity, or the extended north pole
of the body, on the invariable plane is called the \ii{herpolhode}. It is an
involved curve that results from two superposed conical motions. For an
observer on the rotating body, another curve is more interesting. The pole
of the rotating body traces a curve on the energy ellipsoid  which is
itself attached to the body. This curve is called the \ii{polhode}. The
polhode is a closed curve i three dimensions, the herpolhode is an open
curve in two dimensions. The curves were named by \iinn{Louis Poinsot} and
are especially useful for the description of the motion of rotating
irregularly shaped bodies. For an complete visualization, see the excellent
website \protect\url{www.ialms.net/sim/3drigidbodysimulation/}.
The polhode is a circle only if the rigid body has rotational symmetry; the
pole motion is then called \ii{precession}; we encountered it
above.\seepageone{iprecession} As shown in \figureref{ipolarmotion}, the
measured polhode of the Earth is not of the expected shape; this
irregularity is due to several effects, among them to the nonrigidity of
our planet.
Even though the angular momentum of a free tumbling brick is fixed in space,
it is not fixed in the body frame. Can you confirm this?\challengn In the
body frame, the end of the angular momentum of a tumbling brick moves along
still another curve, given by the intersection of a sphere and an ellipsoid,
as \iinn{Jacques Binet} pointed out.
\item Is it true that the Moon in the first quarter in the northern hemisphere
looks like the Moon in the last quarter in the southern
hemisphere?\challengenor{moonyes}
% MK: http://www.skyinmotion.de/de/zeitraffer_einzel.php?NR=12
% Improved Apr 2014
\csepsfnb[p]{icircumpolar}{scale=1}{Long exposures of the stars at night 
one when facing north, above the Gemini telescope in Hawaii, and one above
the Alps that includes the celestial equator, with the geostationary
satellites on it ({\textcopyright}~\protect\iname{Gemini
Observatory/AURA}, \protect\iinn{Michael Kunze}).}
% this is the figure www.ausgo.unsw.edu.au/gallery/telescopes01.html
% Gemini north telescope Mauna Kea
% Dec 2016
\csepsfnb{iearthshadow}{scale=1}{The shadow of the Earth  here a
panoramic photograph taken at the South Pole  shows that the Earth is
round ({\textcopyright}~\protect\iinn{Ian R. Rees}).}
% Old text
\item An impressive confirmation that the Earth is a sphere can be seen at
sunset, if we turn, against our usual habit, our back on the Sun. On the
eastern sky we can then see the impressive rise of the {Earth's shadow}. We
can admire the vast shadow rising over the whole horizon, clearly having the
shape of a segment of a huge circle. \figureref{iearthshadow} shows an
example. In fact,\index{Earth!shadow of} more precise investigations show
that it is not the shadow of the Earth alone, but the shadow of its
ionosphere.\index{shadow!of the Earth}\index{shadow!of
ionosphere}\index{ionosphere!shadow of}
\item How would \figureref{icircumpolar} look if taken at the
Equator?\challengenor{circpol}
% Mar 2015
\item Precision measurements show that not all planets move in exactly the
same plane. Mercury shows the largest deviation. In fact, no planet moves
exactly in an ellipse, nor even in a plane around the Sun. Almost all of
these effects are too small and too complex to explain here.
% Impr. Mar 2015
\item Since the Earth is round, there are many ways to drive from one point on
the Earth to another along a circle segment. This freedom of choice has
interesting consequences for volley balls and for men {watching} women.
Take a volleyball and look at its air inlet. If you want to move the inlet
to a different position with a simple rotation, you can choose the rotation
axis in many different ways. Can you confirm this?\challengn In other
words, when we look in a given direction and then want to look in another,
the eye can accomplish this change in different ways. The option chosen by
the human eye had already been studied by medical scientists in the
eighteenth century. It is called \ii{Listing's `law'}.\footnote{If you are
interested in learning in more detail how nature and the eye cope with the
complexities of three dimensions, see the
\url{schorlab.berkeley.edu/vilis/whatisLL.htm} and
%
% Updated Jun 2011
\url{www.physpharm.fmd.uwo.ca/undergrad/llconsequencesweb/ListingsLaw/perceptual1.htm}
websites.} %
%
It states that all axes that nature chooses lie in one plane. Can you
imagine its position in space?\challengenor{listing} Many men have a real
interest that this mechanism is strictly followed; if not, looking at women
on the beach could cause the muscles moving their eyes to get knotted
up.\index{women!dangers of looking after}
% New in Sep 2017
\csmovfilmrepeatwide{diracconverted}{scale=1}{A steel ball glued in a
mattress can rotate forever.\protect\index{rotation!tethered} (QuickTime
film {\textcopyright}~\protect\iinn{Jason Hise}).}
% New in Sep 2017
\item Imagine to cut open a soft mattress, glue a steel ball into it, and glue
the mattress\index{mattress!with rotating steel ball}\index{ball!rotating in
mattress} together again. Now imagine that we use a magnetic field to
rotate the steel ball glued inside. Intuitively, we think that the ball can
only be rotated by a finite angle, whose value would be limited by the
elasticity of the mattress. But in reality, the steel ball can be rotated
\emph{infinitely often!} This surprising possibility is a consequence of the
tethered rotation shown in \figureref{ifakewheel3}\seepageone{ifakewheel3}
and \figureref{dodecatwist}. Such a continuous rotation in a mattress is
shown in \figureref{diracconverted}. And despite its fascination, nobody
has yet realized the feat. Can you?\challengeres{steelball}
\end{curiosity}
%
% Reread July 2016
\subsection{Legs or wheels?  Again}
% Index OK
The acceleration and deceleration of standard wheeldriven cars is never much
greater than about $1\:g = \csd{9.8}{m/s^{2}}$, the acceleration due to
gravity\index{legs!advantages}\index{legs!vs. wheels}\index{wheel!vs. leg}
on our planet. Higher accelerations are achieved by motorbikes and racing
cars through the use of suspensions that divert weight to the axes and by the
use of spoilers, so that the car is pushed downwards with more than its own
weight. Modern spoilers are so efficient in pushing a car towards the track
that racing cars could race on the roof of a tunnel without falling down.
Through the use of special tyres the downwards forces produced by aerodynamic
effects are transformed into grip; modern racing tyres allow forward, backward
and sideways accelerations (necessary for speed increase, for braking and for
turning corners) of about 1.1 to 1.3 times the load. Engineers once believed
that a factor 1 was the theoretical limit and this limit is still sometimes
found in textbooks; but advances in tyre technology, mostly by making clever
use of interlocking between the tyre and the road surface as in a gear
mechanism, have allowed engineers to achieve these higher values. The highest
accelerations, around $4\, g$, are achieved when part of the tyre melts and
glues to the surface. Special tyres designed to make this happen are used for
dragsters, but high performance radiocontrolled model cars also achieve such
values.
How do wheels compare to using legs? High \iin{jump} athletes can achieve
peak accelerations of about 2 to 4 $g$, cheetahs over $3\,g$, \iin{bushbabies}
up to $13\,g$, \iin{locusts} about $18\,g$, and \iin{fleas} have been measured
to accelerate about $135\,g$.\cite{a36} The maximum
acceleration\index{acceleration!animal record} known for animals is that of
\iin{click beetles},\index{beetle!click} a small insect able to accelerate at
over \csd{2000}{m/s^{2}}$=200\,g$, about the same as an airgun pellet when
fired.\index{leg!performance} Legs are thus definitively more efficient
accelerating devices than wheels  a cheetah can easily beat any car or
motorbike  and evolution developed legs, instead of wheels, to improve the
chances of an animal in danger getting to safety.
\cssmallepsfnb{fbasil}{scale=0.6}{A basilisk lizard (\protect\iie{Basiliscus
basiliscus}) running on water, with a total length of about 25\,cm, showing
how the propulsing leg pushes into the water
({\textcopyright}~\protect\iname{TERRA}).}
In short, legs \emph{outperform} wheels. But there are other reasons for using
legs instead of wheels. (Can you name some?)\challengenor{legstreet} For
example, legs, unlike wheels, allow walking on water.\indexe{Basiliscus
basiliscus} Most famous for this ability is the \ii{basilisk}, % lizard},%
%
\footnote{In the Middle Ages, the term `basilisk' referred to a mythical
monster supposed to appear shortly before the end of the world. Today, it
is a small reptile in the Americas.}
%
a lizard\index{lizard} living in Central America and shown in
\figureref{fbasil}. This reptile is up to \csd{70}{cm} long and has a mass
of about \csd{90}{g}. It looks like a miniature \iie{Tyrannosaurus rex} and
is able to run over water surfaces on its hind legs. The motion has been
studied in detail with highspeed cameras and by measurements using aluminium
models of the animal's feet.\index{legs!on water}\index{walking!on
water}\index{water!walking on}
% The lizards run by pushing the legs against the water very rapidly.
The experiments show that the feet slapping on the water provides only 25\,\%
of the force necessary to run above water;\cite{jeswat} the other 75\,\% is
provided by a pocket of compressed air that the basilisks create between their
feet and the water once the feet are inside the water. In fact, basilisks
mainly walk on air. (Both effects used by basilisks are also found in fast
\iin{canoeing}.)\cite{canoe} It was calculated that humans are also able to
walk on water, provided their feet hit the water with a speed of
\csd{100}{km/h} using the simultaneous physical power of 15 sprinters. Quite
a feat for all those who ever did so.\index{Jesus}\index{gods!and walking on
water}
\cstftlepsfnb{iwaterstrider}{scale=0.3022}{A water strider, total size
about 10\,mm ({\textcopyright}~\protect\iinn{Charles
Lewallen}).}[10mm]{iwaterrobot}{scale=0.88}{A water walking robot, total size
about 20\,mm ({\textcopyright}~\protect\iname{AIP}).}
% Jan 2005
There is a second method of walking and running on water; this second method
even allows its users to remain immobile on top of the water surface. This is
what water striders,\index{water!strider} insects of the family \iie{Gerridae}
with an overall length of up to \csd{15}{mm}, are able to do (together with
several species of spiders),\index{spider!water walking} as shown in
\figureref{iwaterstrider}. Like all insects, the water strider has six legs
(spiders have eight). The water strider uses the back and front legs to hover
over the surface, helped by thousands of tiny hairs attached to its body. The
hairs, together with the surface tension of water, prevent the strider from
getting wet.\index{robot!walking on water}\index{water!walking robot} If you
put shampoo into the water, the water strider sinks and can no longer move.
The water strider uses its large middle legs as oars to advance over the
surface, reaching speeds of up to \csd{1}{m/s} doing so. In short, water
striders actually row over water.
% Aug 2006
The same mechanism is used by the small robots that can move over water and
were developed by \iinn{Metin Sitti} and his group, as shown in
\figureref{iwaterrobot}.\cite{waterrob}
% Mar 2012
Robot design is still in its infancy.\index{robot!walking or running} No robot
can walk or even run as fast as the animal system it tries to copy. For
twolegged robots, the most difficult kind, the speed record is around 3.5 leg
lengths per second. In fact, there is a race going on in robotics
departments: each department tries to build the first robot that is faster,
either in metres per second or in leg lengths per second, than the original
fourlegged animal or twolegged human. The difficulties of realizing this
development goal show how complicated walking motion is and how well nature
has optimized living systems.
% Nov 2004
Legs pose many interesting problems. Engineers know that a staircase is
comfortable to walk only if for each step the depth $l$ plus \emph{twice} the
height $h$ is a constant: $l + 2h=\csd{0.63\pm 0.02}{m}$. This is the
socalled \ii[staircase!formula]{staircase formula}. Why does it
hold?\challengenor{staircasefor}
Most animals have an \emph{even} number of legs. Do you know an exception?
Why not?\challengenor{evenwheels} In fact, one can argue that no animal has
less than four legs. Why is this the case?
% Aug 2005
On the other hand, all animals with two legs have the legs side by side,
whereas most systems with two wheels have them one behind the other. Why is
this not the other way round?\challengn % round is British, around is American
% Feb 2014
\csepsfnb{irunningallometry}{scale=1}{The graph shows how the relative
running speed changes with the mass of terrestrial mammal species, for 142
different species. The graph also illustrates how the running performance
changes above 30\,kg. Filled squares show Rodentia; open squares show
Primata; filled diamonds Proboscidae; open diamonds Marsupialia; filled
triangles Carnivora; open triangles Artiodactyla; filled circles
Perissodactyla; open circles Lagomorpha ({\textcopyright}~\protect\iinn{José
IriarteDíaz}/{JEB}).}
% Feb 2014
Legs are very efficient actuators.\index{legs!efficiency of} As
\figureref{irunningallometry} shows, most small animals can run with about
25 body lengths per second.\cite{iriarte} For comparison, almost no car
achieves such a speed. Only animals that weigh more than about \csd{30}{kg},
including humans, are slower.
% Jul 2006
Legs also provide simple distance rulers: just count your steps. In 2006, it
was discovered that this method is used by certain ant species, such as
\iie{Cataglyphis fortis}. They can count to at least 25\,000, as shown by
\iinn{Matthias Wittlinger} and his team.\cite{wittlinger} These ants use the
ability to find the shortest way back to their home even in structureless
desert terrain.
% Oct 2007
Why do \csd{100}{m} sprinters run\index{athletics}\index{sprint!training}
faster than ordinary people? A thorough investigation\cite{bellizzi} shows
that the speed $v$ of a sprinter is given by
\begin{equation}
v = f \, L_{\hbox{\scriptsize\rm stride}} = f \, L_{\rm c} \, \frac{F_{\rm c}}{W}
\cvend
\end{equation}
where $f$ is the frequency of the legs, $L_{\hbox{\scriptsize\rm stride}}$ is
the stride length, $L_{\rm c}$ is the contact length  the length that the
sprinter advances during the time the foot is in contact with the floor  $W$
the weight of the sprinter, and $F_{\rm c}$ the average force the sprinter
exerts on the floor during contact. It turns out that the frequency $f$ is
almost the same for all sprinters; the only way to be faster than the
competition is to increase the stride length
$L_{\hbox{\scriptsize\rm stride}}$. Also the contact length $L_{\rm c}$
varies little between athletes. Increasing the stride length thus requires
that the athlete hits the ground with strong strokes. This is what athletic
training for sprinters has to achieve.
%
% Nov 2008, Jul 2016
\subsection{Summary on Galilean relativity}
% Index OK
% Feb 2012
The Earth rotates.\index{relativity!Galilean, summary} The acceleration is so
small that we do not feel it. The speed of rotation is large, but we do not
feel it, because there is \emph{no way} to do so.
Undisturbed or inertial motion \emph{cannot} be felt or measured. It is thus
impossible to distinguish motion from rest. The distinction between rest and
motion depends on the observer: \emph{Motion of bodies is relative.} That is
why the soil below our feet seems so stable to us, even though it moves with
high speed across the universe.
Since motion is relative, speed values depend on the observer. Later on will
we discover that one example of motion in nature has a speed value that is
\emph{not} relative: the motion of light.
% After this short overview of motion based on contact,
But we continue first with the study of motion transmitted over distance,
without the use of any contact at all.
% It is easier and simpler to study.
\vignette{classical}
%
%
%
%
\newpage
\chapter{Motion due to gravitation}
\markboth{\thesmallchapter\ motion due to gravitation}%
{\thesmallchapter\ motion due to gravitation}
% This section reread in July 2016
% This section index OK
\begin{quote}\selectlanguage{italian}%
Caddi come\indname{Dante Alighieri}\indname{Alighieri, Dante} corpo
morto cade.\selectlanguage{british}\\
Dante, \bt Inferno/ c.~V, v.~142.%
%
\footnote{`I fell like dead bodies fall.' Dante Alighieri \lived(1265,
Firenze1321, Ravenna), the powerful Italian poet.} %
%
\end{quote}
\csini{T}{he} first and main method to generate motion without any
contact\linebreak hat we discover in our environment is \emph{height}.
Waterfalls, snow, rain,\linebreak he ball of your favourite game and falling
apples all rely on it. It was one of the fundamental discoveries of physics
that height has this property because there is an interaction between every
body and the Earth. \ii[gravitation]{Gravitation} produces an
\emph{acceleration} along the line connecting the centres of gravity of the
body and the Earth. Note that in order to make this statement, it is
necessary to realize that the Earth is a body in the same way as a stone or
the Moon, that this body is finite and that therefore it has a centre and a
mass. Today, these statements are common knowledge, but they are by no means
evident from everyday personal experience.
In several myths about the creation or the organization of the world, such as
the biblical one or the Indian one, the Earth is not an object, but an
imprecisely defined entity, such as an island floating or surrounded by water
with unclear boundaries and unclear method of suspension. Are you
able\challengenor{roundearth} to convince a friend that the Earth is round and
not flat? Can you find another argument apart from the roundness of the
Earth's shadow when it is visible on the Moon, shown in
\figureref{iayiosuper}?
%
% Added title in Aug 2016
\subsection{Gravitation as a limit to uniform motion}
% Index OK
% undisturbed
% uniform
% powerful
% free
% unbounded
% unlimited
% boundless
% ongoing
% sustained
% lasting
% persistent
% steady
% permanent
% Aug 2016
A productive way to define gravitation, or gravity for short,\index{gravity!is
gravitation} appears when we note that no object around us moves along a
straight line. In nature, there is a \emph{limit} to steady, or constant
motion:
\begin{quotation}
\np\csrhd Gravity prevents uniform motion, i.e., it prevents constant and
straight motion.\index{gravity!as limit to motion}
\end{quotation}
In nature, we \emph{never} observe bodies moving at constant speed along a
straight line. Speaking with the vocabulary of kinematics: Gravity introduces
an \emph{acceleration} for every physical body. The gravitation of the objects
in the environment curves the path of a body, changes its speed, or both. This
limit has two aspects. First, gravity prevents unlimited uniform
motion:\index{motion!is never uniform}
\begin{quotation}
\np\csrhd Motion cannot be straight for ever. Motion is not boundless.
\end{quotation}
We will learn later what this means for the universe as a whole. Secondly,
\begin{quotation}
\np\csrhd Motion cannot be constant and straight even during short time
intervals.
\end{quotation}
In other words, if we measure with sufficient precision, we will always find
deviations from uniform motion. (Physicist also say that motion is never
inertial.)\index{motion!is never inertial} These limits apply no matter how
\emph{powerful} the motion is and how \emph{free} the moving body is from
external influence. In nature, gravitation prevents the steady, uniform motion
of atoms, billiard balls, planets, stars and even galaxies.
Gravitation is the first limitation to motion that we discover in nature. We
will discover two additional limits to motion later on in our walk. These
three fundamental limits are illustrated in
\figureref{iphysicsstructure}.\seepageone{iphysicsstructure} To achieve a
\emph{precise} description of motion, we need to take each limit of motion
into account. This is our main aim in the rest of our adventure.
Gravity affects \emph{all} bodies, even if they are distant from each
other. How exactly does gravitation affect two bodies that are far apart? We
ask the experts for measuring distant objects: the astronomers.
% Mar 2012
\csepsfnb{ideklination}{scale=1}{Some important concepts when observing the
stars and at night.}
%
% Improved May 2005, Oct 2009, Mar 2012, reread Jul 2016
\subsection{Gravitation in the sky}
% Index OK
%The expression $\smash{a=GM/r^2}$ for the acceleration
The gravitation of the Earth forces the Moon in an orbit around it. The
gravitation of the Sun forces the Earth in an orbit around it and sets the
length of the year. Similarly, the gravitation of the Sun
determines\index{gravitation!and planets(} the\index{planet!and universal
gravitation} motion of all the other planets across the sky. We usually
imagine to be located at the centre of the Sun and then say that the planets
`orbit the Sun'. The Sun thus prevents the planets from moving in straight
lines and forces them into orbits. How can we check this?
First of all, looking at the sky at night, we can check that the planets
always stay within the \ii{zodiac}, a narrow stripe across the
sky.\seepageone{isixcelestialbodies} The centre line of the zodiac gives the
path of the Sun and is called the \ii{ecliptic}, since the Moon must be
located on it to produce an eclipse. This shows that planets move
(approximately) in a single,\seepageone{ieclipse} common plane.%
%
\footnote{The apparent height of the ecliptic changes with the time of the
year and is the reason for the changing seasons. Therefore\index{season}
seasons are a gravitational effect as well.}
% Mar 2012, Sep 2014
\csepsf{imarsretrograde}{scale=1}{`Planet' means `wanderer'. This composed
image shows the retrograde motion of planet Mars across the sky  the
\protect\iin{Pleiades star cluster} is at the top left  when the planet is
on the other side of the Sun. The pictures were taken about a week apart
and superimposed. The motion is one of the many examples that are fully
explained by universal gravitation ({\textcopyright}~\protect\iinn{Tunc
Tezel}).}
% from apod/ap060422.html
% (NO) one day add the labeled image, also on apod (mouseover)
To learn more about the motion in the sky, astronomers have performed numerous
measurements of the movements of the Moon and the planets. The most
industrious of all was \iinns{Tycho Brahe},%
%
\footnote{Tycho Brahe \livedplace(1546 Scania1601 Prague), famous % Danish
astronomer, builder of Uraniaborg, the astronomical castle. He consumed
almost 10\,\% of the Danish gross national product for his research, which
produced the first star catalogue and the first precise position
measurements of planets.} %
%
who organized an industrialscale search for astronomical facts sponsored by
his king. His measurements were the basis for the research of his young
assistant, the Swabian astronomer \iinns{Johannes Kepler}%
%
\footnote{Johannes Kepler \lived(1571 Weil der Stadt1630 Regensburg) studied
Protestant theology and became a teacher of mathematics, astronomy and
rhetoric. He helped his mother to defend herself successfully in a trial
where she was accused of witchcraft. His first book on astronomy made him
famous, and he became assistant to \iinn{Tycho Brahe} and then, at his
teacher's death, the Imperial Mathematician. He was the first to use
mathematics in the description of astronomical observations, and introduced
the concept and field of `celestial physics'.} %
%
who found the first precise description\seepagethree{exisdefi} of planetary
motion. This is not an easy task, as the observation of
\figureref{imarsretrograde} shows. In his painstaking research on the
movements of the planets in the zodiac, Kepler discovered several `laws',
i.e., patterns or rules.\index{Kepler's laws} The motion of all the planets
follow the same rules, confirming that the Sun determines their orbits. The
three main ones are as follows: %\label{keplerla}
\smallskip
\begin{Strich}
\item[{1.}] Planets move on ellipses with the Sun located at one focus (1609).
\item[{2.}] Planets sweep out equal areas in equal times (1609).
% I used to have 1604 for the first two  probably wrong
\item[{3.}] All planets have the same ratio $T^2/d^3$ between the orbit
duration $T$ and the semimajor axis $d$ (1619).
\end{Strich}
\smallskip
\cssmallepsf{ielli}{scale=1}{The motion of a planet around the Sun, showing
its semimajor axis $d$, which is also the spatial average of its distance
from the Sun.}
\np Kepler's results are illustrated in \figureref{ielli}. The sheer work
required to deduce the three `laws' was enormous. \iname[Kepler,
Johannes]{Kepler} had no calculating machine available. The calculation
technology he used was the recently discovered \iin{logarithms}. Anyone who
has used tables of logarithms to perform calculations can get a feeling for
the amount of work behind these three discoveries.
Finally, in 1684, all observations by Kepler about planets and \iin{stones}
were condensed into an astonishingly simple result by the English physicist
\iinns{Robert Hooke} and a few others:%
%
\footnote{Robert Hooke \lived(16351703), important English physicist and
secretary of the Royal Society. Apart from discovering the inverse square
relation and many others, such as Hooke's `law', he also wrote the \btsim
Micrographia/, a beautifully illustrated exploration of the world of the
very small.} %
%
\begin{quotation}%
% May 2004
\noindent \csrhd Every body of mass $M$ attracts any other body towards its
centre with an acceleration whose magnitude $a$ is given by
\begin{equation}
a = G \,\frac{ M }{ r^2}
\label{nlg}
\end{equation}
where $r$ is the centretocentre distance of the two bodies.
\end{quotation}
This is called \ii{universal gravitation}, or the \emph{universal `law' of
gravitation},\index{gravitation!universal} because it is valid both in the
sky and on Earth, as we will see shortly. The proportionality constant $G$ is
called the \ii{gravitational constant};\index{constant!gravitational} it is
one of the fundamental constants of nature, like the speed of light $c$ or the
quantum of action $\hbar$. More about $G$ will be
said\seepageone{unigrpropzz} shortly.
The effect of gravity thus decreases with increasing distance; the effect
depends on the inverse distance squared of the bodies under consideration. If
bodies are small compared with the distance $r$, or if they are spherical,
expression (\ref{nlg}) is correct as it stands; for nonspherical shapes the
acceleration has to be calculated separately for each part of the bodies and
then added together.
% Rest of section moved here in Aug 2016
\csepsfnb{iellipsebook}{scale=1}{The proof that a planet moves in an ellipse
(magenta) around the Sun, given an inverse square distance relation for
gravitation. The proof  detailed in the text  is based on the relation
SP$+$PF$=$R. Since R is constant, the orbit is an ellipse}
% Oct 2009, Impr. Apr 2010, Nov 2016
Why is the usual planetary orbit an ellipse?\cite{heckman} The simplest
argument is given in \figureref{iellipsebook}. We know
that\index{orbit!elliptical} the acceleration due to gravity varies as
$\smash{a=GM/r^2}$. We also know that an orbiting body of mass $m$ has a
constant energy $E<0$. We then can draw, around the Sun, the circle with
radius $R=GMm/E$, which gives the largest distance that a body with energy
$E$ can be from the Sun. We now project the planet position $P$ onto this
circle, thus constructing a position $C$. We then reflect $C$ along the
tangent to get a position $F$. This last position $F$ is fixed in space and
time, as a simple argument shows. (Can you find it?\challengenor{heckmansol})
As a result of the construction, the distance sum SP+PF is constant in time,
and given by the radius $R=GMm/E$. Since the distance sum is constant, the
orbit is an ellipse, because an ellipse is precisely the curve that appears
when this sum is constant. (Remember that an ellipse can be drawn with a
piece of rope in this way.) Point $F$, like the Sun, is a focus of the
ellipse. The construction thus shows that the motion of a planet defines two
foci and follows an elliptical orbit defined by these two foci. In short, we
have deduced the first of Kepler's `laws' from the expression of universal
gravitation.
