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% Start of ``Fall, Flow and Heat'', volume 1 of Motion Mountain by C. Schiller
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% This file is part of the sources of the Motion Mountain Physics Textbook.
% As a source file, this file is copyright from 2008 to the present by
% Christoph Schiller. This source file is NOT FREE; in particular, it is not
% creative commons and is not open source. All rights remain with Christoph
% Schiller. The earliest versions of this file are copyright 1990 by
% Christoph Schiller.
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% {La tavola delle materie specifica}
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\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{%
Em nossa aventura para saber como se movem as coisas,\phantom{Wp}\\%
a experiência da caminhada e outros movimentos\phantom{Wp}\\%
nos leva a introduzir os conceitos de\phantom{Wp}\\%
velocidade, tempo, distância, massa e temperatura.\phantom{Wp}\\%
Nós aprendemos a utilizálos para\emph{medir as mudanças}\phantom{Wp}\\% % corrects a bug
e percebemos que a natureza tenta minimizálas.\phantom{Wp}\\% % corrects a bug
Nos descobriremos como flutuar no espaço,\phantom{Wp}\\% % vomit comet,
% % astronaut
porque temos pernas ao invés de rodas,\phantom{Wp}\\%
porque a desordem nunca pode ser eliminada,\phantom{Wp}\\% % third princ of thermo
e porque um dos assuntos mais polêmicos\phantom{Wp}\\%
da ciência é o fluxo de águia por um tubo.\phantom{Wp}\\% % 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\se, and move,\\%
% what love has to do with magnets and amber,\\%
% and why we can see the stars.\\%
}
%
\part[Fall, Flow and Heat]{Fall, Flow and Heat}
\thispagestyle{empty}
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%
\index{astronautsee{cosmonaut}}%
\index{electronseealso{positron}}%
\index{electron chargeseealso{positron charge}}%
\index{gravitysee{gravitation}}%
\index{gravitation, universalsee{universal gravitation}}%
\index{gravitationseealso{universal gravitation}}%
%
%
% \chapter{}
\\Capítulo 1 {Porque devemos nos importar com movimento}
% \chapter{}
\markboth{\thesmallchapter\Porque devemos nos importar com movimento}
{\thesmallchapter\ Porque devemos nos importar com movimento}
% Impr. Jan 2011
% Index ok
\begin{quote}
Todo movimento é uma ilusão.\\
%Following
Zenão de Eleia\footnote{Zenão de Eleia (\circa450 {\bce}),\indname{Zenão de Eleia}
um dos principais filósofos da escola Eleatica de filosofia.}
\end{quote}
\csini{W}{ham!} O raio batendo na árvore próxima violentamente perturba a nossa caminha pela floresta tranquila.\linebreak e causa nossos corações a de repente bater mais rápido. No topo da árvore nós vemos uma labareda iniciar e depois sumir. O vento gentil move as folhas ao redor de nós ajuda a restaurar a calma do local.\linebreak \label{zenoill} Em um lugar próximo, a água de um pequeno rio segue o seu complicado caminho pelo vale, refletindo em sua superfície a eterna mudança das formas das nuvens, acima.
%
% \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
Movimento está em todo lugar: amigável e inímica, % horrible
terrível e bela. É fundamental à nossa existência humana. Nós precisamos de movimentação para crescer, apredender, pensar e aproveitar a vida. Nós usamos movimento para andar por uma floresta, % with our legs,
para ouvir seus ruídos % with our eardrums\se,
E para falar sobre isso. Como todos os animais, nós dependemos de movimento para conseguir comida e sobreviver aos perigos da existência.
%
Como todos os seres vivos, nós precisamos de movimento para reproduzir \index{reproduction} para respirar e digerir. Como todos os objetos, movimento nos traz calor.
Movimento é a observação mais fundamental na natureza. Na verdade, a situação é que \textit{tudo} que
acontece no mundo é algum tipo de movimento. Não existem exceções. Movimento é uma parte tão básica de nossas observações que mesmo a origem da palavra está perdida nas trevas da história linguística IndoEuropeia. A fascinação com movimento sempre fez este um assunto perfeito para a curiosidade humana.
%Already at the beginning of written thought, during
Lá pelo % sixth
% Jun 2005: changed to
século quinto {\bce} na Antiga Grécia, ao seu estudo foi dado um nome\cite{russo} \ii{física}.
%
% 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.
Movimento também é importante do ponto de vista da condição humana. O que podemos saber? Daonde
o \iin{mundo} veio? Quem somos nós? De onde nós viemos?\index{origin!human} O que faremos? O que devemos fazer? O que o futuro irá trazer? %Where do people come from? Where do they go?
O que é a \iin[death!origin]{morte}? Aonde a vida leva? Todas estas perguntas são sobre movimento.
O estudo do movimento nos traz respostas que são ao mesmo tempo profundas e surpreendentes.
Movimento é misterioso.\index{Mistério do Movimento} Apesar de ser encontrado em todos os lugares  nas estrelas, nas marés,\cite{hewex} em nossas pálpebras  nem os antigos pensadores, nem as miriades de outros em 25 séculos desde lá conseguiram lançar muita luz neste mistério central:\emph{O que é movimento?} Nos vamos observar que a resposta comum, 'movimento é mudança de posição no tempo', é inadequada. Apenas recentemente uma resposta foi encontrada. Esta é a história de como entrar esta resposta.
% 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.}
Movimento é parte da experiência humana. Se nós imaginarmos a experiência humana como uma ilha, então o destino, simbolizado pelas ondas do mar, nos levou até a sua praia. Próximo ao centro da ilha\indexs{Island,
Experience}\indexs{Experience Island} uma específica montanha surge contra o horizonte.
{}Do seu topo nós podemos ver sobre toda a paisagem e ter uma impressão da relação entre as experiências entre todas as experiências humanas, e em particular, entre os vários tipos de movimento.\index{motion, manifestations}
%
Este é um guia para o que eu chamei de Montanha do Movimento \iin{Motion Mountain} (see
\figureref{imiland9};
% Oct 2008
a less artistic but more exact version is given in
\figureref{iphysicsstructure}).
%
A caminhada é uma das mais bonitas aventuras da mente humana. A primeira \index{motion, manifestations}
questão a perguntar é:
%
\subsection{Existe movimento?}
% Impr. Jan 2011
% 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{UKenglish}\footnote{`\emph{A charada} não existe.
Se uma questão {pode} ser feita, também \emph{pode} ser respondida.'}\\
% 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 Para afiar a mente a respeito da questão da existência do movimento, olhe para
\figureref{ifakerotationleft} ou \figureref{irotsnake} e siga as instruções.\cite{motillus} Em todos os casos as figuras parecem rodar.\index{motion illusions, figures showing}\index{illusions of motion}
% Apr 2006, impr. Jan 2011
Você pode experimentar efeitos imiliares se você caminhar sobre uma calçada de paralelepípedos construída em padrões arqueados ou se você olhar para as numerosas ilusões de ótica de\iname{Kitaoka Akiyoshi} no site\url{www.ritsumei.ac.jp/~akitaoka}.\cite{kitaoka}
% 
Como podemos ter certeza que movimento é diferente destas ou similares\challengenor{motill}ilusões?
Muitos estudiosos argumentaram que movimento não existe de forma alguma.\index{motion!does not exist} Os argumentos deles influenciaram profundamente a investigação do movimento por muitos séculos.\cite{asoc}
% in the past and they still continue do so.
%
% The issues
% raised by them will accompany us throughout our adventure.
%
%
Por exemplo, o filósofo grego\iname[Parmênides of Eleia]{Parmenides}
(nascido
\circa 515 {\bce} em Eleia, uma pequena cidade próxima de Nápoles) %, in southern Italy)
argumentou que desde que nada vem do nada, mudança não pode existir. Ele
ressaltou a \emph{permanência} da natureza e consistentemente\index{permanence of nature} manteve que toda mudança e portanto, todo movimento era uma \iin[motion as an
illusion]{ilusão}.\cite{a5} %
\cstepsfnb{ifakerotationleft}{scale=1}%
{ifakerotationright}{scale=0.9}{Illusions of motion: look at the figure on
the left and slightly move the page, or look at the white dot at the centre of
the figure on the right and move your head back and forward.}
\inames{Heraclitus} \livedca(\circa540\circa480 {\bce}) tinha uma opinião diferente. Ele deixoua clara em sua famosa expressão% {%\tau\acute\upsilon\
%\pi\acute\alpha\nu\tau\alpha\ \varrho\epsilon\acute\iota}
%
\csgreekok{p'anta {\columncolor{hks152}[0pt][2cm]} l
@{\extracolsep{\fill}} l @{}}
%
\toprule
%
\tabheadf{Observation} & \tabhead{Velocity} \\[0.5mm]
%
\midrule
%
\iin[growth of deep sea manganese crust]{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} & $\csd{10}{mm/a}=\csd{0.3}{nm/s}$\\
%
\iin[human growth]{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 Beaufort (light air) & below \csd{1.5}{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 speed]{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}\\
%
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{Voyager} satellite & \csd{14}{km/s}\\
%
\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 & \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{light tower} when passing over the Moon &
\csd{2 \cdot10^{9}}{m/s}\\
%
Highest \iin{proper velocity} ever achieved for electrons by man & \csd{7
\cdot10^{13}}{m/s} \\
%
Highest possible velocity for a light spot or 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
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
% \subsubsubsubsubsubsubsubsection{Properties of Galilean velocity}
{\small
\begin{table}[t]
\small
\centering
\caption{Properties 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\seepagefive{mespde} \\
%
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} 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.