The second of Kepler's `laws', about equal swept areas, implies that planets
move faster when they are near the Sun. It is a simple way to state the
conservation of angular momentum. What does the third `law' state?\challengn
Can you confirm that also the second and third of Kepler's `laws' follow from
Hooke's expression of universal gravity?\challengenor{solconics} Publishing
this result  which was obvious to Hooke  was one of the achievements of
Newton. Try to repeat this achievement; it will show you not only the
difficulties, but also the possibilities of physics, and the joy that puzzles
give.
Newton solved these puzzles with geometric drawings  though in quite a
complex manner. It is well known that Newton was not able to write down, let
alone handle, differential equations at the time he published his results on
gravitation.\cite{a14} In fact, Newton's notation and calculation methods were
poor. (Much poorer than yours!) The English mathematician \inames[Hardy,
Godfrey H.]{Godfrey Hardy}%
%
\footnote{Godfrey Harold Hardy \lived(18771947) was an important % English
number theorist, and the author of the wellknown \btsim A Mathematician's
Apology/. He also `discovered' the famous Indian mathematician
\iinn{Srinivasa Ramanujan}, and brought him to Britain.}
%
used to say that the insistence on using {Newton}'s integral and differential
notation,
%which he developed much later  instead of using the one of his
rather than the earlier and better method, still common today, due to his
rival \iname[Leibniz, Gottfried Wilhelm]{Leibniz}  threw back English
mathematics by 100 years.
To sum up, \iname[Kepler, Johannes]{Kepler}, \iname[Hooke, Robert]{Hooke} and
\iname[Newton, Isaac]{Newton} became famous because they brought order to the
description of planetary motion. They showed that all motion due to gravity
follows from the same description, the inverse square distance. For this
reason, the inverse square distance relation $a= GM/r^2$ is called the
\emph{universal} law of gravity. Achieving this unification of motion
description, though of small practical significance, was widely publicized.
The main reason were the ageold prejudices and fantasies linked with
\iin{astrology}.
In fact, the inverse square distance relation explains many additional
phenomena. It explains the motion and shape of the Milky Way and of the other
galaxies, the motion of many weather phenomena, and explains why the Earth has
an atmosphere but the Moon does not.\index{gravitation!and planets)} (Can you
explain this?)\challengenor{noatmo}
% In fact, universal gravity explains much more about the Moon.
\cssmallepsfnb{iayiosuper}{scale=1.65}{How to compare the radius of the Earth
with that of the Moon during a partial lunar eclipse
({\textcopyright}~\protect\iinn{Anthony Ayiomamitis}).}
% EMAILED FEB 2008  OK for book , also his other ones
%
% Title added Aug 2016
\subsection{Gravitation on Earth}
% Index OK
This inverse square dependence of gravitational acceleration is often, but
incorrectly, called Newton's `law' of gravitation. Indeed, the %English
occultist and physicist \iinn{Isaac Newton} % \lived(16421727)
proved more elegantly than Hooke that the expression agreed with all
astronomical and terrestrial observations. Above all, however, he organized a
better public relations campaign, in which he falsely claimed to be the
originator of the idea.\cite{nauen}
Newton published a simple proof showing that the description of astronomical
gravitation also gives the correct description for \iin{stones} thrown through
the air, down here on `father Earth'. To achieve this, he compared the
acceleration $a_{\rm m}$ of the Moon with that of stones $g$. For the ratio
between these two accelerations, the inverse square relation predicts a value
$g/a_{\rm m} = d_{\rm m}^2 / R^2 $, where $d_{\rm m}$ the distance of the Moon
and $R$ is the radius of the Earth. The Moon's distance can be measured by
triangulation, comparing the position of the Moon against the starry
background from two different points on\label{moondist} Earth.%
%
\footnote{The first precise  but not the first  measurement was achieved
in 1752 by the French astronomers \iname{Lalande} and \iname{La Caille}, who
simultaneously measured the position of the Moon seen from Berlin and from
Le Cap.} %
%
The result is $d_{\rm m}/R = 60 \pm 3$, depending on the orbital position of
the Moon, so that an average ratio $g / a_{\rm m} = 3.6\cdot 10^3$ is
predicted from universal gravity. But both accelerations can also be measured
directly. At the surface of the Earth, \iin{stones} are subject to an
acceleration due to gravitation with magnitude $g=\csd{9.8}{m/s^2}$, as
determined by measuring the time that stones need to fall a given distance.
For the Moon, the definition of acceleration, ${a}=\diffd {v}/\diffd t$, in
the case of circular motion  roughly correct here  gives
$a_{\rm m}= d_{\rm m} (2 \pi / T )^2 $, where $T=\csd{2.4}{Ms}$ is the time
the Moon takes for one orbit around the Earth.%
%
\footnote{This expression for the centripetal
acceleration\index{acceleration!centripetal} is deduced easily by noting
that for an object in circular motion, the magnitude $v$ of the velocity
${\bm v}= \diffd{\bm x}/\diffd t $ is given as $v = 2 \pi r / T$. The
drawing of the vector ${\bm v}$ over time, the socalled
\ii{hodograph},\challengenor{hodo} shows that it behaves exactly like the
position of the object. Therefore the magnitude $a$ of the acceleration
${\bm a}= \diffd {\bm v}/\diffd t $ is given by the corresponding
expression, namely $a=2\pi v /T$.} %
%
%{\footnote{This is deduced easily by noting that for an object in circular
%motion, i.e.,{} for which $x^2+y^2=d^2$, the speed is given as $v = 2 \pi d
% / T$.
%One then has $v_{\rm y }= x (2 \pi / T) $ and $v_{\rm x } =  y (2 \pi /
% T)$.
%One thus gets $a_{\rm x }= (2 \pi / T) v_{\rm y } =  (2 \pi / T)^2 x$ and
%$a_{\rm y }= (2 \pi / T) v_{\rm x } =  (2 \pi /T)^2 y$, so that for
%$a=\sqrt{a_{\rm x }^2+a_{\rm y }^2}$ one gets the result just mentioned.}} %
%
The measurement of the radius of the Earth%
%
% Improved Mar 2018
\footnote{This is the hardest quantity to measure oneself. The most
surprising way to determine the Earth's size is the following: watch a
sunset in the garden of a house, with a stopwatch in hand, such as the one
in your mobile phone.\cite{dawli} When the last ray of the Sun disappears,
start the stopwatch and run upstairs. There, the Sun is still visible; stop
the stopwatch when the Sun disappears again and note the time $t$. Measure
the height difference $h$ between the two eye positions where the Sun was
observed. The Earth's radius $R$ is then given by $R=k\,h/t^2$, with
$k=\csd{378\cdot 10^6}{s^2}$.\challengenor{dawlinsstop}
There is also a simple way to measure the distance to the Moon, once the
size of the Earth is known.\cite{vh} Take a photograph of the Moon when it
is high in the sky, and call $\theta$ its zenith angle, i.e.,{} its angle
from the vertical.\index{zenith!angle} Make another photograph of the Moon a
few hours later, when it is just above the horizon.\index{Moon!size,
apparent} On this picture, unlike the common optical
illusion,\seepageone{moonsizeillus} the Moon is smaller, because it is
further away. With a sketch the reason for this becomes immediately clear.
If $q$ is the ratio of the two angular diameters, the EarthMoon distance
$d_{\rm m}$ is given by the relation
$d_{\rm m}^2=R^2+(2Rq\cos\theta/ (1q^2))^2$. Enjoy finding its derivation
from the sketch.\challengenor{myownfor}
Another possibility is to determine the size of the Moon by comparing it
with the size of the \iin{shadow!Earth, during lunar eclipse}, as
shown in \figureref{iayiosuper}. The distance to the Moon is then computed
from its angular size, about \csd{0.5}{\csdegrees}.} %
%
yields $R=\csd{6.4}{Mm}$, so that the average EarthMoon distance is
$d_{\rm m}=\csd{0.38}{Gm}$. One thus has $g/a_{\rm m} = 3.6\cdot10^3$, in
agreement with the above prediction. With this famous
`\iin[Moon!calculation]{Moon calculation}' we have thus shown that the inverse
square property of gravitation indeed describes both the motion of the Moon
and that of stones. You might want to deduce the value of the product $GM$
for Earth.\challengenor{gmsol}
Universal gravitation thus describes all motion due to gravity  both on
Earth and in the sky. This was an important step towards the unification of
physics. Before this discovery, from the observation that on the Earth all
motion eventually comes to rest, whereas in the sky all motion is eternal,
\iname{Aristotle} and many others had concluded that motion in the
\emph{sublunar} world has \emph{different} properties from motion in the
\emph{translunar} world. Several thinkers had criticized this distinction,
notably the % French
philosopher and rector of the University of
Paris,\cite{a24} \iinn{Jean Buridan}.%
%
\footnote{Jean Buridan \livedca(\circa1295 \circa1366) was also one of the
first modern thinkers to discuss the rotation of the Earth about an axis.}
%
% also \circa1300  after 1358 (his last mention as rector)
%
The Moon calculation was the most important result showing this distinction to
be wrong. This is the reason for calling Hooke's expression (\ref{nlg}) the
\ii[universal gravitation!origin of name]{universal} gravitation.
\csepsfnb{frodin}{scale=1}{A physicist's and an artist's view of the fall of
the Moon: a diagram by {Christiaan Huygens}\protect\indname{Huygens,
Christiaan} %\protect\lived(16291695)
(not to scale) and
% on the right \emph{La terre et la lune},
a marble statue by {Auguste Rodin}.\protect\indname{Rodin, Auguste}%
%\protect\lived(18401917)
} %
% Jun 2007
\cssmallepsfnb{ipendulum}{scale=1}{A precision second pendulum, thus about
1$\,$m in length; almost at the upper end, the vacuum chamber that
compensates for changes in atmospheric pressure; towards the lower end, the
wide construction that compensates for temperature variations of pendulum
length; at the very bottom, the screw that compensates for local variations
of the gravitational acceleration, giving a final precision of about 1$\,$s
per month ({\textcopyright}~\protect\iname{Erwin Sattler OHG}).}
% EMAILED FEB 2008  Sabine.Mueller@erwinsattler.de
Universal gravitation allows us to answer another old question. Why does the
{Moon} not fall from the sky?\index{fall!of Moon}\index{Moon!fall of} Well,
the preceding discussion showed that \ii{fall} is motion due to gravitation.
Therefore the Moon actually \emph{is} falling, with the peculiarity that
instead of falling \emph{towards} the Earth, it is continuously falling
\emph{around} it. \figureref{frodin} illustrates the idea. The Moon is
continuously missing the Earth.%
%
\footnote{Another way to put it is to use the answer of the Dutch physicist
\iinn{Christiaan Huygens} \lived(16291695): the Moon does not fall from the
sky because of the \iin{centrifugal acceleration}. As explained on
\cspageref{centrifug}, % this vol I
this explanation is often out of favour at universities.
There is a beautiful problem connected to the left side of the
figure:\cite{topper} Which points on the surface of the Earth can be hit by
shooting from a mountain? And which points\challengedif{moush} can be hit
by shooting horizontally?}
% Mar 2012
The Moon is not the only object that falls around the Earth.
\figureref{ifloatingastronaut} shows another.
% Mar 2012
\cssmallepsfnb{ifloatingastronaut}{scale=1}{The man in orbit feels no
weight, the blue atmosphere, which is not, does (NASA).}
%
% Reread Jul 2016
\subsection{Properties of gravitation: $G$ and $g$}
% Index OK
Gravitation\label{unigrpropzz} implies\index{gravitation!properties of(} that
the path of a \iin[stones]{stone} is not a \iin{parabola}, as stated
earlier,\seepageone{parabolaxyz} but actually an \ii[ellipse!as
orbit]{ellipse} around the centre of the Earth. %\seepageone{ugorbits}
This happens for exactly the same reason that the planets move in ellipses
around the Sun. Are you able to confirm this
statement?\seepageone{iellipsebook}
%
% Oct 2007
%\subsubsubsubsubsubsubsubsection{Table of g values}
{\small
\begin{table}[t]
\sbox{\cshelpbox}{\small\dirrtabularnolines
%
\begin{tabular}{@{\hspace{0em}}
>{\PBS\raggedright} p{50mm} @{\hspace{2em}}
>{\PBS\raggedright} p{40mm} @{\hspace{0em}}}
%
\toprule
%
\tabheadf{Place} & \tabhead{Value} \\[0.5mm]
%
\midrule % Source unknown
%
Poles & \csd{9.83}{m/s^2} \\
%
Trondheim & \csd{9.8215243}{m/s^2} \\
%
Hamburg & \csd{9.8139443}{m/s^2} \\
%
Munich & \csd{9.8072914}{m/s^2} \\
%
Rome & \csd{9.8034755}{m/s^2} \\
%
Equator & \csd{9.78}{m/s^2} \\
%
Moon & \csd{1.6}{m/s^2} \\
%
Sun & \csd{273}{m/s^2} \\
%
%
\bottomrule
\end{tabular}%
}% end of sbox
%
\centering
\begin{minipage}{\wd\cshelpbox}
\captionsetup{width=\wd\cshelpbox} % needed since Dec 2015
\caption{Some measured\protect\index{acceleration!due to gravity, table of values}
values of the acceleration due to gravity.}%
\label{accgravtab}\usebox{\cshelpbox}
\end{minipage}
\end{table}
}
% May 2004
Universal gravitation allows us to understand the puzzling acceleration value
\hbox{$g=\csd{9.8}{m/s^2}$} we encountered in equation (\ref{kin}). The value
is due to the relation
\begin{equation}
g=GM_{\rm Earth}/R_{\rm Earth}^2 \cp
\label{eq:spherear}
\end{equation}
The expression can be deduced from equation (\ref{nlg}), universal gravity, by
taking the Earth to be spherical. The everyday acceleration of gravity $g$
thus results from the size of the Earth, its mass, and the universal constant
of gravitation $G$. Obviously, the value for $g$ is almost constant on the
surface of the Earth, as shown in \tableref{accgravtab}, because the Earth is
almost a sphere. Expression (\ref{eq:spherear}) also explains why $g$ gets
smaller as one rises above the Earth, and the deviations of the shape of the
Earth from sphericity explain why $g$ is different at the poles and higher on
a plateau. (What would $g$ be on the Moon? On Mars? On
Jupiter?)\challengenor{planetg}
By the way, it is possible to devise a simple machine, other than a
\iin{yoyo}, that slows down the effective acceleration of gravity by a known
amount, so that one can measure its value more easily. Can you imagine this
machine?\challengenor{atwood}
Note that 9.8 is roughly $\pi^{2}$.\index{gravitation!Earth acceleration
value} This is \emph{not} a coincidence: the metre has been chosen in such a
way to make this (roughly) correct. The period $T$ of a swinging
pendulum,\index{p@$\pi$ and gravity} i.e.,{} a back and forward swing, is
given\challengenor{pendper} by%
%
\footnote{Formula (\ref{gape}) is noteworthy mainly for all that is missing.
The period of a pendulum does \emph{not} depend on the mass of the swinging
body. In addition, the period of a pendulum does \emph{not} depend on the
amplitude. (This is true as long as the oscillation angle is smaller than
about \csd{15}{\csdegrees}.) \iname[Galilei, Galileo]{Galileo} discovered
this as a student, when observing a \iin{chandelier} hanging on a long rope
in the dome of Pisa. Using his \iin{heartbeat} as a \iin{clock} he found
that even though the amplitude of the swing got smaller and smaller, the
time for the swing stayed the same.
A \iin[leg]{leg} also moves like a \iin{pendulum}, when one walks normally.
Why then do taller people tend to \iin[walking!speed]{walk} faster? Is the
relation also true for animals of different size?\challengenor{walspeed}} %
%
\begin{equation}
T= 2 \pi \sqrt{
\frac{l}{ g}
} \cv
\label{gape}
\end{equation}
where $l$ is the length of the pendulum, and $g=\csd{9.8}{m/s^2}$ is the
gravitational acceleration. (The pendulum is assumed to consist of a compact
mass attached to a string of negligible mass.) The oscillation time of a
pendulum depends only on the length of the string and on $g$, thus on the
planet it is located on.
If the metre had been defined such that $T/2=\csd{1}{s}$, the value of the
normal acceleration $g$ would have been exactly
\csd{\pi^{2}}{m/s^{2}}$=\csd{9.869\,604\,401\,09}{m/s^2}$.\challengn
%$\pi^{2}m/s^{2}$
%
Indeed, this was the first proposal for the definition of the metre; it was
made in 1673 by Huygens and repeated in 1790 by \iname{Talleyrand}, but was
rejected by the conference that defined the metre because variations in the
value of $g$ with geographical position, temperatureinduced variations of the
length of a pendulum and even air pressure variations induce errors that are
too large to yield a definition of useful precision. (Indeed, all these
effects must be corrected in pendulum clocks, as shown in
\figureref{ipendulum}.)
% May 2004
Finally, the proposal was made to define the metre as $1/40\,000\,000$ of the
circumference of the Earth through the poles, a socalled \ii[meridian!and
metre definition]{meridian}. This proposal was almost identical to  but
much more precise than  the pendulum proposal. The meridian definition of
the metre was then adopted by the French national assembly on 26 March 1791,
with the statement that `a meridian passes under the feet of every human
being, and all meridians are equal'. (Nevertheless, the distance from Equator
to the poles is not exactly \csd{10}{Mm}; that is a strange
story.\cite{metrestory} One of the two geographers who determined the size of
the first metre stick was dishonest. The data he gave for his measurements 
the general method of which is shown in \figureref{imetre}  was fabricated.
Thus the first official metre stick in Paris was shorter than it should be.)
\cssmallepsfnb{imetre}{scale=0.30}{The measurements that lead to the
definition of the metre ({\textcopyright}~\protect\iinn{Ken Alder}).}
% Impr. Feb 2012
Continuing our exploration of the gravitational acceleration $g$, we can still
ask: Why does the Earth have the mass and size it has? And why does $G$ have
the value it has? The first question asks for a history of the Solar System;
it is still unanswered and is topic of research. The second question is
addressed in \appendixref{units1}.
If gravitation is indeed universal, and if all objects really attract each
other, attraction should also occur between any two objects of everyday life.
Gravity must also work \emph{sideways}.\index{gravitation!sideways action of}
This is indeed the case, even though the effects are extremely small. Indeed,
the effects are so small that they were measured only long after universal
gravity had predicted them. On the other hand, measuring this effect is the
only way to determine the gravitational constant $G$. Let us see how to do
it.
We note that measuring the gravitational constant $G$ is also the only way to
determine the mass of the \ii[Earth!mass measurement]{Earth}. The first to do so, in
1798, was the English physicist \iinns{Henry Cavendish}; he used the machine,
ideas and method of \iinns{John Michell} who died when attempting the
experiment. Michell and Cavendish%
%
\footnote{Henry Cavendish \livedplace(1731 Nice1810 London) was one of the
great geniuses of physics; rich, autistic, misogynist, unmarried and
solitary, he found many rules of nature, but never published them. Had he
done so, his name would be much more well known. {John Michell}
\lived(17241793) was church minister, geologist and amateur astronomer.} %
%
called the aim and result of their experiments `weighing the Earth'.
% Jun 2010, Aug 2016
\csepsfnb{ifourmilab}{scale=1}{Home experiments that allow determining the
gravitational constant $G$, weighing the Earth, proving that gravity also
works sideways and showing that gravity curves space. Top left and right: a
torsion balance made of foam and lead, with pétanque (boules) masses as
fixed masses; centre right: a torsion balance made of wood and lead, with
stones as fixed masses; bottom: a time sequence showing how the stones do
attract the lead ({\textcopyright}~\protect\iinn{John Walker}).}
% Jun 2010
\csepsfnb{iuwash}{scale=1}{A modern precision torsion balance experiment to
measure the gravitational constant, performed at the University of
Washington ({\textcopyright}~\protect\iname{EötWash Group}).}
% Alternative:
% http://www.bipm.org/en/scientific/mass/pictures_mass/torsion_balance.html
% Jun 2010
The idea of Michell was to suspended a horizontal handle, with two masses at
the end, at the end of a long metal wire. He then approached two additional
large masses at the two ends of the handle, avoiding any air currents, and
measured how much the handle rotated. \figureref{ifourmilab} shows how to
repeat this experiment in your basement, and \figureref{iuwash} how to
perform it when you have a larger budget.
% Impr. Jun 2010
The value the gravitational constant $G$ found in more elaborate versions of
the MichellCavendish experiments is\index{gravitation!value of constant}
\begin{equation}
G = \csd{6.7\cdot 10^{11}}{Nm^2/kg^2} = \csd{6.7\cdot
10^{11}}{m^3/kg\;s^2} \cp
\end{equation}
Cavendish's experiment was thus the first to confirm that gravity also works
{sideways}. The experiment also allows deducing the mass $M$ of the Earth
from its radius $R$ and the relation $g=GM/R^2$.\challengn Therefore, the
experiment also allows to deduce the average density of the Earth. Finally, as
we will see later on, this experiment proves, if we keep in mind that the
speed of light is finite and invariant, that space is
curved.\seepagetwo{grlaghz} All this is achieved with this simple setup!
% Feb 2012
Cavendish found\cite{caverrr} a mass density of the Earth of 5.5 times that of
water.\index{Earth!density} This was a surprising result, because rock only
has 2.8 times the density of water. What is the origin of the large density
value?\challengn
% New Aug 2016
We note that $G$ has a small value. Above, we mentioned that gravity limits
motion. In fact, we can write the expression for universal gravitation in the
following way:
\begin{equation}
\frac{a r^2}{M} = G > 0
\end{equation}
Gravity prevents uniform motion. In fact, we can say more: \emph{Gravitation
is the smallest possible effect of the environment on a moving body.} All
other effects come on top of gravity. However, it is not easy to put this
statement in a simple\challengedif{simplegformula} formula.
% Impr. Jun 2010
Gravitation between everyday objects is weak. For example, two average people
\csd{1}{m} apart feel an acceleration towards each other that is less than
that exerted by a \iin[fly!common]{common fly} when landing on the
skin.\challengenor{flyskin} Therefore we usually do not notice the attraction
to other people. When we notice it, it is much stronger than that. The
measurement of $G$ thus proves that gravitation cannot be the true cause of
people falling in love, and also that erotic\index{eros} attraction is not of
gravitational origin, but of a different source. The physical basis for
love\index{love!physics of} will be studied later\seepagethree{edynstzys} in
our walk: it\index{gravitation!properties of)} is called
\ii{electromagnetism}.
% Impr. Jan 2015: !!!1 improve this picture, add 3d, search on internet
\cssmallepsf{ipotedef}{scale=1}{The potential and the gradient,
visualized for two spatial dimensions.}[%
\psfrag{f}{\small $\phi(x,y)$}%
\psfrag{g}{\small ${\rm grad\,} \phi$}%
\psfrag{x}{\small $x$}%
\psfrag{y}{\small $y$}%
]
%
% Improve one day  note of Jul 2016
\subsection{The gravitational potential} % added Aug 2009
% Index OK
Gravity\label{potdef} has an important property: all effects of gravitation
can also be described by another observable, namely the
\ii[potential!gravitational]{(gravitational) potential} $\varphi$. We then
have the simple relation that the acceleration is given by the \ii{gradient}
of the potential
\begin{equation}
{\bm a} = {\nabla \varphi} \qqhbox{or} {\bm a} = {\rm grad\,}
\varphi \cp
\label{minussigndef}
\end{equation}
The gradient is just a learned term for `slope along the steepest direction'.
The gradient is defined for any point on a slope, is large for a steep one and
small for a shallow one. The gradient points in the direction of steepest
ascent, as shown in \figureref{ipotedef}.
%
The gradient is abbreviated $\bm{\nabla}$, pronounced `nabla', and is
mathematically defined through the relation ${\bm{\nabla}} \varphi= (\partial
\varphi/\partial x, \partial \varphi/\partial y, \partial \varphi/\partial z)
= {\rm grad\,} \varphi$.%
%
\footnote{In two or more dimensions slopes are written
$\partial \varphi/\partial z$  where $\partial$ is still pronounced `d' 
because in those cases the expression $d \varphi/d z$ has a slightly
different meaning. The details lie outside the scope of this walk.} %
%
The minus sign in (\ref{minussigndef}) is introduced by convention, in order
to have higher potential values at larger heights. In everyday life, when the
spherical shape of the Earth can be neglected, the gravitational potential is
given by
\begin{equation}
\varphi = gh \cp
\end{equation}
The potential $\varphi$ is an interesting quantity; with a single number at
every position in space we can describe the vector aspects of gravitational
acceleration. It automatically gives that gravity in New Zealand acts in the
opposite direction to gravity in Paris. In addition, the potential suggests
the introduction of the socalled \ii{potential energy} $U$ by setting
\begin{equation}
U= m \varphi
\end{equation}
and thus allowing us to determine the change of \emph{kinetic} energy $T$ of a
body falling from a point 1 to a point 2 via
\begin{equation}
T_{1}T_{2}= U_{2}U_{1} \qquad\hbox{or}\qquad
{\te \frac{ 1 }{ 2}}
m_{\rm 1} {\bm v_{\rm 1}}^{2}

{\te \frac{ 1 }{ 2}}
m_{\rm 2} {\bm v_{\rm 2}}^{2}
=
m\varphi_{2}m\varphi_{1}
%
\cp
\end{equation}
In other words, the \ii[energy!conservation]{total energy}, defined as the sum
of kinetic and potential energy, is \emph{conserved} in motion due to gravity.
This is a characteristic property of gravitation. Gravity conserves energy
and momentum.
Not all\seepageone{enconszz} accelerations can be derived from a potential;
systems with this property are called
\ii[system!conservative]{conservative}. Observation shows that accelerations
due to friction are not conservative, but accelerations due to
electromagnetism are.
%
% Later we will see that the
% corresponding sum is conserved for all \emph{fundamental} interactions as
% well.\index{energy is conserved}\index{conservation of energy} We'll come
% back to this topic in more detail shortly.
%
In short, we can either say that gravity can be described by a potential, or
say that it conserves energy and momentum. Both mean the same. When the
nonspherical shape of the Earth can be neglected, the potential energy of an
object at height $h$ is given by
\begin{equation}
U = mgh \cp
\end{equation}
To get a feeling of how much energy this is, answer the following question. A
car with mass \csd{1}{Mg} falls down a cliff of \csd{100}{m}. How much water
can be heated from freezing point to boiling point with the energy of the
car?\challengenor{muchheat}
%
% Reread July 2016
\subsection{The shape of the Earth} % added Aug 2009
% Index OK
% Jan 2005
Universal gravity also explains why the Earth and most planets are (almost)
spherical. Since gravity increases with decreasing distance, a liquid body in
space will always try to form a spherical shape. Seen on a large scale, the
Earth is indeed liquid. We also know that the Earth is cooling down  that
is how the crust and the continents formed. The sphericity of smaller solid
objects encountered in space, such as the Moon, thus means that they used to
be liquid in older times.
The Earth is thus not flat, but roughly spherical. Therefore, the top of two
tall buildings is further apart than their base. Can this effect
be\challengenor{builddist} measured?
Sphericity considerably simplifies the description of motion. For a
spherical\index{Earth!shape(} or a pointlike body of mass $M$, the potential
$\varphi$ is\challengn
\begin{equation}
\varphi =G \,\frac{ M}{ r} \cp
\end{equation}
A potential considerably simplifies the description of motion, since a
potential is additive: given the potential of a point particle, we can
calculate the potential and then the motion around any other irregularly
shaped object.%
%
\footnote{Alternatively, for a general, extended body, the potential is found
by requiring that the \ii{divergence} of its gradient is given by the mass
(or charge) density times some proportionality constant. More precisely, we
have
\begin{equation}
\Delta\varphi= 4 \pi G \varrho
\label{Poiq}
\end{equation}
where $\varrho=\varrho({\bm x},t)$ is the mass volume density of the body
and the socalled \emph{Laplace}\index{Laplace operator} \ii{operator}
$\Delta$, pronounced `delta', is defined as
$\Delta f = \nabla \nabla f = \partial^{2} f/\partial x^{2} + \partial^{2}
f/\partial y^{2} + \partial^{2} f/\partial z^{2}$. Equation (\ref{Poiq}) is
called the \ii{Poisson equation} for the potential $\varphi$. It is named
after \iinns{SiméonDenis Poisson} \lived(17811840), eminent French
mathematician and physicist. The positions at which $\varrho$ is not zero
are called the \ii[potential!sources of]{sources} of the potential.
%
\label{earnself}
%
The socalled \iin{source term} $\Delta\varphi$ of a function is a measure
for how much the function $\varphi(x)$ at a point $x$ differs from the
average value in a region around that point. (Can you show this, by showing
that $\Delta \varphi \sim \bar\phi \phi(x)$?)\challengn In other words, the
Poisson equation (\ref{Poiq}) implies that the actual value of the potential
at a point is the same as the average value around that point minus the mass
density multiplied by $4 \pi G$. In particular, in the case of
\iin[space!empty]{empty space} the potential at a point is equal to the
average of the potential around that point.
Often the concept of \ii{gravitational field} is introduced, defined as
${\bm g}=\nabla \varphi$. We avoid this in our walk, because we will
discover that, following the theory of relativity, gravity is not due to a
field at all; in fact even the concept of gravitational potential turns out
to be only an approximation.} %
%
%It is then easy to calculate the motion of a comet near the
%Sun and all the planets.
Interestingly, the number $d$ of dimensions of space is coded into the
potential $\phi$ of a spherical mass: the dependence of $\phi$ on the radius
$r$ is in fact $1/r^{d2}$.\challengenor{d} The exponent $d2$ has been
checked experimentally to extremely high precision; no deviation of $d$ from
$3$ has ever been found.\cite{dmintwo}
\cssmallepsfnb{iearthshape}{scale=0.15}{The shape of the Earth, with
exaggerated height scale ({\textcopyright}~\protect\iname{GeoForschungsZentrum
Potsdam}).}
The concept of potential helps in understanding the \ii[Earth!shape]{shape} of
the Earth in more detail. Since most of the Earth is still liquid when seen
on a large scale, its\cite{erderot} surface is always horizontal with respect
to the direction determined by the combination of the accelerations of gravity
and rotation. In short, the Earth is \emph{not} a sphere. It is not an
ellipsoid either. The mathematical shape defined by the equilibrium
requirement is called a \ii{geoid}.\cite{something} The geoid shape,
illustrated in \figureref{iearthshape}, differs from a suitably chosen
ellipsoid by at most \csd{50}{m}. Can you describe the geoid
mathematically?\challenge % !!!5
The geoid is an excellent approximation to the actual shape of the Earth; sea
level differs from it by less than 20 metres. The differences can be measured
with satellite radar and are of great interest to geologists and
geographers. For example, it turns out that the \iin{South Pole} is nearer to
the equatorial plane than the \iin{North Pole} by about \csd{30}{m}. This is
probably due to the large land masses in the northern hemisphere.
Above we saw how the inertia of matter, through the
socalled\seepageone{iflattening} `\iin[force!centrifugal]{centrifugal
force}', increases the radius of the Earth at the Equator. In other words,
the Earth is \emph{flattened} at the poles. The Equator has a radius $a$ of
\csd{6.38}{Mm}, whereas the distance $b$ from the poles to the centre of the
Earth is \csd{6.36}{Mm}. The precise flattening $(ab)/a$ has the value
$1/298.3=0.0034$.\seeapp{units1}
%
%precise values, following
%booklet 6356,777 and 6378,163 and 298,04
%
% IERS: 6378,1363 km, 1/298.257
%
As a result, the top of \iin[Chimborazo, Mount]{Mount Chimborazo} in Ecuador,
even though its height is only \csd{6267}{m} above sea level, is about
\csd{20}{km} farther away from the centre of the\index{Chomolungma, Mount}
Earth than the top of \iin[Sagarmatha, Mount]{Mount Sagarmatha}%
%
\footnote{Mount Sagarmatha\index{Chomolungma, Mount} is sometimes also called
\iin[Everest, Mount]{Mount Everest}.} %
%
in Nepal, whose height above sea level is \csd{8850}{m}. The top of Mount
Chimborazo is in fact the point on the surface most distant from the centre of
the Earth.
The shape\label{eastoprotx} of the Earth has another important consequence.
If the Earth stopped rotating (but kept its shape),\index{Earth!stops
rotating} the water of the oceans would flow from the Equator to the poles;
all of Europe would be under water, except for the few mountains of the
\iin{Alps} that are higher than about \csd{4}{km}. The northern parts of
Europe would be covered by between \csd{6}{km} and \csd{10}{km} of water.
Mount Sagarmatha would be over \csd{11}{km} above sea level. We would also
walk inclined. If we take into account the resulting change of shape of the
Earth, the numbers come out somewhat smaller. In addition, the change in
shape would produce extremely strong earthquakes\index{earthquake} and storms.
As long as there are none of these effects, we can be \emph{sure} that the
\iin[Sun!will rise tomorrow]{Sun will indeed rise tomorrow}, despite
what\seepageone{wittgsonne} some philosophers\index{Earth!shape)} pretended.
%
% Apr 2005, impr. Jul 2005
\subsection{Dynamics  how do things move in various dimensions?}
% Index OK
The concept of potential is a powerful tool.\index{dynamics}\index{motion!and
dimensions} If a body can move only along a  straight or curved  line,
the concepts of kinetic and potential energy are sufficient to determine
completely the way the body moves.
In fact, motion in \emph{one dimension} follows directly from energy
conservation.\index{energy!conservation} For a body moving along a given
curve, the speed at every instant is given by energy conservation.
If a body can move in \emph{two dimensions}  i.e., on a flat or curved
surface  \emph{and} if the forces involved are \emph{internal} (which is
always the case in theory, but not in practice), the conservation of angular
momentum can be used. The full motion in two dimensions thus follows from
energy and angular momentum conservation.\index{angular momentum!conservation}
For example, all properties of free fall follow from energy and angular
momentum conservation. (Are you able to show this?)\challengenor{freefall2d}
Again, the potential is essential.
In the case of motion in \emph{three dimensions}, a more general rule for
determining motion is necessary. If more than \emph{two spatial dimensions}
are involved conservation is insufficient to determine how a body moves. It
turns out that general motion follows from a simple principle: the time
average of the difference between kinetic and potential energy must be as
small as possible. This is called the \emph{least action
principle}.\index{principle!of least action}\index{action!principle of
least} We will explain the details\seepageone{actprde} of this calculation
method later. But again, the potential is the main ingredient in the
calculation of change, and thus in the description of any example of motion.
For simple gravitational motions, motion is twodimensional, in a plane. Most
threedimensional problems are outside the scope of this text; in fact, some
of these problems are so hard that they still are subjects of research. In
this adventure, we will explore threedimensional motion only for selected
cases that provide important insights.
% % Mar 2012
% \csepsfnb{ideklination}{scale=1}{Some important concepts when observing the
% stars at night.}
% %
% % Improved May 2005, Oct 2009, Mar 2012, reread Jul 2016
% \subsection{Gravitation in the sky}
% % Index OK
% The expression $\smash{a=GM/r^2}$ for the acceleration\index{gravitation!and
% planets(} due to universal gravity also describes the\index{planets!and
% universal gravitation} motion of all the planets across the sky. We usually
% imagine to be located at the centre of the Sun and say that the planets `orbit
% the Sun'. How can we check this?
% First of all, looking at the sky at night, we can check that the planets
% always stay within the \ii{zodiac}, a narrow stripe across the sky. The
% centre line of the zodiac gives the path of the Sun and is called the
% \ii{ecliptic}, since the Moon must be located on it to produce an
% eclipse.\seepageone{ieclipse} This shows that planets move (approximately) in
% a single, common plane.%
% %
% \footnote{The apparent height of the ecliptic changes with the time of the
% year and is the reason for the changing seasons. Therefore\index{season}
% seasons are a gravitational effect as well.}
% % Mar 2012
% The detailed motion of the planets is not easy to describe. As
% \figureref{ideklination} shows, observing a planet or star requires measuring
% various angles. For a planet, these angles change every night. From the way
% the angles change, one can deduce the motion of the planets.
% %
% A few generations before \iin{Hooke}, using the observations of \ii{Tycho
% Brahe}, the Swabian astronomer \iinn{Johannes Kepler}, in his painstaking
% research on the movements of the planets in the zodiac, had deduced several
% `laws'.\index{Kepler's laws} The three main ones are as
% follows: %\label{keplerla}
% \smallskip
% \begin{Strich}
% \item[{1.}] Planets move on ellipses with the Sun located at one focus (1609).
% \item[{2.}] Planets sweep out equal areas in equal times (1609).
% % I also had 1604 for the first two  probably wrong
% \item[{3.}] All planets have the same ratio $T^2/d^3$ between the orbit
% duration $T$ and the semimajor axis $d$ (1619).
% \end{Strich}
% \smallskip
% \cssmallepsf{ielli}{scale=1}{The motion of a planet around the Sun, showing
% its semimajor axis $d$, which is also the spatial average of its distance
% from the Sun.}
% \np Kepler's results are illustrated in \figureref{ielli}. The sheer work
% required to deduce the three `laws' was enormous. \iname[Kepler,
% Johannes]{Kepler} had no calculating machine available. The calculation
% technology he used was the recently discovered \iin{logarithms}. Anyone who
% has used tables of logarithms to perform calculations can get a feeling for
% the amount of work behind these three discoveries.
% Now comes the central point. The huge volume of work by Brahe and Kepler can
% be summarized in the expression
% \begin{equation}
% a=GM/r^2 \cv \quad{or} {\bm a} =  G M {\bm r}/ r^{3 } \cvend
% \end{equation}
% as Hooke and a few others had stated. Let us see why.
% \csepsfnb{iellipsebook}{scale=1}{The proof that a planet moves in an ellipse
% (magenta) around the Sun, given an inverse square distance relation for
% gravitation (see text).}
% % Oct 2009, Impr. Apr 2010
% Why is the usual planetary orbit an ellipse?\cite{heckman} The simplest
% argument is given in \figureref{iellipsebook}. We know
% that\index{orbit!elliptical} the acceleration due to gravity varies as
% $\smash{a=GM/r^2}$. We also know that an orbiting body of mass $m$ has a
% constant energy $E<0$. We then can draw, around the Sun, the circle with
% radius $R=GMm/E$, which gives the largest distance that a body with energy
% $E$ can be from the Sun. We now project the planet position $P$ onto this
% circle, thus constructing a position $S$. We then reflect $S$ along the
% tangent to get a position $F$. This last position $F$ is constant in time, as
% a simple argument shows. (Can you find it?\challengenor{heckmansol}) As a
% result of the construction, the distance sum OP+PF is constant in time, and
% given by the radius $R=GMm/E$. Since this distance sum is constant, the
% orbit is an ellipse, because an ellipse is precisely the curve that appears
% when this sum is constant. (Remember that an ellipse can be drawn with a
% piece of rope in this way.) Point $F$, like the Sun, is a focus of the
% ellipse. This is the first of Kepler's `laws'.
% Can you confirm that also the other two of Kepler's `laws' follow from Hooke's
% expression of universal gravity?\challengenor{solconics} Publishing this
% result was the main achievement of Newton. Try to repeat his achievement; it
% will show you not only the difficulties, but also the possibilities of
% physics, and the joy that puzzles give.
% The second of Kepler's `laws', about equal swept areas, implies that planets
% move faster when they are near the Sun. It is a simple way to state the
% conservation of angular momentum. What does the third `law' state?\challengn
% Newton solved these puzzles with geometric drawing  though in quite a
% complex manner. It is well known that Newton was not able to write down, let
% alone handle, differential equations at the time he published his results on
% gravitation.\cite{a14} In fact, Newton's notation and calculation methods were
% poor. (Much poorer than yours!) The English mathematician \inames[Hardy,
% Godfrey H.]{Godfrey Hardy}%
% %
% \footnote{Godfrey Harold Hardy \lived(18771947) was an important % English
% number theorist, and the author of the wellknown \btsim A Mathematician's
% Apology/. He also `discovered' the famous Indian mathematician
% \iinn{Srinivasa Ramanujan}, and brought him to Britain.}
% %
% used to say that the insistence on using {Newton}'s integral and differential
% notation,
% %which he developed much later  instead of using the one of his
% rather than the earlier and better method, still common today, due to his
% rival \iname[Leibniz, Gottfried Wilhelm]{Leibniz}  threw back English
% mathematics by 100 years.
% To sum up, \iname[Kepler, Johannes]{Kepler}, \iname[Hooke, Robert]{Hooke} and
% \iname[Newton, Isaac]{Newton} became famous because they brought order to the
% description of planetary motion. They showed that all motion due to gravity
% follows from the same description, the inverse square distance. For this
% reason, the inverse square distance relation $a= GM/r^2$ is called the
% \emph{universal} law of gravity. Achieving this unification of motion
% description, though of small practical significance, was widely publicized.
% The main reason were the ageold prejudices and fantasies linked with
% \iin{astrology}.
% In fact, the inverse square distance relation explains many additional
% phenomena. It explains the motion and shape of the Milky Way and of the other
% galaxies, the motion of many weather phenomena, and explains why the Earth has
% an atmosphere but the Moon does not. (Can you explain
% this?)\challengenor{noatmo} In fact, universal gravity explains much more
% about the\index{gravitation!and planets)} Moon.
%
% Reread Jul 2016
\subsection{The Moon}
% Index OK
% Jun 2005
How long\index{Moon(} is a day on the Moon? The answer is roughly 29
Earthdays. That is the time that it takes for an observer on the Moon to see
the Sun again in the same position in the sky.
% Changed from 0.25 to 0.25*1.25 in Feb 2014, to have roughly same size as the
% following maps
\csmovfilmrepeat{lunation3}{scale=0.3125}{The change of the moon during the
month, showing its libration (QuickTime film
{\textcopyright}~\protect\iinn{Martin Elsässer})}
% EMAILED FEB 2008  melsaess@opentext.com
One often hears that the Moon always shows the same side to the Earth. But
this is wrong. As one can check with the naked eye, a given feature in the
centre of the face of the Moon at full Moon is not at the centre one week
later. The various motions leading to this change are called
\ii[libration]{librations}; they are shown in the film in
\figureref{lunation3}.
% \footnote{The film is in
% DivX 5 AVI format and requires a software plugin in Acrobat Reader that can
% play it.}
The motions appear mainly because the Moon does not describe a circular, but
an elliptical orbit around the Earth and because the axis of the Moon is
slightly inclined, when compared with that of its rotation around the Earth.
% and due to some smaller effects.
As a result, only around 45\,\% of the Moon's surface is permanently hidden
from Earth.
% maybe reduce file size
% Feb 2014: changed from 0.8 to 1 (larger image)
\csepsfnb{fhiddenmoonside}{scale=1}{High resolution maps (not photographs) of
the near side (left) and far side (right) of the moon, showing how often the
latter saved the Earth from meteorite impacts (courtesy USGS).}
The first photographs of the hidden\index{Moon!hidden part} area of the Moon
were taken in the 1960s by a Soviet artificial satellite; modern satellites
provided exact maps, as shown in \figureref{fhiddenmoonside}. (Just zoom
into the figure for fun.)\challengn The hidden surface is much more irregular
than the visible one, as the hidden side is the one that intercepts most
asteroids attracted by the Earth. Thus the gravitation of the Moon helps to
deflect asteroids from the Earth. The number of animal life extinctions is
thus reduced to a small, but not negligible number. In other words, the
gravitational attraction of the Moon has saved the human race from extinction
many times over.%
%
% Corrected in Jan 2014
\footnote{The web pages \url{www.minorplanetcenter.net/iau/lists/Closest.html}
and
% needs http:
\href{http://www.minorplanetcenter.net/iau/lists/InnerPlot.html}{InnerPlot.html}
give an impression of the number of objects that almost hit the Earth every
year. Without the Moon, we would have many additional catastrophes.} %
The trips to the Moon in the 1970s also showed that the Moon originated from
the Earth itself: long ago, an object hit the Earth almost tangentially and
threw a sizeable fraction of material up into the sky. This is the only
mechanism able to explain the large size of the Moon, its low iron content, as
well as its general material composition.\cite{moonconf}
The Moon is receding from the Earth at \csd{3.8}{cm} a year.\cite{gutzwi} This
result confirms the old deduction that the \iin[tide!slowing Moon]{tides}
slow down the Earth's rotation. Can you imagine how this measurement
was\challengenor{moondi} performed? Since the Moon slows down the Earth, the
Earth also changes shape due to this effect. (Remember that the shape of the
Earth depends on its speed of rotation.) These changes in shape influence the
\iin[tectonism]{tectonic activity} of the Earth, and maybe also the drift of
the continents.\index{earthquake!and Moon}
The Moon has many effects on animal life. A famous example is the midge
\iie{Clunio}, which lives on coasts with pronounced \iin[tide!and hatching
insects]{tides}.\cite{a15} Clunio spends between six and twelve weeks as a
larva, %, under water, % not
sure then hatches and lives for only one or two hours as an adult flying
insect, during which time it reproduces. The midges will only reproduce if
they hatch during the low tide phase of a \emph{spring tide}. Spring tides
are the especially strong \iin[tide!spring]{tides} during the full and new
moons, when the solar and lunar effects combine, and occur only every 14.8
days. In 1995, \iinn{Dietrich Neumann} showed that the larvae have two
builtin clocks, a circadian and a circalunar one, which together control the
hatching to precisely those few hours when the insect can reproduce. He also
showed that the circalunar clock is synchronized by the brightness of the Moon
at night. In other words, the larvae monitor the Moon at night and then
decide when to hatch: they are the smallest known \iin[astronomer!smallest
known]{astronomers}.
If {insects} can have \iin{circalunar} cycles, it should come as no surprise
that \iin{women} also have such a cycle;\index{cycle!menstrual} however, in
this case the precise origin of the cycle length is still unknown and a topic
of research.\cite{menstr}
The Moon also helps to stabilize the tilt of the Earth's axis, keeping it more
or less fixed relative to the plane of motion around the Sun. Without the
Moon, the axis would change its direction irregularly, we would not have a
regular day and night rhythm, we would have extremely large climate changes,
and the evolution of life would have been impossible.\cite{obli} Without the
Moon, the Earth would also rotate much faster and we would have much less
clement weather.\cite{notmoon} The Moon's main remaining effect on the Earth,
the precession of its axis, is responsible for the ice\index{Moon)}
ages.\seepageone{moonprecsn}
% Nov 2016
The orbit of the Moon\index{Moon!orbit} is still a topic of research. It is
still not clear why the Moon orbit is at a \csd{5}{\csdegrees} to the ecliptic
and how the orbit changed since the Moon formed. Possibly, the collision that
led to the formation of the Moon tilted the rotation axis of the Earth and the
original Moon; then over thousands of millions of years,\cite{moonstorynew}
both axes moved in complicated ways towards the ecliptic, one more than the
other. During this evolution, the distance to the Moon is estimated to have
increased by a factor of 15.
% % Dec 2006 seems wrong
% % Updated in Feb 2005
% Furthermore, the Moon shields the Earth from cosmic radiation
% by\index{cosmic rays} greatly increasing the Earth's magnetic field. In
% other
% words, the Moon is of central importance for the evolution of life.
% Understanding how often Earthsized planets have Moonsized satellites is
% thus
% important for the estimation of the probability that life exists on other
% planets.\cite{moonmake} So far, it seems that large satellites are rare;
% there
% are only four known moons that are larger than that of the Earth, but they
% circle much larger planets, namely Jupiter and Saturn. Indeed, the
% formation
% of satellites is still an area of research. But let us return to the
% effects
% of gravitation in the sky.
% www.windows.ucar.edu/tour/link=/our_solar_system/moons_table.html
%
% Reread Jul 2016
\subsection{Orbits  conic sections and more}
% Index ok
The path\label{ugorbits} of a body continuously orbiting another under the
influence of gravity is an \ii{ellipse} with the central body at one focus. A
circular orbit is also possible, a circle being a special case of an ellipse.
Single encounters of two objects can also be \ii[parabola]{parabolas} or
\ii[hyperbola]{hyperbolas}, as shown in \figureref{iuniorbit}.
% Feb 2005
Circles, ellipses, parabolas and hyperbolas are collectively known as
\ii{conic sections}. Indeed each of these curves can be produced by cutting a
cone with a knife. Are you able to confirm this?\challengn
% Caption improved in Mar 2013
\csepsfnb{iuniorbit}{scale=1}{The possible orbits, due to universal gravity,
of a small mass around a \emph{single} large mass (left) and a few recent
examples of measured orbits (right), namely those of some extrasolar planets
and of the Earth, all drawn around their respective central star, with
distances given in astronomical units ({\textcopyright}~\protect\iinn{Geoffrey
Marcy}).}
If orbits are mostly ellipses, it follows that comets \emph{return}. The
English astronomer \iinn{Edmund Halley} \lived(16561742) was the first to
draw this conclusion and to predict the return of a \iin{comet}. It arrived
at the predicted date in 1756, after his death, and is now named after him.
%This result
%finally settled a long dispute on whether comets were heavenly bodies or only
%images on the sky.
The period of Halley's comet\index{comet!Halley's} is between 74 and 80 years;
the first recorded sighting was 22 centuries ago, and it has been seen at
every one of its 30 passages since, the last time in 1986.
Depending on the initial energy and the initial angular momentum of the body
with respect to the central planet, paths are either \emph{elliptic},
\emph{parabolic} or \emph{hyperbolic}. Can you determine the conditions for
the energy and the angular momentum needed for these paths to
appear?\challengenor{elparhycond}
In practice, parabolic orbits do not exist in nature. (Some comets seem to
approach this case when moving around the Sun; but almost all comets follow
elliptical paths  as long as they are far from other planets.). Hyperbolic
paths do exist; artificial satellites follow them when they are shot towards a
planet, usually with the aim of changing the direction of the satellite's
journey across the Solar System.
% May 2005
Why does the inverse square `law' lead to conic sections?\label{conicsec}
First, for two bodies, the total angular momentum $L$ is a constant:
\begin{equation}
L = m r^2 \dot \phi = m r^2 \left (\frac{\diffd\phi}{\diffd t} \right )
\end{equation}
and therefore the motion lies in a plane. Also the energy $E$ is a constant
\begin{equation}
E = {\te \frac{1}{2}} m \left (\frac{\diffd r}{\diffd t} \right )^2 +
{\te \frac{1}{2}} m \left (r\frac{\diffd\phi}{\diffd t} \right )^2
 G \frac{mM}{r} \cp
\end{equation}
Together, the two equations imply that\challengn
\begin{equation}
r= \frac{L^2}{Gm^2M} \; % checked
%
\frac{1}{1+
\sqrt{1+\frac{\hbox{$2EL^2$}}{\hbox{$G^2 m^3 M^2$}} } \cos \phi}
% checked
\cp
\end{equation}
% May 2005
Now, any curve defined by the general expression
\begin{equation}
r = \frac{C}{1 + e \cos \phi} \quad\hbox{or}\quad r = \frac{C}{1  e \cos
\phi}
\end{equation}
is an ellipse for $0 < e < 1$, a parabola for $e=1$ and a hyperbola for $e
>1$, one focus being at the origin. The quantity $e$, called the
\ii{eccentricity}, describes how squeezed the curve is. In other words, a
body in orbit around a central mass follows a conic section.
% Apr 2006
In all orbits, also the heavy mass moves. In fact, both bodies orbit around
the common centre of mass. Both bodies follow the same type of curve 
ellipse, parabola or hyperbola  but the sizes of the two curves
differ.\challengn
% Feb 2005
If more than two objects move under mutual gravitation, many additional
possibilities for motions appear. The classification and the motions are
quite complex. In fact, this socalled \emph{manybody problem}
is\index{problem!manybody(}\index{manybody problem(} still a topic of
research, both for astronomers and for mathematicians. Let us look at a few
observations.
When several planets circle a star, they also attract each other. Planets
thus do not move in perfect ellipses. The largest deviation is a
\iin[perihelion!shift]{perihelion shift}, as shown in
\figureref{ieaorbit}.\seepageone{ieaorbit} It is observed for Mercury and a
few other planets, including the Earth. Other deviations from elliptical
paths appear during a single orbit. In 1846, the observed deviations of the
motion of the planet Uranus from the path predicted by universal gravity were
used to predict the existence of another planet, Neptune, which was discovered
shortly afterwards.
We have seen\seepageone{negmass} that mass is always positive and that
gravitation is thus always attractive; there is \emph{no} \ii{antigravity}.
Can gravity be used for \ii{levitation} nevertheless, using more than two
bodies? Yes; there are two examples.%
%
\footnote{Levitation is discussed in detail in\seepagethree{levdis} the
section on electrodynamics.} %
%
The first are the \iin{geostationary satellites}, which are used for easy
transmission of television and other signals from and towards Earth.
% % WORKS  MAR 2013
% \csmovfilm{geo91313175}{scale=1}{The motion of geostationary satellites when
% filmed from the Earth, showing the location of the celesial Equator (mov film
% {\textcopyright}~\protect\iinn{Michael Kunze}).} % please send him mail
% % WORKS  MAR 2013
% \csmpgfilm{geo91313175}{scale=1}{The motion of geostationary satellites when
% filmed from the Earth, showing the location of the celesial Equator (mpg film
% {\textcopyright}~\protect\iinn{Michael Kunze}).} % please send him mail
% http://apod.nasa.gov/apod/ap120411.html I have permission!
% WORKS  MAR 2013
\csmp4filmwide{geo91313175}{scale=1}{Geostationary satellites, seen here in
the upper left quadrant, move against the other stars and show the location of
the celestial Equator. (MP4
film {\textcopyright}~\protect\iinn{Michael Kunze})} % please send him mail
% % Nov 2012 % I do not take this film, even though it is interesting
% \csmpgmov?{astramovie}{scale=1}{The motion of geostationary satellites when
% filmed from the Earth ({\textcopyright}~\protect\iname{Astra})}
% % wn.com/Astra_1M
\csepsfnb{ilagr}{scale=1}{Geostationary satellites (left)
and the main stable Lagrangian points (right).}
%
%
The \ii[libration!Lagrangian points]{Lagrangian libration points} are the
second example. Named after their discoverer, these are points in space near
a twobody system, such as MoonEarth or EarthSun, in which small objects
have a stable equilibrium position. An overview is given in
\figureref{ilagr}. Can you find their precise position, remembering to take
rotation into account?\challengenor{l123} There are three additional
Lagrangian points on the EarthMoon line (or Sunplanet line). How many of
them are stable?\challengedif{l123d}
% the middle one; levyleblond says the outer two are stable
There are thousands of asteroids, called \ii{Trojan asteroids},
at\index{asteroid!Trojan} and around the Lagrangian points of the SunJupiter
system. In 1990, a Trojan asteroid for the MarsSun system was discovered.
Finally, in 1997, an `almost Trojan' asteroid was found that follows the Earth
on its way around the Sun (it is only transitionary and follows a somewhat
more complex orbit). This `second companion' of the Earth has a diameter of
\csd{5}{km}.\cite{libea} Similarly, on the main Lagrangian points of the
EarthMoon system a high concentration of dust has been observed.
% NOT THIS CANDIDATE: http://ifa.hawaii.edu/~barnes
%
% Mar 2014; OK Mar 2015
\csepsfnb{iirregularorbit}{scale=1}{An example of irregular orbit, partly
measured and partly calculated, due to the gravitational attraction of
\emph{several} masses: the orbit of the temporary Earth quasisatellite 2003
YN107 in geocentric coordinates. This asteroid, with a diameter of 20(10)\,m,
became orbitally trapped near the Earth around 1995 and remained so until
2006. The black circle represents the Moon's orbit around the Earth.
({\textcopyright}~\protect\iinn{Seppo Mikkola}).}
% Mar 2014
Astronomers know that many other objects follow irregular orbits, especially
asteroids. For example, asteroid 2003 YN107\cite{mikkola} followed an
irregular orbit, shown in \figureref{iirregularorbit}, that accompanied the
Earth for a number of years.
To sum up, the single equation ${\bm a} =  G M {\bm r}/ r^{3 }$ correctly
describes a large number of phenomena in the sky.\index{universality of
gravity} The first person to make clear that this expression describes
\emph{everything} happening in the sky was \iinn{Pierre~Simon Laplace} %
in his famous treatise %% Traité de % part of title or not? yes,
%% says the bnf
\btsim Traité de mécanique céleste/. When \iname{Napoleon} told him that he
found no mention about the creator in the book, Laplace gave a famous, one
sentence summary of his book: \emph{Je n'ai pas eu besoin de cette hypothèse.}
`I had no need for this hypothesis.'\index{gods!and Laplace} In particular,
Laplace studied the stability of the Solar System, the eccentricity of the
lunar orbit, and the eccentricities of the planetary orbits, always getting
full agreement between calculation and measurement.
These results are quite a feat for the simple expression of universal
gravitation; they also explain why it is called `universal'. But how
\emph{accurate} is the formula? Since astronomy allows the most precise
measurements of gravitational motion, it also provides the most stringent
tests. %Pluto was found by mistake, I thought
In 1849, \iinn{Urbain Le~Verrier} concluded after intensive study that there
was only one known example of a discrepancy between observation and universal
gravity, namely one observation for the planet \iin{Mercury}. (Nowadays a few
more are known.) The point of least distance to the Sun of the orbit of
planet Mercury, its \ii{perihelion}, rotates around the Sun at a rate that is
slightly less than that predicted: he found a tiny difference, around
\csd{38}{\csseconds} per century.\cite{invrev} (This was corrected to
\csd{43}{\csseconds} per century in 1882 by \iinn{Simon Newcomb}.) Le Verrier
thought that the difference was due to a planet between Mercury and the Sun,
\ii{Vulcan}, which he chased for many years without success. Indeed, Vulcan
does not exist. The correct explanation of the difference had to wait
for\seepagetwo{perishift} \iinn{Albert
Einstein}.\index{problem!manybody)}\index{manybody problem)}
\csepsfnb{i4tides}{scale=1}{Tides at SaintValéry en Caux on 20 September
2005 ({\textcopyright}~\protect\iinn{Gilles Régnier}).}
% EMAILED FEB 2008  gilles@gillesregnier.com
%
% Reread Jul 2016
\subsection{Tides}
% Index OK
\label{tides}%
%
Why do physics texts always talk about \iin[tide!importance
of]{tides}?\cite{tideanim} Because, as general relativity will show, tides
prove that space is curved! It is thus useful to study them in a bit more
detail. \figureref{i4tides} how striking tides can be. Gravitation explains
the sea {tides} as results of the attraction of the ocean water by the Moon
and the Sun. Tides are interesting; even though the amplitude of the tides is
only about \csd{0.5}{m} on the open sea, it can be up to \csd{20}{m} at
special places near the coast. Can you imagine why?\challengenor{reso} The
\emph{soil} is also lifted and lowered by the Sun and the Moon, by about
\csd{0.3}{m}, as satellite measurements show.\cite{gravimetry} Even the
\emph{atmosphere} is subject to tides, and the corresponding pressure
variations can be filtered out from the weather pressure
measurements.\cite{FalkRuppel}
% Both OK Mar 2015
\cstftlepsfpsfragboth{itidea}{scale=1}{Tidal deformations due to
gravity.}[%
\psfrag{t0}{\small $t=0$}%
\psfrag{t1}{\small $t_{1}$}%
] % OK PSFRAG, APR 2014
{itidebare}{scale=1}{The origin of tides.}[]
Tides appear for any \emph{extended} body moving in the gravitational field of
another. To understand the origin of tides, picture a body in orbit, like the
Earth, and imagine its components, such as the segments of
\figureref{itidea}, as being held together by springs. Universal gravity
implies that orbits are slower the more distant they are from a central body.
As a result, the segment on the outside of the orbit would like to be slower
than the central one; but it is \emph{pulled} by the rest of the body through
the springs. In contrast, the inside segment would like to orbit more rapidly
but is \emph{retained} by the others. Being slowed down, the inside segments
want to fall towards the Sun. In sum, both segments feel a pull away from the
centre of the body, limited by the springs that stop the deformation.
Therefore, \emph{extended bodies are deformed in the direction of the field
inhomogeneity.}
For example, as a result of tidal forces, the Moon always has (roughly) the
same face to the Earth. In addition, its radius in direction of the Earth is
larger by about \csd{5}{m} than the radius perpendicular to it.
% the internet says so
If the inner springs are too weak, the body is torn into pieces; in this way a
\emph{ring} of fragments can form, such as the asteroid ring between Mars and
Jupiter or the rings around \iin{Saturn}.\index{rings!astronomical, and
tides}
% Imrpoved Apr 2006
Let us return to the Earth. If a body is surrounded by water, it will form
bulges in the direction of the applied gravitational field. In order to
measure and compare the strength of the tides from the Sun and the Moon, we
reduce tidal effects to their bare minimum. As shown in
\figureref{itidebare}, we can study the deformation of a body due to gravity
by studying the arrangement of four bodies. We can study the free fall case,
because orbital motion and free fall are equivalent. Now, gravity makes some
of the pieces approach and others diverge, depending on their relative
positions. The figure makes clear that the strength of the deformation 
water has no builtin springs  depends on the change of gravitational
acceleration with distance; in other words, the \emph{relative} acceleration
that leads to the tides is proportional to the derivative of the gravitational
acceleration.
Using the numbers from \appendixref{units1},\seepageone{units1} the
gravitational accelerations from the Sun and the Moon measured on Earth are
\begin{align}
&a_{\rm Sun}=\frac { GM_{\rm Sun} }{d^2_{\rm Sun}} = \csd{5.9}{mm/s^2} \non
%
&a_{\rm Moon}=\frac { GM_{\rm Moon} }{d^2_{\rm Moon}} = \csd{0.033}{mm/s^2}
%\label{eq:asun}
\end{align}
and thus the attraction from the Moon is about 178 times weaker than that from
the Sun.
When two nearby bodies fall near a large mass, the relative acceleration is
proportional to their distance, and follows $da= (da/dr) \, dr$. The
proportionality factor $da/dr= \nabla a $, called the \emph{tidal
acceleration} (gradient), is the true measure of tidal
effects.\indexs{acceleration!tidal} Near a large spherical mass $M$, it is
given by\challengn
\begin{equation}
\frac {da}{dr} % _{\rm rel}
=  \frac { 2 GM }{r^3}
%\label{eq:asdfasds}
\end{equation}
which yields the values
\begin{align}
&\frac { da_{\rm Sun %, rel
}}{dr}=  \frac {2 GM_{\rm Sun} }{d^3_{\rm Sun}} = \csd{0.8 \cdot
10^{13}}{/s^2} \non
%
&\frac { da_{\rm Moon %, rel
}}{dr}= \frac {2 GM_{\rm Moon} }{d^3_{\rm Moon}} =
\csd{1.7 \cdot 10^{13}}{/s^2} \cp
%\label{eq:asun2}
\end{align}
In other words, despite the much weaker pull of the Moon, its tides are
predicted to be over \emph{twice as strong} as the tides from the Sun; this is
indeed observed. When Sun, Moon and Earth are aligned, the two tides add up;
these socalled \ii{spring tides} are especially strong and happen every 14.8
days, at full and new moon.
% Dec 2006, from Andrew Young
Tides lead to a pretty puzzle. Moon tides are much stronger than Sun tides.
This implies that the Moon is much denser than the Sun.\index{Moon!density and
tides}\index{Sun!density and tides} Why?\challengenor{tidedensity}
% MAr 2012
\csepsf{itideintro}{scale=1}{The Earth, the Moon and the friction effects of
the tides (not to scale).}
Tides\label{tidfryy} also produce \emph{friction}, as shown in
\figureref{itideintro}.\index{friction!produced by tides}\index{tide!and
friction} The friction leads to a slowing of the Earth's rotation.
Nowadays, the slowdown can be measured by precise clocks (even though short
time variations due to other effects, such as the weather,\cite{dayle} are
often larger). The results fit well with fossil results showing that 400
million years ago, in the \iin{Devonian} period,\seepagetwo{univhist} a year
had 400 days, and a day about 22 hours.\index{year!number of days in the
past}\index{day!length of} It is also estimated that 900 million years ago,
each of the 481 days of a year were 18.2 hours long. The friction at the
basis of this slowdown also results in an increase in the distance of the Moon
from the Earth by about \csd{3.8}{cm} per year. Are you able to explain
why?\challengenor{moondepa}
% Apr 2006
\cssmallepsfnb{iiovolcanism}{scale=1}{A spectacular result of tides:
volcanism on Io (NASA).}
% Apr 2006 (info is from Fathi Namouni); ``no prediction'' in Aug 2006
As mentioned above, the tidal motion of the soil is also responsible for the
triggering of \emph{earthquakes}. Thus the Moon\index{Moon!dangers of} can
have also dangerous effects on Earth. (Unfortunately, knowing the mechanism
does not allow predicting earthquakes.) The most fascinating example of tidal
effects is seen on Jupiter's satellite \iin{Io}. Its tides are so strong that
they induce intense volcanic activity, as shown in \figureref{iiovolcanism},
with eruption plumes as high as \csd{500}{km}. If tides are even stronger,
they can destroy the body altogether, as happened to the body between Mars and
Jupiter that formed the \iin{planetoids}, or (possibly) to the moons that led
to Saturn's rings.
In summary, tides are due to relative accelerations of nearby mass points.
This has an important consequence. In the chapter on general
relativity\seepagetwo{gravyty} we will find that time multiplied by the speed
of light plays the same role as length. Time then becomes an additional
dimension, as shown in \figureref{itidecur}. Using this similarity, two free
particles moving in the same direction correspond to parallel lines in
spacetime. Two particles falling sidebyside also correspond to parallel
lines. Tides show that such particles approach each other. In other words,
tides imply that parallel lines approach each other.\seepagetwo{tidesrel} But
parallel lines can approach each other \emph{only} if spacetime is {curved}.
In short, tides imply \emph{curved} spacetime and space. This simple
reasoning could have been performed in the eighteenth century; however, it
took another 200 years and \iinn{Albert Einstein}'s genius to uncover it.
% OK psfrag of APR 2014; OK Mar 2015
\cstftlepsfpsfragboth{itidecur}{scale=1}{Particles falling sidebyside
approach over
time.}[\psfrag{x}{$x$}\psfrag{t}{$t$}\psfrag{t1}{$t_{1}$}\psfrag{t2}{$t_{2}$}]%
{ilightgravbend}{scale=1}{Masses bend
light.}[\psfrag{a}{$\alpha$}\psfrag{b}{$b$}\psfrag{M}{$M$}]
%
% Impr. July 2016
\subsection{Can light fall?}
% Index OK
\begin{quote}\selectlanguage{german}%
Die Maxime, jederzeit selbst zu denken, ist die Aufklärung.\\
\iinn{Immanuel Kant}\selectlanguage{british}%
% bad pagination if footonote one line higher
\footnote{The maxim to think at all times for oneself is the
\iin{enlightenment}.}%
\end{quote}
\label{solli}%
%
\np Towards\label{canlightfall} the end of the seventeenth century people
discovered that light\index{fall!of light}\index{light!fall of} has a finite
velocity  a story which we will tell in detail\index{mass!deflects light}
later.\seepagetwo{specialrelat} An entity that moves with infinite velocity
cannot be affected by gravity, as there is no time to produce an effect. An
entity with a finite speed, however, should feel gravity and thus fall.
%\cssmallepsf{ilightgravbend}{scale=1}{Masses bend light.}
Does its speed increase when light reaches the surface of the Earth? For
almost three centuries people had no means of detecting any such effect; so
the question was not investigated. Then, in 1801, the Prussian astronomer
\iinn{Johann Soldner} \lived(17761833) was the first to put the question in a
different way.\cite{soldnera} Being an astronomer, he was used to measuring
stars and their observation angles. He realized that light passing near a
massive body would be \emph{deflected} due to gravity.\index{light!deflection
near masses}
Soldner studied a body on a hyperbolic path, moving with velocity $c$ past a
spherical mass $M$ at distance $b$ (measured from the centre), as shown in
\figureref{ilightgravbend}. Soldner deduced the deflection
angle\challenge % !!!5
\begin{equation}
\alpha_{\rm univ.\ grav.}= \frac { 2 }{ b }\frac { G M }{ c^{2} } \cp
\label{eq:sold}
\end{equation}
The value of the angle is largest when the motion is just grazing the mass
$M$. For light deflected by the mass of the Sun, the angle turns out to be at
most a tiny \csd{0.88}{\csseconds}$=\;$\csd{4.3}{\muunit rad}. In Soldner's
time, this angle was too small to be measured. Thus the issue was forgotten.
Had it been pursued, general relativity would have begun as an experimental
science, and not as the theoretical effort of \iinn{Albert Einstein}! Why?
The\seepagetwo{soldi} value just calculated is \emph{different} from the
measured value. The first measurement took place in 1919;%
%
\footnote{By the way, how would you measure the deflection of light near the
bright\challengenor{deflsuntri} Sun?}
%
it found the correct dependence on the distance, but found a deflection of up
to \csd{1.75}{\csseconds}, exactly double that of expression (\ref{eq:sold}).
The reason is not easy to find; in fact, it is due to the curvature of space,
as we will see. In summary, light can fall, but the issue hides some
surprises.
%
% Improved completely in August 2014
\subsection{Mass: inertial and gravitational}
% Index OK
Mass\index{mass!identity of gravitational and inertial(} describes how an
object interacts with others. In our walk, we have encountered two of its
aspects. \emph{Inertial mass} is\index{mass!inertial, definition} the
property that keeps objects moving and that offers resistance to a change in
their motion. \emph{Gravitational mass} is\index{mass!gravitational,
definition} the property responsible for the acceleration of bodies nearby
(the active aspect) or of being accelerated by objects nearby (the passive
aspect). For example, the active aspect of the mass of the Earth determines
the surface acceleration of bodies; the passive aspect of the bodies allows us
to weigh them in order to measure their mass using distances only, e.g.~on a
scale or a balance. The gravitational mass is the basis of \ii{weight}, the
difficulty of lifting things.%
%
\footnote{What are the weight values shown by a balance for a person of
\csd{85}{kg} \iin{juggling} three balls of \csd{0.3}{kg}\challenge
each?} % !!!2 have solution in 2016 email!
Is the gravitational mass of a body equal to its inertial mass? A rough
answer is given by the experience that an object that is difficult to move is
also difficult to lift. The simplest experiment is to take two bodies of
different masses and let them fall. If the acceleration is the same for all
bodies, inertial mass is equal to (passive) gravitational mass, because in the
relation $ma = \nabla (GMm/r)$ the lefthand $m$ is actually the inertial
mass, and the righthand $m$ is actually the gravitational mass.
\label{galmimg}
%
Already in the seventeenth century \iname[Galilei, Galileo]{Galileo} had made
widely known an even older argument showing without a single experiment that
the gravitational acceleration is indeed \emph{the same} for all bodies. If
larger masses fell more rapidly than smaller ones, then the following paradox
would appear. Any body can be seen as being composed of a large fragment
attached to a small fragment. If small bodies really fell less rapidly, the
small fragment would slow the large fragment down, so that the complete body
would have to fall \emph{less} rapidly than the larger fragment (or break into
pieces). At the same time, the body being larger than its fragment, it should
fall \emph{more} rapidly than that fragment. This is obviously impossible:
all masses must fall with the same acceleration.
Many accurate experiments have been performed since \iname[Galilei,
Galileo]{Galileo}'s original discussion. In all of them the independence of
the acceleration of free fall from mass and material composition has been
confirmed with the precision they allowed.\cite{mimg} In other words,
experiments confirm:
\begin{quotation}
\npcsrhd Gravitational mass and inertial mass are \emph{equal}.
\end{quotation}
What is the origin of this mysterious equality?
% Improved August 2014
The equality of gravitational and inertial mass is not a mystery at all.
% This socalled `mystery' is an example of disinformation.
% , now common across the whole world of physics
% education.
Let us go back to the definition of mass\seepageone{mass1} as a negative
inverse acceleration ratio. We mentioned that the physical origin of the
accelerations does not play a role in the definition because the origin does
not appear in the expression. In other words, the value of the mass is {by
definition} independent of the interaction. That means in particular that
inertial mass, based on and measured with the electromagnetic interaction, and
gravitational mass\index{mass!identity of gravitational and inertial} are
identical \emph{by definition}.
% Dec 2016
\csepsfnb{imassequality}{scale=1}{The falling ball is in inertial motion
for a falling observer and in gravitational motion for an observer on the
ground. Therefore, inertial mass is the same as gravitational mass.}
% Dec 2016
The best proof of the equality of inertial and gravitational mass is
illustrated in \figureref{imassequality}: it shows that the two concepts only
differ by the viewpoint of the observer. Inertial mass and gravitational mass
describe the same observation.
% Improved August 2014
We also note that we have not defined a separate concept of `passive
gravitational mass'. (This concept is sometimes found in research papers.)
The mass being accelerated by gravitation is the inertial mass. Worse, there
is no way to define a `passive gravitational mass' that\index{mass!no passive
gravitational} differs from inertial mass. Try it!\challengenor{passgrmass}
All methods that measure a passive gravitational mass, such as weighing an
object, cannot be distinguished from the methods that determine inertial mass
from its reaction to acceleration. Indeed, all these methods use the same
nongravitational mechanisms. Bathroom scales are a typical example.
% Improved August 2014
Indeed, if the `passive gravitational mass' were\index{mass!no passive
gravitational} different from the inertial mass, we would have strange
consequences. Not only is it hard to distinguish the two in an experiment;
for those bodies for which it were different we would get into trouble with
energy conservation.\challengn
% Improved August 2014
In fact, also assuming that (`active') `gravitational mass' differs from
inertial mass gets us into trouble.
% Another way of looking at the issue is as follows.
How could `gravitational mass' differ from inertial mass? Would the
difference depend on relative velocity, time, position, composition or on mass
itself? No. Each of these possibilities contradicts either energy or
momentum conservation.\challengenor{massissues}
% Improved August 2014
In summary, it is no wonder that all measurements confirm the equality of all
mass types: there is no other option  as Galileo pointed out. The lack of
other options is due to the fundamental equivalence of all mass definitions:
%
\begin{quotation}
\noindent \csrhd Mass ratios are acceleration ratios.
\end{quotation}
%
The topic is usually rehashed in general relativity,\seepagetwo{grinma} with
no new results, because the definition of mass remains the same.
Gravitational and inertial masses remain equal. In short:
%
\begin{quotation}
\noindent \csrhd Mass is a unique property of each body.
\end{quotation}
%
% OLD NONSENSE:
%
% Both types of masses really make sense only for universe with more than one
% body, even though inertial mass only needs the existence of other bodies
% with vanishing mass, whereas gravitational mass needs larger bodies to be
% defined.
%
% Inertial mass is its fundamental, or absolute aspect. Gravitational
% mass is relative; it specifies how a mass relates to other masses.
%
Another, deeper issue remains, though. What is the \emph{origin} of mass?
Why does it exist? This simple but deep question cannot be answered by
classical physics. We will need some patience to find
out.\index{mass!identity of gravitational and inertial)}
%
% Impr. Jul 2016
\subsection{Curiosities and fun challenges about gravitation}
% Index OK
\begin{quote}
% Fallen ist weder ein Fehler noch eine Schande;\\
% Liegen bleiben ist beides.\footnote{`Falling is neither a fault nor a
% shame; keep lying is both.'
\selectlanguage{german}Fallen ist weder gefährlich noch eine Schande; Liegen
bleiben ist beides.\selectlanguage{british}\footnote{`Falling is neither
dangerous nor a shame; to keep lying is both.' Konrad Adenauer
\livedplace(1876 Köln1967 Rhöndorf), West German Chancellor.}\\
\iinn{Konrad Adenauer}
\end{quote}
\begin{curiosity}
% Dec 2005
\item[] Gravity on the Moon is only one sixth of that on the Earth. Why does
this imply that it is difficult to walk quickly and to run on the Moon (as
can be seen in the \csac{TV} images recorded there)?
% Oct 2017
\item Understand and explain the following statement: a beam balance measures
mass, a spring scale measures weight.\challengn
% Jun 2016
\csepsfnb{iminimoonnasa}{scale=1}{The calculated obit of the
quasisatellite 2016 HO3, a temporary companion of the Earth
(courtesy~NASA).}
% Jun 2016
\item Does the Earth have other satellites apart from the Moon and the
artificial satellites shot into orbit up by rockets? Yes. The Earth has a
number of minisatellites and a large number of
quasisatellites.\index{satellite!of the Earth}\index{quasisatellite} An
especially longlived quasisatellite, an asteroid called 2016 HO3, has a
size of about \csd{60}{m} and was discovered in 2016. As shown in
\figureref{iminimoonnasa}, it orbits the Earth and will continue to do so
for another few hundred years, at a distance from 40 to 100 times that of
the Moon.
% Nov 2016
\item Show that a sphere bouncing  without energy loss  down an inclined
plane, hits the plane in spots whose distances increase by a constant amount
at every bounce.
% Oct 2009
\item Is the acceleration due to gravity constant over time? Not really.
Every day, it is estimated that \csd{10^8}{kg} of material fall onto the
Earth in the form of \iin{meteorites} and \iin[asteroid!falling on
Earth]{asteroids}. (Examples can be seen in \figureref{iperseids} and
\figureref{imeteorite}.) Nevertheless, it is unknown whether the mass of
the Earth increases with time (due to collection of meteorites and cosmic
dust) or decreases (due to gas loss). If you find a way to settle the
issue, publish it.\index{mass!of Earth, time variation}\index{Earth!mass,
time variation}
% Oct 2016, have permission per email
\csepsfnb{iperseids}{scale=1}{A composite photograph of the Perseid meteor
shower that is visible every year in mid August. In that month, the Earth
crosses the cloud of debris stemming from comet SwiftTuttle, and the
source of the meteors appears to lie in the constellation of Perseus,
because that is the direction in which the Earth is moving in mid
August. The effect and the picture are thus similar to what is seen on the
windscreen when driving by car while it
is\protect\seepageone{isnowmotion} snowing.
({\textcopyright}~\protect\iinn{Brad Goldpaint} at
\protect\url{goldpaintphotography.com}).}
% Feb 2014, have permission per email
\csepsfnb{imeteorite}{scale=1}{Two photographs, taken about a second apart,
showing a meteor breakup ({\textcopyright}~\protect\iinn{Robert
Mikaelyan}).}
\item Incidentally, discovering objects hitting the Earth is not at all easy.
Astronomers like to point out that an \iin[asteroid!difficulty of
noticing]{asteroid} as large as the one that led to the extinction of the
\iin{dinosaurs} could hit the Earth without any astronomer noticing in
advance, if the direction is slightly unusual, such as from the south, where
few telescopes are located.
% moved here and improved in June 2007
\item Several humans have survived free falls from aeroplanes for a thousand
metres or more, even though they had no parachute. A minority of them even
did so without any harm at all. How was this possible?\challengenor{fallsu}
\cssmallepsfnb{iscopaesasso2}{scale=0.2}{Brooms fall more rapidly than
stones ({\textcopyright}~\protect\iinn{Luca Gastaldi}).}
% EMAILED FEB 2008
\item Imagine that\index{coin!puzzle}\index{puzzle!coin} you have twelve coins
of identical appearance, of which one is a forgery. The forged one has a
different mass from the eleven genuine ones. How can you decide which is
the forged one and whether it is lighter or heavier, using a simple balance
only three times?\challengn
You have nine identicallylooking
spheres,\index{sphere!puzzle}\index{puzzle!sphere} all of the same mass,
except one, which is heavier. Can you determine which one, using the
balance only two times?
% May 2005
\item For a physicist, \ii{antigravity} is repulsive gravity  it does not
exist in nature. Nevertheless, the term `antigravity' is used incorrectly
by many people, as a short search on the internet shows. Some people call
any effect that \emph{overcomes} gravity, `antigravity'. However, this
definition implies that tables and chairs are
\iin[antigravity!device]{antigravity devices}. Following the definition,
most of the wood, steel and concrete producers are in the antigravity
business. The internet definition makes absolutely no sense.
% Sep 2007
\item What is the cheapest way to switch gravity off for 25
seconds?\challengenor{vomitcomet}
\item Do all objects on Earth fall with the same acceleration of
\csd{9.8}{m/s^{2}}, assuming that air resistance can be neglected? No;
every housekeeper knows that. You can check this by yourself. As shown in
\figureref{iscopaesasso2}, a \iin[brooms]{broom} angled at around
\csd{35}{\csdegrees} hits the floor before a \iin[stones]{stone}, as the
sounds of impact confirm. Are you able to explain why?\challengenor{broom}
% Apr 2005
\item Also bungee jumpers are accelerated more strongly than $g$. For a
bungee cord of mass $m$ and a jumper of mass $M$, the maximum acceleration
$a$ is
\begin{equation}
a= g \left (1 + \frac{m}{8M} \left ( 4 + \frac{m}{M}\right)
\right ) \cp
\end{equation}
Can you deduce the relation from \figureref{ibungee}?\challengenor{bungee}
% Apr 2005
\cstftlepsf{ibungee}{scale=1}{The starting situation for a bungee jumper.}
% (OK) increase height of left figure
[30mm]{ibalancetrick}{scale=1}{An honest balance?}
% It seems that `a honest' is wrong
\item Guess: What is the mass of a ball of cork with a radius of
\csd{1}{m}?\challengenor{cork}
\item Guess: One thousand \csd{1}{mm} diameter steel balls are collected.
What is the mass?\challengenor{cork2}
% Impr. Jun 2007
\item How can you use your observations made during your travels with a
bathroom scale to show that the Earth is not flat?\challengenor{notflat}
\item Both the Earth and the Moon attract bodies. The centre of mass of the
EarthMoon system is \csd{4800}{km} away from the centre of the Earth,
quite near its surface. Why do bodies on Earth
still % add figure  no, do not.
fall towards the centre of the Earth?\challengenor{eamocm}
\item Does every spherical body fall with the same acceleration? No. If the
mass of the object is comparable to that of the Earth, the distance
decreases in a different way. Can you confirm this statement?\challengn
\figureref{ibalancetrick} shows a related puzzle.
% due to the gravitational field of the other body, which usually is
% neglected
What then is wrong about \iname[Galilei, Galileo]{Galileo}'s
argument\seepageone{galmimg} about the constancy of acceleration of free
fall?
\csepsfnb{iairresitovercome}{scale=1}{Reducing air resistance increases
the terminal speed: left, the 2007 speed skiing world record holder
{Simone Origone} with 69.83\,m/s
% EMAILED FEB 2008  s.origone@tiscali.it
and right, the 2007 speed world record holder for bicycles on snow {Éric
Barone} with 61.73\,m/s ({\textcopyright}~\protect\iinn{Simone Origone},
\protect\iinn{Éric Barone}).}
% EMAILED FEB 2008  eric@ericbarone.com
\item What is the fastest speed that a human can achieve making use of
gravitational acceleration? There are various methods that try this; a few
are shown in \figureref{iairresitovercome}. Terminal speed of free
falling skydivers can be even higher, but no reliable record speed value
exists. The last word is not spoken yet, as all these records will be
surpassed in the coming years. It is important to require normal altitude;
at stratospheric altitudes, speed values can be four times the speed values
at low altitude.\seepagetwo{kittingerspeed}
\item It is easy to put a mass of a kilogram onto a table. Twenty kilograms
is harder. A thousand is impossible. However, \csd{6\cdot 10^{24}}{kg} is
easy. Why?\challengenor{table}
\item The friction\seepageone{tidfryy} between the Earth and the Moon slows
down the rotation of both. The Moon stopped rotating millions of years ago,
and the Earth is on its way to doing so as well. When the Earth stops
rotating, the Moon will stop moving away from Earth. How far will the Moon
be from the Earth at that time?\challenge % !!!5
Afterwards however, even further in the future, the Moon will move back
towards the Earth, due to the friction between the EarthMoon system and
the Sun. Even though this effect would only take place if the Sun burned
for ever, which is known to be false, can you explain
it?\challengenor{cmoondist}
\item When you run towards the east, you \emph{lose weight}. There are two
different reasons for this:\index{running!reduces weight} the `centrifugal'
acceleration increases so that the force with which you are pulled down
diminishes, and the Coriolis force appears, with a similar result. Can you
estimate the size of the two effects?\challenge % !!!5
% Feb 2014
\item Laboratories use two types of \iin{ultracentrifuges}: \emph{preparative}
ultracentrifuges isolate viruses, organelles and biomolecules, whereas
\emph{analytical} ultracentrifuges measure shape and mass of macromolecules.
The fastest commercially available models achieve \csd{200\,000}{rpm}, or
\csd{3.3}{kHz}, and a centrifugal acceleration of $\smash{10^6} \cdot g$.
\item What is the relation between the time a \iin[stones]{stone} takes
falling through a distance $l$ and the time a pendulum takes swinging though
half a circle of radius $l$?\challengenor{pisw} (This problem is due to
\iname[Galilei, Galileo]{Galileo}.) How many digits of the number $\pi$ can
one expect to determine in this way?
\item Why can a spacecraft accelerate through the \ii{slingshot effect} when
going round a planet, despite momentum conservation?\challengenor{slingshot}
% Mar 2006:
It is speculated that the same effect is also the reason for the few
exceptionally fast stars that are observed in the galaxy. For example, the
star HE04575439 moves with \csd{720}{km/s}, which\cite{heber} is much
higher than the 100 to \csd{200}{km/s} of most stars in the Milky Way. It
seems that the role of the accelerating centre was taken by a black hole.
\item The orbit of a planet around the Sun has many interesting
properties.\cite{circpla} What is the \iin{hodograph} of the orbit? What is
the hodograph for parabolic and hyperbolic orbits?\challengenor{hodoci}
% Dec 2006
\cssmallepsfnb{igalsats}{scale=0.8}{The four satellites of Jupiter discovered
by Galileo and their motion ({\textcopyright}~\protect\iinn{Robin Scagell}).}
% EMAILED FEB 2008  robin@galaxypix.com
% Dec 2006
\item The \ii{Galilean satellites} of Jupiter,\label{medicisat} shown in
\figureref{igalsats}, % on \cspageref{igalsats}, %this vol I
can be seen with small amateur telescopes. Galileo discovered them in 1610
and called them the \ii{Medicean satellites}. (Today, they are named, in
order of increasing distance from Jupiter, as \iin{Io}, \iin{Europa},
\iin{Ganymede} and \iin{Callisto}.) They are almost mythical objects. They
were the first bodies found that obviously did not orbit the Earth; thus
Galileo used them to deduce that the Earth is not at the centre of the
universe. The satellites have also been candidates to be the first
\ii{standard clock}, as their motion can be predicted to high accuracy, so
that the `standard time' could be read off from their position. Finally,
due to this high accuracy, in 1676, the speed of light was first measured
with their help, as told in the section on special
relativity.\seepagetwo{finitelspd}
% Jun 2005
\item A simple, but difficult question: if all bodies attract each other, why
don't or didn't all stars fall towards each other?\challengenor{newton}
Indeed, the inverse square expression of universal gravity has a limitation:
it does not allow one to make sensible statements about the matter in the
{universe}.\index{universe!description by universal gravitation} Universal
gravity predicts that a homogeneous mass distribution is unstable; indeed,
an inhomogeneous distribution is observed. However, universal gravity does
not predict the average mass density, the darkness at night, the observed
speeds of the distant galaxies, etc. In fact, `universal' gravity does not
explain or predict a single property of the universe. To do this, we need
general relativity.\seepagetwo{whyseestars}
\item The acceleration $g$ due to gravity at a depth of \csd{3000}{km} is
\csd{10.05}{m/s^{2}}, over 2\,\% more than at the surface of the Earth. How
is this possible?\cite{hodg} Also, on the Tibetan plateau, $g$ is influenced
by the material below it.
% {higher} % !.!3 how much? Never found a clear explanation, Mar 2010
% than the sea level value of \csd{9.81}{m/s^{2}},
% even though the plateau is
% more distant from the centre of the Earth than sea level is. How
% is this possible?\challengenor{plateau}
\csepsf{imoonmotion}{scale=1}{Which of the two Moon paths is correct?}
\item When the Moon circles the Sun,\index{Moon!path around Sun} does its path
have sections \emph{concave} towards the Sun, as shown at the right of
\figureref{imoonmotion}, or not, as shown on the
left?\challengenor{concave} (Independent of this issue, both paths in the
diagram disguise that the Moon path does \emph{not} lie in the same plane as
the path of the Earth around the Sun.)
\item You can prove that objects \emph{attract each other} (and that they are
not only attracted by the Earth) with a simple experiment that anybody can
perform at home, as described on the
\url{www.fourmilab.ch/gravitation/foobar} website.
\item It is instructive to calculate the \emph{escape velocity} from the
Earth,\index{escape velocity}\index{velocity!escape} i.e.,{} that velocity
with which a body must be thrown so that it never falls back. It turns out
to be around \csd{11}{km/s}. (This was called the \ii[velocity!second
cosmic]{second cosmic velocity} in the past; the \ii[velocity!first
cosmic]{first cosmic velocity} was the name given to the lowest speed for an
orbit, \csd{7.9}{km/s}.) The exact value of the escape velocity depends on
the latitude of the thrower, and on the direction of the throw.\challengn
(Why?)
What is the escape velocity from the Solar System? (It was once called the
\ii[velocity!third cosmic]{third cosmic velocity}.) By the way, the escape
velocity from our galaxy is over \csd{500}{km/s}.
%
What would happen if a planet or a system were so heavy that the escape
velocity from it would be larger than the speed of light?\challengenor{bh1}
\item What is the largest \iin[asteroid!puzzle]{asteroid} one can escape from
by jumping?\challengenor{aste}
\item For bodies of irregular shape, the \iin{centre of gravity} of a
body\index{mass!centre of}\index{gravity!centre of} is \emph{not} the same
as the \iin{centre of mass}. Are you able to confirm
this?\challengenor{handle} (Hint: Find and use the simplest example
possible.)
\item Can gravity produce repulsion? What happens to a small test body on the
inside of a large Cshaped mass? Is it pushed towards the centre of
mass?\challenge % !!!5
% EVEN THOUGH I THINK HE IS RIGHT, I TOOK THE DISCUSSION OUT.
% BACK IN IN JULY 2016
%
\item A heavily disputed argument\index{mass!inertial, puzzle} for the
equality of inertial and\index{mass!gravitational, puzzle} gravitational
mass was given by \iname[Chubykalo, A.E.]{Chubykalo},
% valeri @ canera.reduaz.mx
and \iname[Vlaev, S.J.]{Vlaev}.\cite{meqm} The total kinetic energy $T$ of
two bodies circling around their common centre of mass, like the Earth and
the Moon, is given by $T= G m M / 2 R$, where the two quantities $m$ and $M$
are the \emph{gravitational} masses of the two bodies and $R$ their
distance. From this expression, in which the inertial masses do \emph{not}
appear on the right side, they deduce that the inertial and gravitational
mass must be proportional to each other. Can you see how? Is the reasoning
correct?\challengenor{massdispute}
\item The \ii[shape!of the Earth]{shape} of the Earth\index{Earth!shape} is
not a sphere.\cite{pluli} As a consequence, a plumb line usually does not
point to the centre of the Earth. What is the largest deviation in
degrees?\challenge % !!!5
\item Owing to the slightly flattened shape of the Earth,\index{Earth!shape}
the source of the Mississippi is about \csd{20}{km} nearer to the centre of
the Earth than its mouth; the water effectively runs uphill. How can this
be?\challengenor{missis}
\cssmallepsfnb{ianalemma}{scale=0.5}{The analemma over Delphi, taken
between January and December 2002 ({\textcopyright}~\protect\iinn{Anthony
Ayiomamitis}).}
% EMAILED FEB 2008  OK!
\item If you look at the sky every day at 6 a.m., the Sun's position varies
during the year. The result of photographing the Sun on the same film is
shown in \figureref{ianalemma}. The curve, called the \iin{analemma}, is
due to two combined effects: the inclination of the Earth's axis and the
elliptical shape of the Earth's orbit around the Sun. The top and the
(hidden) bottom points of the analemma correspond to the solstices.
% May 2007:
How does the analemma look if photographed every day at local
noon?\challengenor{noonanalemma} Why is it not a straight line pointing
exactly south?
% Impr. Feb 2012
\item The \iin[constellations]{constellation} in which the Sun stands at noon
(at the centre of the time zone) is supposedly called the `{zodiacal sign}'
of that day. Astrologers\index{astrology} say there are twelve of them,
namely \iin{Aries}, \iin{Taurus}, \iin{Gemini}, \iin{Cancer}, \iin{Leo},
\iin{Virgo}, \iin{Libra}, \iin{Scorpius}, \iin{Sagittarius},
\iin{Capricornus}, \iin{Aquarius} and \iin{Pisces} and that each takes
(quite precisely) a twelfth of a year or a twelfth of the ecliptic. Any
check with a calendar shows that at present, the midday Sun is never in the
zodiacal sign during the days usually connected to it. The relation has
shifted by about a month since it was defined, due to the
precession\seepageone{ieaorbit} of the Earth's axis. A check with a map of
the star sky shows that the twelve constellations do not have the same
length and that on the ecliptic there are fourteen of them, not twelve.
There is \ii{Ophiuchus} or \ii{Serpentarius}, the \iin{serpent bearer}
{constellation}, between Scorpius and Sagittarius, and \ii{Cetus}, the
\iin[whale!constellation]{whale}, between Aquarius and Pisces. In fact, not
a single astronomical statement about zodiacal signs is correct.\cite{tyson}
To put it clearly, \iin{astrology}, in contrast to its name, is \emph{not}
about stars. (In German, the word `Strolch', meaning `rogue' or
`scoundrel', is derived from the word `astrologer'.)
% there was another language where it means `crook'  I cannot recall it
\item For a long time, it was thought that there is no additional planet in
our Solar System outside \iin{Neptune} and \iin{Pluto},\cite{xplanet}
because their orbits show no disturbances from another body. Today, the
view has changed. It is known that there are only eight planets: Pluto is
not a planet, but the first of a set of smaller objects in the socalled
\ii{Kuiper belt}.
% and \iin{Oort cloud}.
% (Astronomers have also agreed to continue to call
% Pluto a `planet' despite this evidence, to avoid debates.)
Kuiper belt objects are regularly discovered; over 1000 are known today.
\csepsfnb{isedna}{scale=1}{The orbit of Sedna in comparison with the orbits
of the planets in the Solar System (NASA).}
% Aug 2014
\csepsfnb{ikuiperoort}{scale=1}{The \protect\iin{Kuiper belt}, containing
mainly planetoids, and the \protect\iin{Oort cloud} orbit, containing
comets, around the Solar System (\protect\iname{NASA},
\protect\iname{JPL}, \protect\iinn{Donald Yeoman}).}
In 2003, two major Kuiper objects were discovered; one, called \ii{Sedna},
is almost as large as Pluto, the other, called \ii{Eris}, is even larger
than Pluto and has a moon.\cite{sedna} Both have strongly elliptical orbits
(see \figureref{isedna}). Since Pluto and Eris, like the asteroid
\iin{Ceres}, have cleaned their orbit from debris, these three objects are
now classified as \ii{dwarf planets}.\index{planet!dwarf}
% Sometimes these bodies change
% trajectory due to the attraction of a nearby planet: that is the birth of a
% new \iin{comet}.
% New in Dec 2014
Outside the Kuiper belt, the Solar System is surrounded by the socalled
\ii{Oort cloud}. In contrast to the flattened \iin{Kuiper belt}, the Oort
cloud is spherical in shape and has a radius of up to 50\,000\,AU, as shown
in \figureref{isedna} and \figureref{ikuiperoort}. The Oort cloud
consists of a huge number of icy objects consisting of mainly of water, and
to a lesser degree, of methane and ammonia. Objects from the Oort cloud
that enter the inner Solar System become comets;\index{comet!origin of} in
the distant past, such objects have brought water onto the Earth.
\item In astronomy new examples of motion are regularly discovered even in the
present century. Sometimes there are also false alarms. One example was
the alleged fall of \ii[comet!mini]{mini comets} on the Earth. They were
supposedly made of a few dozen kilograms of ice, hitting the Earth every few
seconds.\cite{snowball} It is now known not to happen.
\item Universal gravity allows only elliptical, parabolic or hyperbolic
orbits. It is impossible for a small object approaching a large one to be
captured. At least, that is what we have learned so far. Nevertheless, all
astronomy books tell stories of \iin[capture!in universal gravity]{capture}
in our Solar System; for example, several outer \iin{satellites} of
\iin{Saturn} have been captured. How is this
possible?\challengenor{capture}
% Apr 2005
\item How would a tunnel have to be shaped in order that a stone would fall
through it without touching the walls? (Assume constant density.) If the
Earth did not rotate, the tunnel would be a straight line through its
centre, and the stone would fall down and up again, in a oscillating motion.
For a rotating Earth, the problem is much more difficult. What is the shape
when the tunnel starts at the Equator?\challengenor{eqell}
% Jun 2005
\item The International Space Station\index{Space Station!International}
circles the Earth every 90 minutes at an altitude of about \csd{380}{km}.
You can see where it is from the website \url{www.heavensabove.com}. By
the way, whenever it is just above the horizon, the station is the third
brightest object in the night sky, superseded only by the Moon and Venus.
Have a look at it.\challengn
% Apr 2006, Dec 2015
\item Is it true that the centre of mass of the Solar System, its
\iin{barycentre}, is always inside the Sun?\challengenor{sungracen} Even
though the Sun or a star move very little when planets move around them,
this motion can be detected with precision measurements making use of the
Doppler effect for light or radio waves.\seepagetwo{reldopplereff} Jupiter,
for example, produces a speed change of \csd{13}{m/s} in the Sun, the Earth
\csd{1}{m/s}. The first planets outside the Solar System, around the pulsar
PSR1257+12 and around the normal Gtype star Pegasi 51, were discovered in
this way, in 1992 and 1995. In the meantime,
% over 2000{\pres ent} % end 2015
% over 3200 % mid 2016
several thousand
socalled \iin[exoplanet!discoveries]{exoplanets} have been discovered with
this and other methods. Some have even masses comparable to that of the
Earth.
% May 2016
This research also showed that exoplanets are more numerous than stars, and
that earthlike planets are rare.
\item Not all points on the Earth receive the same number of daylight hours
during a year. The effects are difficult to spot, though. Can you find
one?\challengedif{gzgz}
\item Can the phase of the Moon\index{Moon!phase} have a measurable effect on
the human body, for example through tidal effects?\challengenor{moonhumbod}
\cssmallepsfnb{iselene}{scale=1.5}{The phases of the Moon and of Venus, as
observed from Athens in summer 2007 ({\textcopyright}~\protect\iinn{Anthony
Ayiomamitis}).}
% EMAILED FEB 2008  OK also for paper!
% Mar 2012
\csepsfnb{ivenusphases}{scale=1}{Universal gravitation also explains the
observations of Venus,\protect\index{Venus} the evening and morning star.
In particular, universal gravitation, and the elliptical orbits it implies,
explains its phases and its change of angular size. The pictures shown here
were taken in 2004 and 2005. The observations can easily be made with a
binocular or a small telescope ({\textcopyright}~\protect\iname{Wah!}; film
available at \protect\url{apod.nasa.gov/apod/ap060110.html}).}
% Improved Oct 2012
\item There is an important difference between the \iin{heliocentric system}
and the old idea that all planets turn around the
Earth.\index{system!heliocentric}\index{system!geocentric} The heliocentric
system states that certain planets, such as Mercury and Venus, can be
\emph{between} the Earth and the Sun at certain times, and \emph{behind} the
Sun at other times. In contrast, the \iin{geocentric system} states that
they are always in between. Why did such an important difference not
immediately invalidate the geocentric system? And how did the observation
of phases, shown in \figureref{iselene} and \figureref{ivenusphases},
invalidate the geocentric system?\challengenor{heliocentrtr}
\item The strangest reformulation of the description of motion given by
$m{\bm a}= {\bm \nabla} U$ is the almost absurd looking\cite{rayfor}
equation
\begin{equation}{\bm \nabla} v = \diffd {\bm v} / \diffd s \end{equation}
where $s$ is the motion path length. It is called the \ii{ray form} of the
equation of motion. Can you find an example of its
application?\challengenor{rayfor}
\item Seen from \iin{Neptune}, the size of the \iin{Sun} is the same as that
of \iin{Jupiter} seen from the Earth at the time of its closest approach.
True?\challengenor{trn}
% psfrag OK April 2014
\cssmallepsf{ishellgrav}{scale=1}{The vanishing of gravitational force inside
a spherical shell of matter.}[%
\psfrag{dm}{\small $\diffd m$}%
\psfrag{dM}{\small $\diffd M$}%
\psfrag{r}{\small $r$}%
\psfrag{R}{\small $R$}%
\psfrag{m}{\small $m$}%
]
\item The gravitational acceleration for a particle inside a spherical shell
is zero. The vanishing\cite{silvima} of gravity\index{shell!gravity inside
matter} in\index{matter!shell, gravity inside} this\index{gravity!inside
matter shells} case is independent of the particle shape and its position,
and independent of the thickness of the shell. %
%
%
Can you find the argument using \figureref{ishellgrav}?\challengenor{shell}
This works only because of the $1/r^{2}$ dependence of gravity. Can you
show that the result does not hold for nonspherical shells? Note that the
vanishing of gravity inside a spherical shell usually does not hold if other
matter is found outside the shell. How could one eliminate the effects of
outside matter?\challengenor{shelleli}
%
% MY BIG ERROR, AROUND 1997:
%
% On the other hand, experiments show that the
% result is wrong: stones do not float inside circular boxes and food does not
% float inside the stomach. What is the matter here?\seepage{shellprob2}
% Dec 2012
\csepsfnb{ilesagegraph}{scale=1}{Le Sage's own illustration of his model,
showing the smaller density of `ultramondane corpuscules' between the
attracting bodies and the higher density outside them
({\textcopyright}~\protect\iname{Wikimedia})}
\item What\label{lesagegrav} is gravity?\index{gravitation!essence of} This
simple question has a long history. In 1690,\indname{Fatio de Duillier,
Nicolas} {Nicolas Fatio~de~Duillier} and in 1747, \iinn{GeorgesLouis Le
Sage} proposed\cite{lesage} an explanation for the $1/r^{2}$ dependence.
Le Sage argued that the world is full of small particles  he called them
`corpuscules ultramondains'  flying around randomly and hitting all
objects. Single objects do not feel the hits, since they are hit
continuously and randomly from all directions. But when two objects are
near to each other, they produce shadows for part of the flux to the other
body, resulting in an attraction, as shown in
\figureref{ilesagegraph}.\index{shadow!and attraction of bodies} Can you
show that such an attraction has a $1/r^{2}$ dependence?\challengn
However, Le Sage's proposal has a number of problems. First, the argument
only works if the collisions are inelastic. (Why?)\challengn However, that
would mean that all bodies would heat up with time, as \iinn{JeanMarc
LévyLeblond} explains.\cite{hewex}
% page 82 of his physique en questions  mecanique
%
% Apr 2005, Nov 2012
Secondly, a moving body in free space would be hit by more or faster
particles in the front than in the back; as a result, the body should be
decelerated. Finally, gravity would depend on size, but in a strange way.
In particular, three bodies lying on a line should \emph{not} produce
shadows, as no such shadows are observed; but the naive model predicts such
shadows.
Despite all criticisms, the idea that gravity is due to particles has
regularly resurfaced in physics research ever since. In the most recent
version, the hypothetical particles are called \ii{gravitons}. On the other
hand, no such particles have never been observed. We will understand the
origin of gravitation in the final part of our mountain
ascent. %\label{oldsciamaidea}
\item For which bodies does gravity decrease as you approach
them?\challenge % !!!5
\item Could one put a satellite into orbit using a cannon? Does the answer
depend on the direction in which one shoots?\challengenor{cannonsat}
% July 2004
\item Two old computer users share experiences. `I threw my Pentium III and
Pentium IV out of the window.' `And?' `The Pentium III was faster.'
\item How often does the\index{Earth!rise} Earth rise and fall when seen from
the Moon?\challengenor{moonre} Does the Earth show phases?
\item What is the \iin[weight!of the Moon]{weight of the
Moon}?\challenge % !!!5
How does it compare\index{Moon!weight of} with the weight of the
\iin[Alps!weight of the]{Alps}?
\item If a star is made of high density material, the speed of a planet
orbiting near to it could be greater than the speed of
light.\challengenor{dense} How does nature avoid this strange possibility?
\item What will happen to the Solar System in the future?\index{Solar
System!future of} This question is surprisingly hard to answer. The main
expert of this topic, French planetary scientist \iinn{Jacques Laskar},
simulated a few hundred million years of evolution using computeraided
calculus.\cite{sussol} He found that the planetary orbits are stable, but
that there is clear evidence of chaos\seepageone{solchaos} in the evolution
of the Solar System, at a small level. The various planets influence each
other in subtle and still poorly understood ways. Effects in the past are
also being studied, such as the energy change of Jupiter due to its ejection
of smaller asteroids from the Solar System, or energy gains of Neptune.
There is still a lot of research to be done in this field.
\item One of the open problems of the Solar System is the description of
planet distances discovered in 1766 by \iinn{Johann~Daniel Titius}
\lived(17291796) and publicized by \iinn{Johann~Elert Bode}
\lived(17471826). Titius discovered that planetary distances $d$ from the
Sun can be\index{Bode's rule}\index{Titius's rule} approximated by
\begin{equation}
d = a+ 2^n \, b \qhbox{with} a=\csd{0.4}{AU} \ , \ b=\csd{0.3}{AU}
% \label{eq:titius}
\end{equation}
where distances are measured in \iin{astronomical unit}s and $n$ is the
number of the planet. The resulting approximation is compared with
observations in \tableref{ggghhh}.
%
% \subsubsubsubsubsubsubsubsection{Bode formula table}
%
{\small
\begin{table}[t]
\sbox{\cshelpbox}{\small\dirrtabularnolines%
\begin{tabular}{@{\hspace{0em}}l@{\hspace{3mm}}r@{\hspace{3mm}}c@{\hspace{5mm}}l@{\hspace{0em}}}
%
\toprule
%
\tabheadf{Planet} & \tabhead{$n$} & \tabhead{predicted} &
\tabhead{measured} \\
%
& & \multicolumn{2}{@{\hspace{0em}}c@{\hspace{0em}}}{\tabhead{distance in
AU}}\\[0.5mm]
%
\midrule
Mercury & $\infty$ & 0.4 & 0.4\\
Venus & 0 & 0.7 & 0.7 \\
Earth & 1 & 1.0 & 1.0 \\
Mars & 2 & 1.6 & 1.5 \\
Planetoids & 3 & 2.8 & 2.2 to 3.2 \\
Jupiter & 4 & 5.2 & 5.2 \\
Saturn & 5 & 10.0 & 9.5 \\
Uranus & 6 & 19.6 & 19.2 \\
Neptune & 7 & 38.8 & 30.1 \\
Pluto & 8 & 77.2 & 39.5 \\
\bottomrule
\end{tabular}%
}% end of sbox
%
\centering
\begin{minipage}{\wd\cshelpbox}
\captionsetup{width=\wd\cshelpbox} % needed since Dec 2015
\caption[An unexplained property of nature: the TitiusBode rule.]{An
unexplained property of nature:\protect\index{planet!distance values, table}
planet distances from the Sun and the values resulting from the TitiusBode
rule.}\label{ggghhh}\noindent\usebox{\cshelpbox}
\end{minipage}
\end{table}
}
% Aug 2009
\csswffilmrepeat{ast2004}{scale=0.5}{The motion of the planetoids compared to
that of the planets (Shockwave animation
{\textcopyright}~\protect\iinn{HansChristian Greier})}
% EMAILED AUG 2009
Interestingly, the last three planets, as well as the planetoids, were
discovered \emph{after} Bode's and Titius' deaths; the rule had successfully
predicted Uranus' distance, as well as that of the planetoids. Despite these
successes  and the failure for the last two planets  nobody has yet found
a model for the formation of the planets that explains Titius' rule.
% Impr Jun 2005 with info from Wikipedia
The large satellites of Jupiter and of Uranus have regular spacing, but not
according to the TitiusBode rule.\index{Jupiter!moons of}
% Added in Oct 2001, improved with Namouni in April 2006
Explaining or disproving the rule is one of the challenges that remains in
classical mechanics.
% It is known that the rule must be a consequence of the
% formation of satellite systems. The bodies not following a fixed rule, such
% as the outer planets of the Sun or the outer moons of Jupiter, are believed
% not to be part of the original system but to have been captured later.
Some researchers\cite{yesbodelaw} maintain that the rule is a consequence of
scale invariance, others maintain that it is an accident or even a red
herring.\cite{nobodelaw} The last interpretation is also suggested by the
nonTitiusBode behaviour of practically all extrasolar planets. The issue
is not closed.
% Oct 2016
\cssmallepsfnbfp{isixcelestialbodies}{scale=1}{These are the six celestial
bodies that are visible at night with the naked eye and whose positions vary
over the course of the year. The nearly vertical line connecting them is
the\protect\index{ecliptic!illustration of} \emph{ecliptic}, the narrow
stripe around it the \emph{zodiac}.\index{zodiac!illustration of} Together
with the Sun, the seven celestial bodies were used to name the days of the
week. ({\textcopyright}~\protect\iinn{Alex Cherney})}
% May 2005
\item Around\label{babylo} 3000 years ago, the \iin{Babylonians} had measured
the orbital times of the seven celestial bodies that move across the
sky. Ordered from longest to shortest, they wrote them down in
\tableref{babiggghhh}. Six of the celestial bodies are visible in the
beautiful \figureref{isixcelestialbodies}.
%
% \subsubsubsubsubsubsubsubsection{Babilonian table}
%
{\small
\begin{table}[t]
\sbox{\cshelpbox}{\small\dirrtabularnolines%
\begin{tabular}{@{\hspace{0em}}l@{\hspace{5mm}}l@{\hspace{0em}}}
%
\toprule
%
\tabheadf{Body} & \tabhead{Period} \\[0.5mm]
%
\midrule
Saturn & \csd{29}{a} \\
Jupiter & \csd{12}{a} \\
Mars & \csd{687}{d} \\
Sun & \csd{365}{d} \\
Venus & \csd{224}{d} \\
Mercury & \csd{88}{d} \\
Moon & \csd{29}{d} \\
\bottomrule
\end{tabular}%
}% end of sbox
%
\centering
\begin{minipage}{\wd\cshelpbox}
\captionsetup{width=\wd\cshelpbox} % needed since Dec 2015
\caption{The orbital\protect\index{planet!orbit periods, Babylonian
table} periods known to the Babylonians.}\label{babiggghhh}%
\noindent\usebox{\cshelpbox}
\end{minipage}
\end{table}
}
The Babylonians also introduced the week and the division of the day into 24
hours.\index{week!days, order of} They %(and later the Egyptians)
dedicated every one of the 168 hours of the week to a celestial body,
following the order of \tableref{babiggghhh}. They also dedicated the whole
day to that celestial body that corresponds to the first hour of that day.
The first day of the week was dedicated to Saturn; the present ordering of the
other days of the week then follows from \tableref{babiggghhh}.\challengn This
story was told by \iname{Cassius Dio} \livedca(\circa160\circa
230).\cite{cassius} Towards the end of Antiquity, the ordering was taken up by
the Roman empire. In Germanic languages, including English, the Latin names
of the celestial bodies were replaced by the corresponding Germanic gods. The
order Saturday, Sunday, Monday, Tuesday, Wednesday, Thursday and Friday is
thus a consequence of both the astronomical measurements and the astrological
superstitions of the ancients.
\item In 1722, % and in 1736
the great mathematician \iinn{Leonhard Euler} made a mistake in his
calculation that led him to conclude that if a tunnel, or better, a deep
hole were built from one pole\index{tunnel!through the
Earth}\index{hole!through the Earth} of the Earth to the other, a
\iin[stones]{stone} falling into it would arrive at the Earth's centre and
then immediately turn and go back up. \iname{Voltaire} made fun of this
conclusion for many years. Can you correct Euler and show that the real
motion is an oscillation from one pole to the other, and can you calculate
the time a poletopole fall would take (assuming homogeneous
density)?\challengenor{peri42}
What would be the oscillation time for an arbitrary straight
surfacetosurface tunnel of length $l$, thus \emph{not} going from pole to
pole?\challengenor{ststun}
The previous challenges circumvented the effects of the Earth's rotation.
The topic becomes much\cite{simoson} more interesting if rotation is
included. What would be the shape of a tunnel so that a stone falling
through it never touches the wall?\challengenor{holeinearth}
% Feb 2005
\item \figureref{ieclipse} shows a photograph of a solar eclipse taken from
the Russian space station \emph{Mir} and a photograph taken at the centre of
the shadow from the Earth. Indeed, a global view of a phenomenon can be
quite different from a local one. What is the speed of the
shadow?\challengenor{shadowspeed}
% Dec 2005
\item In 2005, satellite measurements have shown that the water in the Amazon
river presses down the land up to \csd{75}{mm} more in the season when it is
full of water than in the season when it is almost empty.\cite{amazonas}
% Janvier 2007
\csepsfnb{ieclipse}{scale=1}{The solar eclipse of 11 August 1999,
photographed by \protect\iinn{JeanPierre Haigneré}, member of the
\protect\iin{Mir} 27 crew, and the (enhanced) solar eclipse of 29 March 2006
({\textcopyright}~\protect\iname{CNES}
%
and \protect\iinn{Laurent Laveder}{/}\protect\iname{PixHeaven.net}).}
% EMAILED FEB 2008  laurent.laveder@laposte.net
% Oct 2012
\cssmallepsfnb{iearthwire}{scale=1}{A wire attached to the Earth's Equator.}
% Dec 2006, Dec 2016
\item Imagine that wires existed that do not break. How long would such a
wire have to be so that, when attached to the Equator, it would stand
upright in the air, as shown\index{space!elevator}
in\index{elevator!space}\index{Equator!wire}\index{wire!at Equator}
\figureref{iearthwire}?\challengenor{uprightwire} Could one build an
elevator into space in this way?
% Dec 2006
\item Usually there are roughly two \iin[tide!once or twice per day]{tides}
per day. But there are places, such as on the coast of \iin{Vietnam}, where
there is only \emph{one} tide per day. See
\url{www.jason.oceanobs.com/html/applications/marees/marees_m2k1_fr.html}.
Why?\challenge % !!!5 why only one tide per day?
% May 2017
% !!!1 add "amphidromic points", where the phase lines of the tides meet,
% and the amplitude goes to zero.
% !!!1 add the concept of a Kelvin wave.
% !!!1 cite the book "sea level science" by David Pugh and Philip Woodworth
% Dec 2006
\item It is sufficient to use the concept of centrifugal force to show that
the rings of Saturn cannot be made of massive material, but must be made of
separate pieces. Can you find out how?\challengenor{roche}
% % Nov 2006  out in 2016: double
% \item A painting is hanging on the wall of Dr.~Dolittle's waiting room. He
% hung up the painting using two nails, and wound the picture wire around the
% nails in such a way that the painting would fall if either nail were pulled
% out.\index{painting puzzle} How did Dr.~Dolittle do it?\challengn
% Mar 2007
\item Why did Mars lose its atmosphere? %\challengeres{marsunknown}
Nobody knows. It has recently been shown that the solar wind is too weak
for this to happen. This is one of the many open riddles of the solar
system.
% Mar 2017
\item All bodies in the Solar System orbit the Sun in the same direction. All?
No; there are exceptions. One intriguing asteroid that orbits the Sun near
Jupiter in the wrong direction was discovered in 2015: it has a size of
\csd{3}{km}. For an animation of its astonishing orbit, opposite to all
Trojan asteroids, see
\url{www.astro.uwo.ca/~wiegert}. % /2015BZ509/2015BZ50919.mp4}.
% Sep 2008
\item The observed motion due to gravity can be shown to be the
\emph{simplest} possible, in the following sense. If we measure change of a
falling object with the expression
$\int mv^2/2  mgh\, \diffd t $,\seepageone{leappi} then a constant
acceleration due to gravity \emph{minimizes} the change in every example of
fall. Can you confirm this?\challengn
% March 2009
\item Motion due to gravity is fun: think about \iin{roller coasters}. If you
want to know more at how they are built, visit \url{www.vekoma.com}.
\end{curiosity}
\begin{quoteunder}
The scientific theory I like best is that the rings of Saturn are made of
lost airline \iin{luggage}.\\
\iinn{Mark Russel} % See www.markrussel.net  its good
\end{quoteunder}
%
% Nov 2008, reread Jul, Aug 2016, Dec 2016
\subsection{Summary on gravitation}
% Index OK
Spherical bodies of mass $M$ attract other bodies at a distance $r$ by
inducing an acceleration towards them given by $a=GM/r^2$. This expression,
\emph{universal gravitation},\index{gravitation!summary of universal}
describes snowboarders, skiers, paragliders, athletes, couch potatoes,
pendula, stones, canons, rockets, tides, eclipses, planet shapes, planet
motion and much more. Universal gravitation is the first example of a unified
description: it describes \emph{how everything falls}. By the acceleration it
produces, gravitation limits the appearance of uniform motion in nature.
\vignette{classical}
%
%
%
%
\newpage
% Improved in Mar 2014 with KPK ideas
\chapter{Classical mechanics, force and the predictability of motion}
\markboth{\thesmallchapter\ classical mechanics, force and the predictability
of motion}%
{\thesmallchapter\ classical mechanics, force and the predictability of
motion}
% Index OK
% Mar 2014
\csepsf{iwaterparabola}{scale=1}{The parabola shapes formed by accelerated
water beams show that motion in everyday life is predictable
({\textcopyright}~\protect\iname{Oase GmbH}).}
\csini{A}{ll} those types of motion in which the only permanent property
of\linebreak body is mass\index{motion!predictability
of(}\index{predictability!of motion(} define
% the mass of a body is its only
% permanent property form
the field of \ii[mechanics!definition]{mechanics}. The same name is
given\linebreak lso to the experts studying the field. We can think of
mechanics as the athletic part of physics.%
%
\footnote{This is in contrast to the actual origin of the term `mechanics',
which means `machine science'. It derives from the Greek
\csgreekok{mhkan'h}, which means `machine' and even lies at the origin of
the English word `machine' itself. Sometimes the term `mechanics' is used
for the study of motion of \emph{solid} bodies only, excluding, e.g.,
hydrodynamics. This use fell out of favour in physics in the twentieth
century.} %
%
Both in athletics and in mechanics only lengths, times and masses are measured
 and of interest at all.
More specifically, our topic of investigation so far is called
\emph{classical} mechanics,\index{mechanics!classical} to distinguish it from
\ii[quantum mechanics]{quantum} mechanics.\index{mechanics!quantum} The main
difference is that in classical physics arbitrary small values are assumed to
exist, whereas this is not the case in quantum physics.
% The use of real numbers for observable quantities is thus central to
% classical physics.
%
% All observables depending on space and time, such as field strengths,
% densities, currents, are described with the help of continuous (and
% commuting) functions of space and time. This is true even in the case
% of motion change due to \iin{contact}. In physics, a classical description
% is possible only in the domains of mechanics, thermal physics, relativity,
% gravitation and electromagnetism.\index{electromagnetism}\index{gravitation}
% Together they form
% the present, first part of our mountain ascent: \iin {classical physics}.
%
Classical mechanics is often also called \ii{Galilean physics} or
\ii{Newtonian physics}.%
%
\footnote{The basis of classical mechanics, the description of motion using
only space and time, is called \ii{kinematics}. An example is the
description of free fall by
$z(t)=z_{0}+ v_{0} (tt_{0}) \frac{ 1 }{ 2} g (tt_{0})^{2}$. The other,
main part of classical mechanics is the description of motion as a
consequence of interactions between bodies; it is called \ii{dynamics}. An
example of dynamics is the formula of universal gravity. The distinction
between kinematics and dynamics can also be made in relativity,
thermodynamics and electrodynamics.} %
%
Classical mechanics states that motion is \emph{predictable}: it thus states
that there are no surprises in motion. Is this correct in all cases? Is
predictability valid in the presence of friction? Of free will? Are there
really no surprises in nature? These issues merit a discussion; they will
accompany us for a stretch of our adventure.
% (DONE) structure this chapter after this paragraph!
We know that there is more to the world than gravity. Simple observations
make this point: \emph{floors} and \emph{friction}. Neither can be due to
gravity.\index{floor!are not gavitational}\index{friction!not due to gravity}
Floors do not fall, and thus are not described by gravity; and friction is not
observed in the skies, where motion is purely due to gravity.%
%
\footnote{This is not completely correct: in the 1980s, the first case of
gravitational friction was discovered: the emission of gravity waves. We
discuss it in detail\seepagetwo{graveney2} in the chapter on general
relativity. The discovery does not change the main point, however.} %
%
Also on Earth, friction is unrelated to gravity, as you might want to check
yourself.\challengn There must be another interaction responsible for
friction. We shall study it in the third volume. But a few issues merit a
discussion right away.
%
% Impr. thoroughly in Mar 2014, added KPK ideas
\subsection{Should one use force? Power?}
% Index ok
\begin{quote}
The direct use of physical force is so poor a solution [...] that it is
commonly employed only by small children and great nations.\\
\iinn{David Friedman}
\end{quote}
\np Everybody has\index{force!use of(} to %\label{noforce}
take a stand on this question, even students of physics.
% It has been debated in many discussions, also in the field of physics.
Indeed, many types of forces are used and observed in daily life. One speaks
of muscular, gravitational, psychic, sexual, satanic, supernatural, social,
political, economic and many others. Physicists see things in a simpler
way. They call the different types of forces observed between objects
\ii[interaction]{interactions}. The study of the details of all these
interactions will show that, in everyday life, they are of electrical or
gravitational origin.
%
% \subsubsubsubsubsubsubsubsection{Table of forces}
{\small
\begin{table}[t]
\small
\centering
\caption{Some force\protect\index{force!values, table} values in nature.}
\label{forcemetab}
\dirrtabularstar
\begin{tabular*}{\textwidth}{@{}>{\columncolor{hks152}[0pt][1cm]}p{100mm}
@{\extracolsep{\fill}} p{35mm}@{}}
%
\toprule
%
\tabheadf{Observation} & \tabhead{Force} \\[0.5mm]
%
\midrule
%
% finish force table % !.!3 more biology, more on force effects on humans
%
Value measured in a magnetic resonance force microscope & $\csd{820}{zN}$\\
%
Force needed to rip a \iin[DNA, ripping apart]{DNA molecule} apart by pulling
at its two ends & $\csd{600}{pN}$\\
%
Maximum force exerted by human bite & $\csd{2.1}{kN}$\\
% updated Jun 2010, was 1.6; there is a paper on this: 2146 N
%
Typical peak force exerted by sledgehammer & $\csd{2}{kN}$\\
%
Force exerted by quadriceps & up to $\csd{3}{kN}$\\
%
Force sustained by \csd{1}{cm^2} of a good adhesive & up to $\csd{10}{kN}$\\
%
Force needed to tear a good rope used in rock climbing & $\csd{30}{kN}$\\
%
%Force in suspension bridges & \\
%
Maximum force measurable in nature & $\csd{3.0\cdot 10^{43}}{N}$\\
%
\bottomrule
\end{tabular*}
\end{table}
}
For physicists, all change is due to motion. The term force then also takes
on a more restrictive definition. \emph{(Physical) force}
is\indexs{force!physical} defined as the \ii[momentum!change]{change of
momentum with time}, i.e.,{} as
\begin{equation}
{\bm F} =
\frac{\diffd {\bm p} }{ \diffd t}
\cp \label{nl}
\end{equation}
A few measured values are listed in \figureref{forcemetab}.
% Mar 2014
A horse is running so fast that\cite{disessa} the hooves\index{horse} touch
the ground only 20\,\% of the time. What is the load carried by its legs
during\challengenor{horseload} contact?
Force is the \emph{change} of momentum. Since momentum is conserved, we can
also say that force measures the \emph{flow} of momentum.\index{momentum!flow
is force(} As we will see in detail shortly, whenever a force accelerates a
body, momentum flows into it.\index{momentum!as a substance} Indeed, momentum
can be imagined to be some invisible and intangible substance.\cite{disessa}
Force measures how much of this substance flows into or out of a body per unit
time.
\begin{quotation}
\noindent \csrhd Force is momentum flow.\index{force!is momentum flow}
\end{quotation}
% Dec 2016
The conservation of momentum is due to the conservation of this liquid. Like
any liquid, momentum flows through a surface.
Using the Galilean definition of linear momentum ${\bm p}=m\bm v$, we can
rewrite the definition of force\index{force!definition of} (for constant mass)
as
\begin{equation}
{\bm F} = {m \bm a} \cvend \label{nl2}
\end{equation}
%
where ${\bm F}={\bm F}(t,{\bm x})$ is the force acting on an object of mass
$m$ and where ${\bm a}={\bm a}(t,{\bm x})=\diffd {\bm v}/\diffd t=\diffd ^2
{\bm x}/\diffd t^2$ is the
acceleration of the same object, that is to say its change of velocity.%
%
\footnote{This equation was first written down by the % Swiss
mathematician and physicist \iinns{Leonhard Euler} \livedplace(1707
Basel1783 St. Petersburg) in 1747,
% over 70 years after
% \iin{Newton}'s first law and
20 years after the death of Newton, to whom it is usually and falsely
ascribed. It was Euler, one of the greatest mathematicians of all time (and
not Newton), who first understood that this definition of force is useful in
\emph{every} case of motion, whatever the appearance, be it for point
particles or extended objects, and be it rigid, deformable or fluid
bodies.\cite{a14} Surprisingly and in contrast to frequentlymade
statements, equation (\ref{nl2}) is even correct in\seepagetwo{relforce}
relativity.} %
%
The expression states in precise terms that force is what changes the
\emph{velocity} of masses.
%, or if one prefers, that \emph{force is the origin of momentum
% change}.
The quantity is called `force' because it corresponds in many, but \emph{not}
all aspects to everyday muscular force. For example, the more force is used,
the further a stone can be thrown. Equivalently, the more momentum is pumped
into a stone, the further it can be thrown. As another example, the concept
of \ii{weight} describes the flow of momentum due to gravity.
\begin{quotation}
\noindent \csrhd Gravitation constantly pumps momentum into massive
bodies.\index{gravitation!as momentum pump}
\end{quotation}
% Mar 2014
Sand in an hourglass is running, and the hourglass is on a scale. Is the
weight shown on the scale larger, smaller or equal to the weight when the sand
has stopped falling?\challengenor{hourglass}
% Mar 2014
\csepsfnb{imomentumflowdynamic}{scale=1}{The pulling child pumps momentum
into the chariot. In fact, some momentum flows back to the ground due to
dynamic friction (not drawn).}
% Nov 2012
Forces are measured with the help of deformations of
bodies.\index{force!measurement} Everyday force values can be measured by
measuring the extension of a spring. Small force values, of the order of
\csd{1}{nN}, can be detected by measuring the deflection of small levers with
the help of a reflected laser beam.
% % !!!2 add new image of force measurement devices: a spring, a laser deflection
% \csepsf{iforcemeasurement}{scale=1}{Forces are measured with the help of
% deformations. Left: a spring with scale; right: a spring lever read out with
% a laser in an atomic force microscope.}
However, whenever the concept of force is used, it should be remembered that
\emph{physical force is different from everyday force or everyday
effort}.\indexs{effort} Effort is probably best approximated by the concept
of \ii[power!physical]{(physical) power}, usually abbreviated $P$, and
defined (for constant force) as
%
\begin{equation}
P =
\frac{\diffd W }{ \diffd t}
= {\bm F} {\bm v} % no \cdot
\end{equation}
in which \ii[work!physical]{(physical) work} $W$ is defined as $W = {\bm F}
% \cdot
{\bm s} $,
where $\bm s$ is the distance along which the force acts. Physical work is a
form of energy, as you might want to check. Work, as a form of energy, has to
be taken into account when the conservation of energy is checked.
With the definition of work just given you can solve the following puzzles.
What happens to the electricity consumption\challengenor{escal} of an
escalator if you walk on it instead of standing still? What is the effect of
the definition of power for the salary of scientists?\challengedif{powerjoke}
A man who walks carrying a heavy rucksack is hardly doing any work; why then
does he get tired?\challengenor{tiredman}
When students in exams say that the force acting on a thrown stone is least at
the highest point of the trajectory,\cite{Hestenes} it is customary to say
that they are using an incorrect view, namely the socalled
\ii[motion!Aristotelian view]{Aristotelian view}, in which force is
proportional to velocity. Sometimes it is even said that they are using a
different concept of \emph{state} of motion. Critics then add, with a tone of
superiority, how wrong all this is. This is an example of intellectual
disinformation. Every student knows from riding a bicycle, from throwing a
stone or from pulling an object that increased \ii[effort!everyday]{effort}
results in increased speed. The student is right; those theoreticians who
deduce that the student has a mistaken concept of \emph{force} are wrong. In
fact, instead of the \emph{physical} concept of force, the student is just
using the \emph{everyday} version, namely effort. Indeed, the effort exerted
by gravity on a flying stone is least at the highest point of the trajectory.
% (One can also argue that
% Aristotle or the student use a different concept of \emph{state} of
% motion.)
Understanding the difference between physical\index{mechanics!learning} force
and everyday effort is the main hurdle in \iin[learning!mechanics]{learning
mechanics}.%
%
\footnote{This stepping stone is so high that many professional physicists do
not really take it themselves; this is confirmed by the innumerable comments
in papers that state that physical force is defined using mass, and, at the
same time, that mass is defined using force (the latter part of the sentence
being a fundamental mistake).}
% Mar 2014
\csepsfnb{imomentumflow}{scale=1}{The two equivalent descriptions of
situations with zero net force, i.e., with a closed momentum flow.
Compression occurs when momentum flow and momentum point in the same
direction; extension occurs when momentum flow and momentum point in opposite
directions.}
Often the flow of momentum,\index{momentum flow} equation (\ref{nl}), is not
recognized as the definition of force. This is mainly due to an everyday
observation: there seem to be forces without any associated acceleration or
change in momentum, such as in a string under tension or in water at high
\iin{pressure}. When one pushes against a tree, as shown in
\figureref{imomentumflow}, there is no motion, yet a force is applied. If
force is momentum flow, where does the momentum go? It flows into the slight
deformations of the arms and the tree. In fact, when one starts pushing and
thus deforming, the associated momentum change of the molecules, the atoms, or
the electrons of the two bodies can be observed. After the deformation is
established %, and looking at even higher magnification,
a continuous and equal flow of momentum is going on in both directions.
% The nature of this flow will be clarified in our exploration of quantum
% theory.
Because force is net momentum flow,\cite{disessa} the concept of force is not
really needed in the description of motion. But sometimes the concept is
practical. This is the case in everyday life, where it is useful in
situations where net momentum values are small or negligible. For example, it
is useful to define pressure as force per area, even though it is actually a
momentum flow per area. At the microscopic level, momentum alone suffices for
the description of motion.
In the section title we asked about on the usefulness of force and power.
Before we can answer conclusively, we need more arguments. Through its
definition, the concepts of force and power are distinguished clearly from
`mass', `momentum', `energy' and from each other. But where do forces
originate? In other words, which effects in nature have the capacity to
accelerate bodies by pumping momentum into objects? \tableref{motors} gives
an overview.\index{momentum}\index{mass}\index{energy}
%
% June 2010, impr. Mar 2014 with KPK
\subsection{Forces, surfaces and conservation}
% Index ok
We saw\label{forcesuftg} that force is the change of momentum. We also saw
that momentum is conserved.\index{momentum conservation!and force} How do
these statements come together? The answer is the same for all conserved
quantities. We imagine a closed surface that is the boundary of a volume in
space. Conservation implies that the conserved quantity enclosed
\emph{inside} the surface can only change by
\emph{flowing through} that surface.%
%
\footnote{Mathematically, the conservation
of a quantity $q$ is expressed with the help of the volume density $\rho=q/V$,
the current $I=q/t$, and the flow or flux ${\bm j} = \rho {\bm v}$, so that
$j= q/At$. Conservation then implies
\begin{equation}
\frac{\diffd q}{\diffd t} = \int_{V} \frac{\partial \rho}{\partial t}
\diffd V =  \int_{A=\partial V} {\bm j} \diffd {\bm A} =  I
\end{equation}
or, equivalently,
\begin{equation}
\frac{\partial \rho}{\partial t} + {\bm \nabla} {\bm j} = 0 \cp
\end{equation}%
This is the \ii[continuity!equation]{continuity equation} for the quantity
$q$. All this only states that a conserved quantity in a closed volume $V$
can only change by flowing through the surface $A$. This is a typical example
of how complex mathematical expressions can obfuscate the simple physical
content.} %
%
All conserved quantities in nature  such as energy, linear momentum,
electric charge, angular momentum  can only change by flowing through
surfaces. In particular, whenever the momentum of a body changes, this
happens through a surface.\index{momentum conservation!and surface flow}
Momentum change is due to momentum flow.\index{force!acting through surfaces}
In other words, the concept of force
always implies a surface through which momentum flows. %
\begin{quotation}
\noindent \csrhd Force is the flow of momentum through a surface.
\end{quotation}
%
This point is essential in understanding physical force.\cite{fherm} Every
force requires a surface for its definition.
To refine your own concept of force, you can search for the relevant surface
when a rope pulls a chariot, or when an arm pushes a tree, or when a car
accelerates.\challengn It is also helpful to compare the definition of force
with the definition of power: both quantities are flows through surfaces. As
a result, we can say:
\begin{quotation}
\noindent \csrhd A \emph{motor} is a momentum
pump.\index{motor!definition}
\end{quotation}
%
{\small%
% \subsubsubsubsubsubsubsubsection{Motor table}
\begin{table}[tp]
\small
% changed them to fit table:
\def\cstabhlinedown{\rule[1ex]{0mm}{1.5ex}} % lo spazio sopra e sotto \hline
\def\cstabhlineup{\rule{0mm}{3.0ex}}
\caption{Selected
processes\protect\index{motor!table}\protect\index{actuator!table} and
devices changing the motion of bodies.}
\label{motors}\index{actuator}\index{motor!type table}
\vbox to 45\baselineskip{%\leavevmode
\dirrtabularstarnolines
\begin{tabular*}{\textwidth}{@{\hspace{0em}}
>{\PBS\raggedright\hspace{0.0em}}p{50mm}
>{\PBS\raggedright\hspace{0.0em}}p{43mm} @{\extracolsep{\fill}}
>{\PBS\raggedright\hspace{0.0em}}p{41mm} @{\hspace{0em}}}%
%
\toprule
\tabhead{Situations that can lead to acceleration} % no headf or headfp
& \tabhead{Situations that
only lead to deceleration} & \tabhead{Motors and actuators} \\[0.5mm]
%
\midrule
%
\ii{piezoelectricity} & & \\
\quad quartz under applied voltage & thermoluminescence & walking piezo
{tripod}\index{tripod}\\
%
\ii[collision!and acceleration]{collisions}\cstabhlineup & & \\
\quad satellite in planet encounter & car crash & {rocket
motor}\index{rocket motor} \\
\quad growth of mountains & meteorite crash & swimming of larvae\\
%
\ii{magnetic effects}\cstabhlineup & & \\
\quad compass needle near magnet & electromagnetic braking & electromagnetic
gun\\
%
\quad magnetostriction & transformer losses & {linear motor}\index{motor!linear}\\
%
\quad current in wire near magnet & electric heating &
{galvanometer}\index{galvanometer}\\
%
\ii{electric effects}\cstabhlineup & & \\
\quad rubbed comb near hair & friction between solids & {electrostatic
motor}\index{motor!electrostatic}\\
%
\quad bombs & fire & muscles, sperm flagella\\
%
\quad cathode ray tube & electron microscope & Brownian motor\\
%
\ii{light}\cstabhlineup & & \\
\quad levitating objects by light & light bath stopping atoms & (true) {light
mill}\index{light!mill}\\
%
\quad solar sail for satellites & light pressure inside stars & solar cell
\\
%
\ii{elasticity}\cstabhlineup & & \\
\quad bow and arrow & trouser suspenders & {ultrasound
motor}\index{ultrasound!motor} \\
%
\quad bent trees standing up again & pillow, air bag &
{bimorphs}\index{bimorphs}\\
%
\ii{osmosis}\cstabhlineup & & \\
\quad water rising in trees & salt conservation of food & osmotic pendulum \\
%
\quad electroosmosis %\cite{electroosmo}
& & tunable Xray screening\\
%
\ii[heat]{heat \& pressure}\cstabhlineup & & \\
\quad freezing champagne bottle & surfboard water resistance& hydraulic
engines \\
%
\quad tea kettle & quicksand & steam engine \\
%
\quad barometer & parachute & air gun, sail \\
%
\quad earthquakes & sliding resistance & seismometer \\
%
\quad attraction of passing trains & shock absorbers & water turbine \\
%
%blowing between parallel pieces of paper, & & water turbine\\
%
%fluids pushing onto bodies, & wind resistance & \\
%
%
%
%buoyancy and boiling of water, & shock absorbers & \\
%
%freezing water & water resistance & \\
%
\ii{nuclei}\cstabhlineup & & \\
\quad radioactivity & plunging into the Sun & supernova explosion \\
%
\ii{biology}\cstabhlineup & & \\
\quad bamboo growth & decreasing blood vessel diameter
& molecular motors\\
% cell division & ...\\
%
%
% rowing ? walking
%
\ii{gravitation}\cstabhlineup & & \\
\quad falling & emission of gravity waves & {pulley}\index{pulley}\\
%
\bottomrule
\end{tabular*}
\vss % !.!4 not the perfect solution for a long table
}
\end{table}
}
%
% Impr. Mar 2014 with KPK
\subsection{Friction and motion}
% Index ok
% Aug 2016
\csepsfnb{ifrictioncoll}{scale=1}{Frictionbased
processes (courtesy \protect\iname{Wikimedia}).}
% (OK) add friction figure collection: Pablo Casals playing cello,
% running man (Juantorena?), ice skater, skewed tree in wind on a cliff, etc.
Every example of motion, from the motion that lets us choose the direction of
our gaze to the motion that carries a butterfly through the landscape, can be
put into one of the two leftmost columns of \tableref{motors}. Physically,
those two columns are separated by the following criterion: in the first
class, the acceleration of a body can be in a different direction from its
velocity. The second class of examples produces only accelerations that are
exactly \emph{opposed} to the velocity of the moving body, as seen from the
frame of reference of the braking medium. Such a resisting force is called
\ii{friction}, \ii{drag} or a \ii{damping}. All examples in the second class
are types of friction. Just check.\challengn Some examples of processes based
on friction are given in \figureref{ifrictioncoll}.
% April 2009
Here is a puzzle on cycling: does side wind brake  and
why?\challengenor{sidewind} % Yes!
Friction can be so strong that all motion of a body against its environment is
made impossible. This type of friction, called \ii[friction!static]{static
friction} or \ii[friction!sticking]{sticking friction}, is common and
important: without it, turning the wheels of bicycles, trains or cars would
have no effect.
% Jul 2006 improved after reader comment
Without static friction, wheels driven by a motor would have no grip.
Similarly, not a single \iin[screw!and friction]{screw} would stay tightened
and no \iin[hair!clip]{hair clip} would work. We could neither run nor walk
in a forest, as the soil would be more slippery than polished
ice.\index{motion!based on friction} In fact not only our own motion, but all
\ii[motion!voluntary]{voluntary motion} of living beings is \emph{based} on
\iin{friction}. The same is the case for all selfmoving machines. Without
static friction, the propellers in ships, aeroplanes and helicopters would not
have any effect and the wings of aeroplanes would produce no lift to keep them
in the air. (Why?)\challengenor{boundary}
In short, static friction is necessary whenever we or an engine want to move
against the environment.\index{motion!passive}\index{motion!voluntary}
%
% Impr. Mar 2014 with KPK
\subsection{Friction, sport, machines and predictability}
% Index ok
Once an object moves through its environment, it is hindered by another type
of friction; it is called \ii[friction!dynamic]{dynamic friction} and acts
between all bodies in relative motion.%
%
\footnote{There might be one exception. Recent research suggest that maybe in
certain crystalline systems, such as tungsten bodies on silicon, under ideal
conditions gliding friction can be extremely small and possibly even vanish in
certain directions of motion.\cite{fricrese} This socalled
\ii{superlubrication} is presently a topic of research.} %
%
Without dynamic friction, falling bodies would always rebound to the same
height, without ever\cite{fricake} coming to a stop; neither
\iin[parachute!needs friction]{parachutes} nor brakes would work; and even
worse, we would have no memory, as we will see later.
All motion examples in the second column of \tableref{motors} include
friction. In these examples, macroscopic energy is not conserved: the systems
are \ii[system!dissipative]{dissipative}. In the first column, macroscopic
energy is constant: the systems are \ii[system!conservative]{conservative}.
\csepsf{iairresistnew}{scale=1}{Shapes and air/water resistance.} %
% drag coefficient for dolphins put to 0.035
% (penguins with 0.0368 exist)
The\label{potentialdef} first two columns can also be distinguished using a
more abstract, mathematical criterion: on the left are accelerations that can
be derived from a potential, on the right, decelerations that can not. As in
the case of gravitation, the description of any kind of motion is much
simplified by the use of a potential: at every position in space, one needs
only the single value of the potential to calculate the trajectory of an
object, instead of the three values of the acceleration or the force.
Moreover, the magnitude of the velocity of an object at any point can be
calculated directly from energy conservation.
The processes from the second column \emph{cannot} be described by a
potential. These are the cases where it is best to use force if we want to
describe the motion of the system. For example, the \iin{friction} or
\iin{drag} force $F$ due to \ii{wind resistance} of a body is \emph{roughly}
given by
\begin{equation}
F= \frac{1}{2} c_{\rm w} \varrho A v^2
\end{equation}
where $A$ is the area of its crosssection and $v$ its velocity relative to
the air, $\varrho$ is the density of air. The \ii[drag!coefficient]{drag
coefficient} $c_{\rm w}$ is a pure number that depends on the shape of the
moving object. A few examples are given in \figureref{iairresistnew}. The
formula is valid for all fluids, not only for air, below the speed of sound,
as long as the drag is due to turbulence. This is usually the case in air and
in water. (At very low velocities, when the fluid motion is not turbulent but
laminar,\seepageone{lamtubrflo} drag is called \ii[drag!viscous]{viscous} and
follows an (almost) linear relation with speed.) You may check that drag, or
aerodynamic resistance \emph{cannot} be derived from a\challengn potential.%
%
\footnote{Such a statement about friction is correct only in three dimensions,
as is the case in nature; in the case of a single dimension, a potential can
\emph{always} be found.\challengenor{onedifri}}
% ostraciidae: 0.157 (4 C) to 0.167 (0C)
The drag coefficient $c_{\rm w}$ is a measured quantity. Calculating drag
coefficients with computers, given the shape of the body and the properties of
the fluid, is one of the most difficult tasks of science; the problem is still
not solved.
%
An aerodynamic car has a value between 0.25 and 0.3; many sports cars share
with vans values of 0.44 and higher,
% Oct 2006
and racing car values can be as high as 1, depending on the amount of the
force that is used to keep the car fastened to the ground. The lowest known
values are for dolphins and penguins.%
%
\footnote{%
% Oct 2014
It is unclear whether there is, in nature, a smallest possible value for the
drag coefficient.
% % Oct 2014
% In practice, the drag coefficient $c_{\rm w}$ seems to be always larger
% than 0.016 % 8, % Probably not true! Settle this issue
% the value for the optimally streamlined \iin[shape!optimal hydrodynamic]{tear
% shape}.\index{tear shape}
The topic of aerodynamic shapes is also interesting for fluid bodies. They
are kept together by \ii{surface tension}. For example, surface tension keeps
the wet hairs of a soaked brush together. Surface tension also determines the
shape of \iin{rain drops}.\index{drop!of rain} Experiments show that their
shape is spherical for drops smaller than \csd{2}{mm} diameter, and that
larger rain drops are \emph{lens} shaped, with the flat part towards the
bottom.\seepagefive{dropphoto} The usual tear shape is \emph{not} encountered
in nature; something vaguely similar to it appears during drop\cite{droplens}
detachment, but \emph{never} during drop fall.} %
Wind resistance is also of importance to humans,\index{athletics!and
drag}\index{sport!and drag} in particular in athletics.\cite{runningcw} It is
estimated that \csd{100}{m} sprinters spend between 3\,\% and 6\,\% of their
power overcoming drag. This leads to varying sprint times $t_{\rm w}$ when
wind of speed $w$ is involved, related by the expression
\begin{equation}
\frac{t_{0}}{t_{\rm w}}= 1.030.03 \left (1
\frac{w t_{\rm w} }{ \csd{100}{m} }
\right )^{2} \cvend
\end{equation}
where the more conservative estimate of 3\,\% is used. An opposing wind speed
of \csd{2}{m/s} gives an increase in time of \csd{0.13}{s}, enough to change
a potential world record into an `only' excellent result. (Are you able to
deduce the
$c_{\rm w}$ value for running humans from the formula?)\challenge % !!!5
% F = 1/2 cw rho A v^2
% P = F v
% t = l/v = l ( P / 1/2 cw rho A) ^(1/3)
% ...?
Likewise, parachuting\index{parachte!and drag} exists due to wind resistance.
Can you determine how the speed of a falling body, with or without parachute,
changes with time, assuming \emph{constant} shape and drag
coefficient?\challengenor{parac}
In contrast, \iin[friction!static]{static friction} has different properties.
It is proportional to the force pressing the two bodies together. Why?
Studying the situation in more detail,\cite{fricbk} sticking friction is found
to be proportional to the actual contact
area. %As friction expert Bowden says, `
It turns out that putting two solids into contact is rather like turning
Switzerland upside down and putting it onto Austria; the area of contact is
much smaller than that estimated macroscopically. The important point is that
the area of actual contact is proportional to the \emph{normal} force, i.e.,
the force component that is perpendicular to the surface.\index{force!normal}
The study of what happens in that contact area is still a topic of {research};
researchers are investigating the issues using instruments such as
\iin[microscope!atomic force]{atomic force microscopes},
\iin[microscope!lateral force]{lateral force microscopes} and
\iin{triboscopes}. These efforts resulted in computer \iin[hard
discs!friction in]{hard discs} which last longer, as the friction between disc
and the reading head is a central quantity in determining the lifetime.
All forms of friction are accompanied by an increase in the temperature of the
moving body. The reason became clear after the discovery of atoms. Friction
is not observed in few  e.g.~2, 3, or 4  particle systems. Friction only
appears in systems with \emph{many} particles, usually millions or more. Such
systems are called \ii[system!dissipative]{dissipative}. Both
the\indexs{system!dissipative} temperature changes and friction itself are
due to the motion of large numbers of microscopic particles against each
other. This motion is not included in the Galilean description. When it is
included, friction and energy loss disappear, and potentials can then be used
throughout. Positive accelerations  of microscopic magnitude  then also
appear, and motion is found to be conserved.
In short, all motion is conservative on a microscopic scale. On a microscopic
scale it is thus possible and most practical to describe
\emph{all} motion without the concept of\label{HHZ} force.%
%
%
\footnote{The first scientist who eliminated force from the description of
nature was \iinns{Heinrich~Rudolf Hertz} \livedplace(1857 Hamburg1894
Bonn), the famous discoverer of electromagnetic waves, in his textbook on
mechanics, \bt Die Prinzipien der Mechanik/ \pu Barth/ \yr 1894/ republished
by \pu Wissenschaftliche Buchgesellschaft/ % Darmstadt/
\yrend 1963/ His idea was strongly criticized at that time; only a
generation later, when quantum mechanics quietly got rid of the concept for
good, did the idea become commonly accepted. (Many have speculated about
the role Hertz would have played in the development of quantum mechanics and
general relativity, had he not died so young.) In his book, Hertz also
formulated the \iin[principle!of the straightest path]{principle of the
straightest path}: particles follow \iin{geodesics}. This same
description is one of the pillars of general relativity, as we will see
later on.} %
%
%It was thus the microscopic study of matter, especially quantum
%mechanics, which proved that force is an unnecessary concept for
%the description of nature. %
%
%In summary, the concept of force is only useful in macroscopic situations
%where motion seems to be created or destroyed.
The moral of the story is twofold: First, one should use force and power only
in one situation: in the case of friction, and only when one does not want to
go into the details.%
%
\footnote{But the cost is high; in the case of human relations the evaluation
should be somewhat more discerning, as research on violence\cite{gilligan} has
shown.}
%
%A beautiful book on the control of force is ..
%
Secondly, friction is not an obstacle to predictability.\index{force!use
of)}\index{momentum!flow is force)} Motion remains predictable.
\begin{quoteunder}\selectlanguage{french}%
Et qu'avonsnous besoin de ce moteur, quand l'étude réfléchie de la nature
nous prouve que le mouvement perpétuel est la première de ses
lois~?\selectlanguage{british}\footnote{`And whatfor do we need this motor,
when the reasoned study of nature proves to us that perpetual motion is the
first of its laws?'}\\
% 1791, page 520, said by the comte de Bressac
\iinn{Donatien~de Sade} \btsim Justine, ou les malheurs de la vertu/.
\end{quoteunder}
%
% Impr. Mar 2014
\subsection{Complete states  initial conditions}
% Index ok
\begin{quote}
Quid sit futurum cras, fuge quaerere ...\footnote{`What future will be
tomorrow, never ask ...' Horace is Quintus Horatius Flaccus
\lived(658
{\bce}), the\cite{chilhadetto2} great Roman poet.}\\
{Horace}, \bt Odi/ lib.~I, ode 9,
v.~13.\indname{Horace, in full Quintus Horatius Flaccus}
\end{quote}
\label{initiacond}
%
\np Let us continue our\index{conditions!initial, definition of} exploration
of the predictability of motion. We often describe the motion of a body by
specifying the time dependence of its position, for example as
\begin{equation}
{\bm x(t)}= {\bm x_{0}} + {\bm v_{0}}(tt_{0}) + {\te
\frac{1 }{ 2}} {\bm a_{0}} (tt_{0})^{2}+ {\te \frac{1 }{ 6}}
{\bm j_{0}} (tt_{0})^{3} + ... \cp
%\label{initial}
\end{equation}
The quantities with an index $0$, such as the starting position ${\bm x_{0}}$,
the starting velocity ${\bm v_{0}}$, etc., are called \ii{initial conditions}.
Initial conditions are necessary for any description of motion. Different
physical systems have different initial conditions. Initial conditions thus
specify the \ii{individuality} of a given system. Initial conditions also
allow us to distinguish the present situation of a system from that at any
previous time: initial conditions specify the \emph{changing aspects} of a
system. Equivalently, they summarize the \ii[past!of a system]{past} of a
system.
Initial conditions are thus precisely the properties we have been
seeking\seepageone{statedef} for a description of the \ii[state!of a
system]{state} of a system. To find a complete description of states we thus
need only a complete description of initial conditions, which we can thus
righty call also \ii[state!initial]{initial states}. It turns out that for
gravitation, as for all other microscopic interactions, there is \emph{no}
need for initial acceleration ${\bm a_{0}}$, initial jerk ${\bm j_{0}}$, or
higherorder initial quantities. In nature, acceleration and \iin{jerk}
depend only on the properties of objects and their environment; they do not
depend on the past. For example, the expression $a=GM/r^{2}$ of universal
gravity, giving the acceleration of a small body near a large one, does not
depend on the past, but only on the environment. The same happens for the
other fundamental interactions, as we will find out shortly.
The \emph{complete state} of a moving mass point\seepageone{mapode} is thus
described by specifying its position and its momentum at all instants of
time.\indexs{state!of a mass point, complete} Thus we have now achieved a
\emph{complete} description of the \emph{intrinsic properties} of point
objects, namely by their mass, and of their \emph{states of motion}, namely by
their momentum, energy, position and time. For \emph{extended rigid} objects
we also need orientation %, angular velocity
and angular momentum. This is the full list for rigid objects; no other state
observables are needed.
Can you specify the necessary state observables in the cases of extended
elastic bodies and of fluids?\challengenor{elbodqu} Can you give an example of
an intrinsic property that we have so far missed?\challengenor{intprops}
The set of all possible states of a system is given a special name: it is
called the \ii{phase space}. We will use the concept repeatedly. Like any
space, it has a number of dimensions. Can you specify this number for a
system consisting of $N$ point particles?\challengenor{phasespsol}
It is interesting to recall an older challenge and ask again: does the
\iin{universe} have initial conditions? Does it have a phase
space?\challengenor{uniic}
Given that we now have a description of both properties and states for point
objects, extended rigid objects and deformable bodies, can we predict all
motion? Not yet. There are situations in nature where the motion of an
object depends on characteristics other than its mass; motion can depend on
its colour (can you find an example?),\challengenor{colmotion} on its
temperature, and on a few other properties that we will soon discover. And
for each intrinsic property there are state observables to discover. Each
additional intrinsic property is the basis of a field of physical enquiry.
% Feb 2012
Speed was the basis for mechanics, temperature is the basis for
thermodynamics, charge is the basis for electrodynamics, etc.
%
We must therefore conclude that as yet we do not have a complete description
of motion.
%That will happen during the upcoming parts of our adventure.
\begin{quoteunder}
An optimist is somebody who thinks that the future is uncertain.\\
Anonymous
\end{quoteunder}
%
% April 2014
\subsection{Do surprises exist? Is the future determined?}
% subsubsection{Evolution, time and determinism}
% Index ok
\begin{quote}
\selectlanguage{german}Die Ereignisse der Zukunft \emph{können} wir nicht aus
den gegenwärtigen erschlie\ss en. Der Glaube an den Kausalnexus ist
ein Aberglaube.\selectlanguage{british}%
%
\footnote{`We cannot infer the events of the future from those of the
present. Belief in the causal nexus is superstition.' % Pears translation
Our adventure, however, will confirm the everyday observation that this
statement is wrong.}
%
\\
Ludwig Wittgenstein, \bt Tractatus/ 5.1361\indname{Wittgenstein, Ludwig}
\end{quote}
\begin{quote}
Freedom is the recognition of necessity.\\
Friedrich Engels \lived(18201895)\indname{Engels, Friedrich}
\end{quote}
\label{surpdet}
%
\np If, after climbing a tree, we jump down, we cannot halt the jump in the
middle of the trajectory; once the jump has begun, it is unavoidable and
determined, like all \iin{passive motion}. However, when we begin to move an
arm, we can stop or change its motion from a hit to a caress.
\iin[motion!voluntary]{Voluntary motion} does not seem unavoidable or
predetermined. Which of these two cases is the general one?\challengn
%
% Note: the question is anthropomorphic!
%
Let us start with the example that we can describe most precisely so far: the
fall of a body. Once %\label{deter}
%
% is \phi for the potential a good choice?
%
the gravitational potential $\phi$ acting on a particle is given and taken
into account, we can use the expression
\begin{equation}
{\bm a}(x) =\nabla \phi =G M {\bm r}/ r^{3} \cvend
\label{eveqt1}
\end{equation}%
%
and we can use the state at a given time, given by initial conditions such as
\begin{align}
{\bm x}(t_{0}) \qhbox{and}
{\bm v}(t_{0}) \cvend
%\label{eveqt}
\end{align}%
%
to determine the motion of the particle in advance. Indeed, with these two
pieces of information, we can calculate the complete trajectory ${\bm x}(t)$.
% In particular, we can thus determine the future motion.
% paragraph rewritten by Carol Martinez in Summer 2007
An equation that has the potential to predict the course of events is called
an \ii[evolution!equation, definition]{evolution equation}. Equation
(\ref{eveqt1}), for example, is an evolution equation for the fall of the
object. (Note that the term `evolution' has different meanings in physics and
in biology.) An evolution equation embraces the observation that not all
types of change are observed in nature, but only certain specific cases. Not
all imaginable sequences of events are observed, but only a limited number of
them. In particular, equation (\ref{eveqt1}) embraces the idea that from one
instant to the next, falling objects change their motion based on the
gravitational potential acting on them.
Evolution equations do not exist only for motion due to gravity, but for
motion due to all forces in nature. Given an evolution equation and initial
state, the whole motion of a\index{future!fixed} system is thus
\emph{uniquely fixed}, a property of motion often called \ii{determinism}.
For example, astronomers can calculate the position of planets with high
precision for thousands of years in advance.
% Owing to this possibility, an equation such as (\ref{eveqt1}) is
% called an \ii[evolution equations]{evolution equation} for the motion of the
% object. (Note that the term `{evolution}' has different meanings in physics
% and in biology.) An evolution equation always expresses the observation
% that
% not all types of change are observed in nature, but only certain specific
% cases. Not all imaginable sequences of events are observed, but only a
% limited number of them. In particular, equation (\ref{eveqt1}) expresses
% that
% from one instant to the next, objects change their motion based on the
% potential acting on them. Thus, given an evolution equation and initial
% state, the whole motion of a\index{future, fixed}
% system is \emph{uniquely fixed}; this property of motion is often %
% % (but not unanimously)
% called \ii{determinism}. Since this term is often used with different
% meanings,
% Imp. April 2014
% (Is repeated in volume 5)
Let us carefully distinguish determinism from several similar concepts, to
avoid misunderstandings. Motion can be deterministic and at the same time be
\ii[unpredictability!practical]{unpredictable in
practice}.\label{unpredpracxx} The unpredictability of
motion\seepagefive{qmclocks} can have four origins:
\begin{Strich}
\item[{1.}] an impracticably large number of particles involved, including
situations with friction,
\item[{2.}] insufficient information about initial conditions, and
\item[{3.}] the mathematical complexity of the evolution equations,
\item[{4.}] strange shapes of spacetime.
\end{Strich}
\np For example, in case of the \ii[weather!unpredictability of]{weather} the
first three conditions are fulfilled at the same time.
%
% Feb 2012
It is hard to predict the weather over periods longer than about a week or
two. (In 1942, Hitler made once again a fool of himself across Germany by
requesting a precise weather
forecast for the following twelve months.) % Told by my grandfather
%
Despite the difficulty of prediction, weather change is still deterministic.
As another example, near \ii{black holes} all four origins apply together.
We will discuss black holes in the section on general relativity. Despite
being unpredictable, motion is deterministic near black holes.
Motion can be both\index{randomness} deterministic and time \emph{random},
i.e.,{} with different outcomes in similar experiments. A roulette ball's
motion\seepageone{roulettemoney} is deterministic, but it is also random.%
%
\footnote{Mathematicians have developed a large number of tests to determine
whether a collection of numbers\index{randomness} may be called
\emph{random}; roulette results pass all these tests  in honest casinos
only, however. Such tests typically check the equal distribution of
numbers, of pairs of numbers, of triples of numbers, etc. Other tests are
the $\chi^{2}$ test, the Monte Carlo test(s), and the \iin[gorilla test for
random numbers]{gorilla test}.\cite{randomtests}} %
%
As we will see later, quantum systems fall into this
category,\seepagefour{probab} as do all examples of irreversible motion, such
as a drop of ink spreading out in clear water.
%
% Sep 2012
Also the fall of a die\index{die throw} is both deterministic and random. In
fact, studies on how to predict the result of a die throw with the help of a
computer are making rapid progress;\cite{kapitaniak} these studies also show
how to throw a die in order to increase the odds to get a desired result.
%
In all such cases the randomness and the irreproducibility are only apparent;
they disappear when the description of states and initial conditions in the
microscopic domain are included.\index{irreversibility!of
motion}\index{reversibility!of motion} In short, determinism does not
contradict \emph{(macroscopic) irreversibility}. However, on the microscopic
scale, deterministic motion is always reversible.
A final concept to be distinguished from determinism is \ii{acausality}.
Causality is the requirement that a cause must precede the effect. This is
trivial in Galilean physics,\index{causality!of motion} but becomes of
importance in special relativity, where causality implies that the speed of
light is a limit for the spreading of effects. Indeed, it seems impossible to
have deterministic motion (of matter and energy) which is
\ii[acausality]{acausal}, in other words, faster than light. Can you confirm
this?\challengenor{causali} This topic will be looked at more deeply in the
section on special relativity.
Saying that motion is `deterministic' means that it is fixed in the future
\emph{and also in the past}. It is sometimes stated that predictions of
\emph{future} observations are the crucial test for a successful description
of nature. Owing to our often impressive ability to influence the future,
this is not necessarily a good test.
%Predictions are after all only statements of
%certain characteristics of nature.
Any theory must, first of all, describe \emph{past} observations correctly.
It is our lack of freedom to change the past that results in our lack of
choice in the description of nature that is so central to physics. In this
sense, the term `\iin[initial condition!unfortunate term]{initial condition}'
is an unfortunate choice, because in fact, initial conditions summarize the
\emph{past} of a system.%
%
% Feb 2012
\footnote{The problems with the term `initial conditions' become clear near
the big bang: at the big bang, the universe has no past, but it is often
said that it has initial conditions. This contradiction will be explored
later in our adventure.} %
%
The central ingredient of a deterministic description is that all motion can
be reduced to an evolution equation plus one specific state. This state can
be either initial, intermediate, or final. Deterministic motion is uniquely
specified into the past and into the future.
To get a clear concept of determinism, it is useful to remind ourselves why
the concept of `time' is introduced in our description of the world. We
introduce time because we observe first that we are able to define sequences
in observations, and second, that unrestricted change is impossible. This is
in contrast to films, where one person can walk through a door and exit into
another continent or another century. In nature we do not observe
metamorphoses, such as people changing into toasters or dogs into
toothbrushes. We are able to introduce `time' only because the sequential
changes we observe are extremely restricted. If nature were not reproducible,
time could not be used.\challengenor{tidefi} In short, determinism expresses
the observation that \emph{sequential changes are restricted to a single
possibility}.\indexs{determinism}
Since determinism is connected to the use of the concept of time, new
questions arise whenever the concept of time changes, as happens in special
relativity, in general relativity and in theoretical high energy physics.
There is a lot of fun ahead.
In summary, every description of nature that uses the concept of time, such as
that of everyday life, that of classical physics and that of quantum
mechanics, is intrinsically and inescapably deterministic, since it connects
observations of the past and the future, \emph{eliminating} alternatives. In
short,
\begin{quotation}
\npcsrhd {The use of time implies determinism, and vice versa.}
\end{quotation}
When drawing metaphysical conclusions, as is so popular nowadays when
discussing quantum theory, one should never forget this
connection.\seepagefive{qmclocks} Whoever uses clocks but denies determinism
is nurturing a \iin{split personality}!%
%
\footnote{That can be a lot of fun though.} %
%
The future is determined.
%
% Impr. Mar 2014, Jul 2016
\subsection{Free will}
% Index OK
\begin{quote}
You do have the ability to surprise yourself.\\
Richard Bandler and John Grinder\indname{Bandler, Richard}\indname{Grinder,
John}
\end{quote}
\np The\index{will, free!(} idea that motion is determined often produces
fear, because we are taught to associate determinism with lack of freedom. On
the other hand, we do experience freedom in our actions and call it \emph{free
will}. We know that it is necessary for our creativity and for our
happiness. Therefore it seems that determinism is opposed to happiness.
But what precisely is free will? Much ink has been consumed trying to find a
precise definition. One can try to define free will as the arbitrariness of
the choice of initial conditions. However, initial conditions must themselves
result from the evolution equations, so that there is in fact no freedom in
their choice. One can try to define free will from the idea of
unpredictability, or from similar properties, such as uncomputability. But
these definitions face the same simple problem: whatever the definition, there
is \emph{no way} to prove experimentally that an action was performed freely.
The possible definitions are useless. In short, because free will cannot be
defined, it \emph{cannot} be observed. (Psychologists also have a lot of
additional data to support this conclusion, but that is another topic.)
No process that is \emph{gradual}  in contrast to
\ii[process!sudden]{sudden}  can be due to free will; gradual processes are
described by time and are deterministic. In this sense, the question about
free will becomes one about the existence of sudden changes in nature. This
will be a recurring topic in the rest of this walk. Can nature
\iin[surprises!in nature]{surprise} us?
%
%
% Even worse, one can easily show that free will is in \emph{contrast} with
% the
% way nature works. There is a famous situation which makes the point. `Tell
% me what I'll do and I'll let you go unharmed; otherwise I will kill you.'
% There is a simple answer which makes any future action following this rule
% impossible. Can you find it?\challenge
% The situation shows that it is impossible\ldots
%
%
In everyday life, nature does not. Sudden changes are not observed. Of
course, we still have to investigate this question in other domains, in the
very small and in the very large. Indeed, we will change our opinion several
times during our adventure, but the conclusion remains.
%Only people have the ability to surprise. The rest of nature doesn't.
%We mention it here in passing, but
%actually it is a central topic of our whole walk.
We note that the lack of surprises in everyday life is built deep into our
nature: evolution has developed \iin{curiosity} because everything that we
discover is useful afterwards. If nature continually surprised us, curiosity
would make no sense.
Many observations contradict the existence of surprises: in the beginning of
our walk we defined time using the continuity of motion; later on we expressed
this by saying that time is a consequence of the conservation of energy.
Conservation is the opposite of surprise. By the way, a challenge remains:
can you show that time would not be definable even if surprises existed only
\emph{rarely}?\challengenor{raresurp}
In summary, so far we have no evidence that surprises exist in nature. Time
exists because nature is deterministic. Free will cannot be defined with the
precision required by physics. Given that there are no sudden changes, there
is only one consistent conclusion: free will is a \emph{feeling}, in
particular of independence of others, of independence from fear and of
accepting the consequences of one's actions.%
%
\footnote{That free will is a feeling can also be confirmed by careful
introspection. Indeed, the idea of free will always arises \emph{after} an
action has been started. It is a beautiful experiment to sit down in a quiet
environment, with the intention to make, within an unspecified number of
minutes, a small gesture, such as closing a hand. If you carefully observe,
in all detail, what happens inside yourself around the very moment of
decision,\challengn you find either a mechanism that led to the decision, or a
diffuse, unclear mist. You never find free will. Such an experiment is a
beautiful way to experience deeply the wonders of the self. Experiences of
this kind might also be one of the origins of human \iin{spirituality}, as
they show the connection everybody has with the rest of nature.} %
%
Free will is a strange name for a feeling of satisfaction.\cite{Hellingio}
%
%
This solves the apparent paradox; free will, being a feeling, exists as a
human experience, even though all objects move without any possibility of
choice. There is no contradiction.
Even if human action is determined, it is still authentic.\cite{authfr} So why
is determinism so frightening? That is a question everybody has to ask
themselves. What difference does determinism imply for your life, for the
actions, the choices,\challengn the responsibilities and the pleasures you
encounter?%
%
\footnote{If nature's `laws' are deterministic, are they in contrast
with\index{morals}\index{ethics} moral or ethical
`laws'?\challengenor{ethicla} Can people still be held responsible for their
actions?}
%
If you conclude that being determined is different from being free, you should
change your life! Fear of determinism usually stems from refusal to take the
world the way it is. Paradoxically, it is precisely % he
% Dec 2014
the person who insists on the existence of free will who is running away from
responsibility.\index{will, free!)}
%
% Impr. Mar 2014, Jul 2016
\subsection{Summary on predictability}
% Index OK
Despite difficulties to predict specific cases, all motion we encountered so
far is both deterministic and predictable. Even friction is predictable, in
principle, if we take into account the microscopic details of matter.
In short, classical mechanics states that the future is determined. In fact,
we will discover that \emph{all} motion in nature, even in the domains of
quantum theory and general relativity, is predictable.
Motion is predictable.\index{motion!is and must be predictable} This is not a
surprising result. If motion were not predictable, we could not have
introduced the concepts of `motion' and `time' in the first place. We can
only talk and think about motion because it is predictable.
%
%1020304050607078
%
% % {La fisica classica, seconda parte}
%
% % versione 8.762 full OED spellcheck see above
% % minmax done Nov 2004
% % tables checked Sep 2005
% % quotes checked Sep 2005
% % steel papers  done Nov 2004
% % green edit Nov 2004
% % triple exclamation marks: total 108, !.!1 (6), !.!2 (24) Jun 2007
% % references ordered Feb 2004
% % `Fakt dass': 2 ok Feb 2004
% % `Aszent': 6 ok Feb 2004
% % people data in Jun 2005
% % every \emph and par in index never
% % (ldots)(\.\.\.) checked Aug 2005
% % professional editor: Alec Edg. Jul 2005
% % checked [az][azAZ] Aug 2003
% % capitalized \bt downcased \ti Aug 2003
% % figures with triple excl. Jul 2014
% % formula letters fully explained May 2004
% % challenge solutions ordered Apr 2006
% % missing challenge solutions triple !'ed Apr 2006
% % psfragged it all Apr 2006
%
% % to get to 8.80
% %  add explanation of Hamiltonian
% %  explain why action S is supposed to be positive
% %  (NO) add new action results: the two types of action
% %  add planimeter to explain integration, or as a curiosity
% %  more humor
% %  more pictures
%
%
% % Jun 2007 ver 8.760 started adding many new photographs
% % Apr 2006 ver 8.742 many details, corrections by readers
% % Apr 2005 ver 8.724 added first AE corrections, changed many details
% % Feb 2005 ver 8.716 improved minimal entropy part
% % Nov 2004 ver 8.706 improved readability throughout, added laser
% % loudspeaker
% % Oct 2004 ver 8.700 added quantum of information
% % Nov 2003 ver 8.672 added minimum entropy
% % Jul 2003 ver 8.662 corrected blbl, improved oscillation part
% % Nov 2002 many small corrections
% % May 2002 green correction done
% % Apr 2002 Adrian's corrections typed in
% % Oct 2001 corrected paper, added many details,
% % added much better action explanation
%
\pagestyle{fimovieheadings} % da non togliere
% \makeatletter
% % May 2003: this should stop the previous film
% % \special{!userdict begin
% % /starthook{gsave 0 0 moveto 100 100 rlineto grestore}def end}
% \makeatother
% %\renewcommand{\baselinestretch}{1.7}\normalsize % only for test reading
%
%
%
%
%
%
% \newpage\gdef\cschaptermark{Galilean Motion}%
% % \chap ter{}
% \sect ion{Global descriptions of motion  the simplicity of complexity}
% % \chap ter{}
% \markboth{\thesection\ the global simplicity of complexity}{the global
% simplicity of complexity}
% Impr. Apr. 2014, Jul 2016
\subsection{From predictability to global descriptions of motion}
% Index OK
\begin{quote}
\csgreekok{Ple\~in >an'agke, z\~hn o>uk >an'agkh.}\footnote{Navigare
necesse, vivere non necesse. `To navigate is necessary, to live is not.'
\iname[Pompeius, Gnaeus]{Gnaeus Pompeius Magnus} \lived(10648 {\bce}) is
cited in this way by \iname{Plutarchus}
\livedca(\circa45\circa125).\cite{chilhadetto2b}}\\
{Pompeius}
% frase 833 motto delle citta` anseatiche
\end{quote}
% \csini{A}{ll} over the Earth  even in Australia  people observe that
% stones fall `down'. This\linebreak ncient observation led to the discovery
% of\label{leasa} the universal `law' of gravity. To find it,\linebreak ll
% that was necessary was to look for a description of gravity that was valid
% globally. The only additional observation that needs to be recognized in
% order to deduce the result $a=GM/r^{2}$ is the variation of gravity with
% height.
% Feb 2012
Physicists aim\index{motion!global descriptions(} to talk about motion with
the highest precision possible. Predictability is an aspect of precision.
The highest predictability  and thus the highest precision  is possible
when motion is described as globally as possible.
All over the Earth  even in Australia  people observe that stones fall
`down'. This ancient observation led to the discovery of\label{leasa} % the
universal gravity. To find it, all that was necessary was to look for a
description of gravity that was valid \emph{globally}. The only additional
observation that needs to be recognized in order to deduce the result
$a=GM/r^{2}$ is the variation of gravity with height.
\cstftlepsf{iglobalview}{scale=1}{What shape of rail allows the black stone
to glide most rapidly from point A to the lower point B?}
[10mm]{iglobalobs}{scale=1}{Can motion be described in a manner common to all
observers?}
In short, thinking \emph{globally} helps us to make our description of motion
more precise and our predictions more useful. How can we describe motion as
globally as possible? It turns out that there are six approaches to this
question, each of which will be helpful on our way to the top of Motion
Mountain.\label{sixglobalkk} We first give an overview; then we explore each
of them.
\begin{Strich}
\item[{1.}] \ii[action!principle]{Action principles} or
\ii[principle!variational]{variational principles}, the first global
approach to motion, arise when we overcome a fundamental limitation of what
we have learned so far. When we predict the motion of a particle from its
current acceleration with an evolution equation, we are using the most
\emph{local} description of motion possible.\seepageone{surpdet} We use the
acceleration of a particle at a certain place and time to determine its
position and motion \emph{just after} that moment and \emph{in the immediate
neighbourhood} of that place. Evolution equations thus have a mental
`horizon' of radius zero.
The contrast to evolution equations are {variational principles}. A famous
example is illustrated in \figureref{iglobalview}. The challenge is to
find the path that allows the fastest possible gliding motion from a high
point to a distant low point. The sought path is the \ii{brachistochrone},
from ancient Greek for `shortest time',\challengedif{tachypath} This puzzle
asks about a property of motion as a whole, for \emph{all times} and
\emph{all positions}. The global approach required by questions such as
this one will lead us to a description of motion which is simple, precise
and fascinating: the socalled principle of \ii[laziness!cosmic, principle
of]{cosmic laziness}, also known\index{principle!of
laziness}\index{principle!of least action} as the principle of \emph{least
action.}
\medskip
\item[{2.}] \emph{Relativity}, the second global approach to
motion,\indexs{relativity} emerges when we compare the various descriptions
of the same system produced by \emph{all possible observers}. For example,
the observations by somebody falling from a cliff  as shown in
\figureref{iglobalobs}  a passenger in a roller coaster, and an observer
on the ground will usually differ. The relationships between these
observations, the socalled \emph{symmetry transformations}, lead us to a
global description of motion, valid for everybody. Later, this approach
will lead us to Einstein's special and general theory of relativity.
\cstftlepsf{ihanged}{scale=1}{What happens when one rope is
cut?}[10mm]{icompastraight}{scale=1}{A famous mechanism, the
\protect\ii{PeaucellierLipkin linkage}, consists of (grey) rods and (red)
joints and allows drawing a straight line with a compass: fix point F, put
a {pencil} into joint P, and then move C with a compass along a circle.}
\medskip
\item[{3.}] \emph{Mechanics of\indexs{mechanics} extended and rigid
bodies\index{body!rigid},} rather than mass points, is required to
understand the objects, plants and animals of everyday life. For such
bodies, we want to understand how \emph{all parts} of them move. As an
example, the counterintuitive result of the experiment in
\figureref{ihanged} shows why this topic is worthwhile.\challengn The
rapidly rotating wheel suspended on only one end of the axis remains almost
horizontal, but slowly rotates around the rope.
In order to design machines, it is essential to understand how a group of
rigid bodies interact with one another. For example, take the
{PeaucellierLipkin linkage} shown in
\figureref{icompastraight}.\cite{peaucel} A joint F is fixed on a wall.
Two movable rods lead to two opposite corners of a movable rhombus, whose
rods connect to the other two corners C and P. This mechanism has several
astonishing properties. First of all, it implicitly defines a circle of
radius $R$ so that one always has the relation $r_{\rm C}=R^2/r_{\rm P}$
between the distances of joints C and P from the centre of this circle.
This is called an \emph{inversion at a circle}.\index{inversion!at circle}
Can you find this special circle?\challengenor{peaulipone} % added Jun 2013
% AE This requires a bit more explanation: mark the distances r_C and (DONE)
Secondly, if you put a pencil in joint P, and let joint C follow a certain
circle, the pencil P draws a straight line. Can you find that
circle?\challengenor{peauliptwo} % added Jun 2013
The mechanism thus allows\index{straight line!drawing with a compass}
drawing a \emph{straight} line with the help of a \iin{compass}.
A\label{fosti} famous machine challenge\cite{fosterni} is to devise a wooden
carriage, with
gearwheels\index{carriage!southpointing}\index{southpointing carriage}
that connect the wheels to an arrow, with the property that, whatever path
the carriage takes, the arrow always points south (see
\figureref{iarrowwagon}).\challengedif{southcarr} The solution to this
puzzle will even be useful in helping us to understand general relativity,
as we will see. Such a wagon allows measuring the curvature of a surface
and of\seepagetwo{riegym} space.
% August 2016
Another important machine part is the \ii[gearbox!differential]{differential
gearbox}. Without it, cars could not follow bends on the road. Can you
explain it to your friends?\challengn
% Nov 2013
\cssmallepsfnb{igearhopp}{scale=1}{The gears found in young plant hoppers
({\textcopyright}~\protect\iinn{Malcolm Burrows}).}
% Nov 2013
Also nature uses machine parts. In 2011, screws\index{screw!in
nature}\index{gears!in nature} and nuts were found in a joint of a weevil
beetle, \iie{Trigonopterus oblongus}.\cite{weevbeet} In 2013, the first
example of biological \emph{gears} have been discovered: in young plant
hoppers of the species \iie{Issus coleoptratus}, toothed gears ensure that
the two back legs jump synchronously.\cite{natgears} \figureref{igearhopp}
shows some details. You might enjoy the video on this discovery available
at \url{www.youtube.com/watch?v=Q8fyUOxD2EA}.
Another interesting example of rigid motion is the way that human movements,
such as the general motions of an arm, are composed from a small number of
basic motions.\cite{huar} All these examples are from the fascinating field
of engineering; unfortunately, we will have little time to explore this
topic in our hike.
\csepsf{iarrowwagon}{scale=1}{A southpointing carriage: whatever the path
it follows, the arrow on it always points south.}
\medskip
\item[{4.}] The next global approach to motion is the description of
\ii[body!extended, nonrigid]{nonrigid extended bodies}. For example,
\ii[fluid!mechanics]{fluid mechanics} studies\index{mechanics!fluid} the
flow of fluids (like honey, water or air) around solid bodies (like spoons,
ships, sails or wings). The aim is to understand how \emph{all parts} of the
fluid move. Fluid mechanics thus describes how insects, birds and
aeroplanes fly,%
%
\footnote{The mechanisms of insect flight are still a subject of active
research. Traditionally, fluid dynamics has concentrated on large
systems, like boats, ships and aeroplanes. Indeed, the smallest
humanmade object that can fly in a controlled way  say, a
radiocontrolled plane or helicopter  is much larger and heavier than
many flying objects that evolution has engineered. It turns out that
controlling the flight of small things requires more knowledge and more
tricks than controlling the flight of large things. There is more about
this topic on \cspageref{smallfluids} in Volume V.} %
%
why sailingboats can sail against the wind, what happens\cite{egghb} when a
hardboiled egg is made to spin on a thin layer of water, or how a
bottle\index{bottle!empty rapidly} full of wine can be emptied in the
fastest way possible.\challengenor{eggb}
% Mar 2015: both figs ok
\cstftlepsf{ichim}{scale=1}{How and where does a falling brick chimney
break?}[10mm] {iglobalstat}{scale=1}{Why do hotair balloons stay
inflated? How can you measure the weight of a bicycle rider using only a
ruler?}
As well as fluids, we can study the behaviour of \emph{deformable solids}.
This area of research is called \ii[continuum!mechanics]{continuum
mechanics}.\index{mechanics!continuum} It deals with deformations and
oscillations of extended structures. It seeks to explain, for example, why
bells are made in particular shapes; how large bodies  such as the falling
chimneys\challengenor{chimney} shown in \figureref{ichim}  or small
bodies  such as diamonds  break when under stress; and how cats can turn
themselves the right way up as they fall. During the course of our journey
we will repeatedly encounter issues from this field, which impinges even
upon general relativity and the world of elementary particles.
% Oct 2007
\csepsfnb{iglobalself}{scale=1}{Why do marguerites  or oxeye daisies,
\protect\iie{Leucanthemum vulgare}  usually have around 21 (left and
centre) or around 34 (right) petals?
({\textcopyright}~\protect\iname{Anonymous}, \protect\iinn{Giorgio
Di~Iorio}
% EMAILED FEB 2008  gioischia@hotmail.it
and \protect\iinn{Thomas Lüthi})}
% EMAILED FEB 2008  luthi@mymail.ch  OK!
\medskip
\item[{5.}] \ii[statistical mechanics]{Statistical mechanics} is the study of
the motion of \emph{huge numbers of particles}. Statistical mechanics is
yet another global approach to the study of motion. The concepts needed to
describe gases, such as temperature, entropy and pressure (see
\figureref{iglobalstat}), are essential tools of this discipline. In
particular, the concepts of statistical physics help us to understand why
some processes in nature do not occur backwards. These concepts will also
help us take our first steps towards the understanding of black holes.
\medskip
\item[{6.}] The last global approach to motion, \ii{selforganization},
involves all of the abovementioned viewpoints \emph{at the same time}.
Such an approach is needed to understand everyday experience, and
\emph{life} itself. Why does a flower form a specific number of petals, as
shown in \figureref{iglobalself}? How does an embryo differentiate in the
womb? What makes our hearts beat?
%
%Other questions are why all people grow to roughly the same end size, why
%most people have five fingers, how certain swarms of fireflies manage to
%blink simultaneously, or why neighboring plants
%and trees often start to blossom together.
%
% Why does muesli separate?
%
How do mountains ridges and cloud patterns emerge? How do stars and galaxies
evolve?
% How are sea waves formed by the wind?
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\raise 0.1ex\hbox{?} }} %
% the evolution of stars and galaxies, or {\mbox{%
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All these phenomena are examples of \ii{selforganization} processes; life
scientists speak of \ii{growth} processes. Whatever we call them, all these
processes are characterized by the spontaneous appearance of patterns, shapes
and cycles. Selforganized processes are a common research theme across many
disciplines, including biology, chemistry, medicine, geology and engineering.
\end{Strich}
% CS: (OK)
% AE: Is 'growth processes' an appropriate term for all these examples (like
% AE: sea waves)?
% Impr. June 2011, Jul 2016
\np We will now explore the six global approaches to motion. We will begin
with the first approach, namely, the description of motion using a variational
principle. This beautiful method for describing, understanding and predicting
motion was the result of several centuries of collective effort, and is the
highlight of Galilean physics.\index{Galilean
physics!highlight}\index{physics!Galilean, highlight}
% particle dynamics.
Variational principles also provide the basis for all the other global
approaches just mentioned. They are also needed for all the further
descriptions of motion that we will explore
afterwards.\index{motion!predictability of)}\index{predictability!of
motion)}\index{motion!global descriptions)}
\vignette{classical}
\clearpage\endinput
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