% 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 to one of them.
%
% \subsubsubsubsubsubsubsubsection{Table of speed measurement methods}
%
{\small
\begin{table}[t]
\small
\centering
\caption{Speed measurement 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
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. The same 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
\csepsftwfull{icruisespeed}{scale=1}{How wing load and sealevel cruise
speed scales with weight in flying objects, 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.\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{allometric
scaling} relations hold for many other properties of moving systems, as we
will see later on.
Velocity is a profound\index{velocity is not Galilean} subject for an
additional reason: we will discover that all 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}
%
% \clearpage % to ship out all graphics and tables
%
%
%
\subsection{What is time?}
\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} 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
\ii{sequence} of observations. \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]{instants} and we call the sequence of all instants \ii{time}.
An observation that is considered the smallest part of a sequence,
i.e.,{} not itself a sequence, is called an \ii{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\se, or duration. Some measured values are given in
\tableref{durmetab}.\footnote{A year is abbreviated a\indexs{a@a (year)}
(Latin
`annus').} \ii{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.
%
% \subsubsubsubsubsubsubsubsection{Table of times}
{\small
\begin{table}[t]
\small
\centering
\caption{Selected time 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, 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's age} & $\csd{4.6}{Ga}$\\
%
Age of oldest stars\index{star age} & $\csd{13.7}{Ga}$ \\
%
Age of most protons\index{proton age} in your body & $\csd{13.7}{Ga}$ \\
%
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\se, and philosophers. All find:
%
\begin{quotation}
\noindent \csrhd \emph{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{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 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 lorries. 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{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[clocks]{clock}. We can thus answer the question of the section
title:
\begin{quotation}
\noindent \csrhd \ii[time!deduction 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! 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. Every clock
reminds us that in order to understand time, we need to understand motion.
% Cheap literature often suggests the opposite, in contrast to the facts.
A \ii{clock} is thus a moving system whose position can be read. Of course, a
\emph{precise} clock is a system moving as regularly as possible, with as
little outside disturbance as possible. 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\se, and \ii[time measurement, ideal]{ideal} way to measure
time. Can you see it?\challengenor{timenat}
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, 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 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 unique in everyday life.
%
% \subsubsubsubsubsubsubsubsection{Properties of Galilean time}
%
{\small
\begin{table}[t]
\small
\caption{Properties 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}&
\seepagefive{mespde} \\
Can have vanishing duration &\iin{continuity} & \iin{denseness},
\iin{completeness} & \seepagefive{topocont} \\
Allow durations to be added&\iin{additivity} & \iin{metricity}&
\seepagefive{mespde} \\
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}
}
The properties of everyday time, also listed in \tableref{galt}, correspond to
the precise version of our everyday experience of time. This concept is
called \ii{Galilean time}\emph{;} all its properties can be expressed
simultaneously by describing time with the help of \iin{real numbers}. In
fact, 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} Every instant of time can be
described by a real number, often abbreviated $t$, and the duration of a
sequence of events is given by the difference between the values for the final
and the starting event.
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}
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 time just listed is approximate, and none is strictly
correct. 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}).}
%
\subsection{Clocks}
% Jun 2007
A \ii{clock} is 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}. Almost all 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'}.
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 fundamental limits for the precision of
clocks.\seepagefive{kkkk}
%
% \subsubsubsubsubsubsubsubsection{Biological rhythms} % also appears in vol 5
%
{\small
\begin{table}[p]
\small
\caption{Examples of biological 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 quater poweer of the mass (after data
from the \protect\iname{EMBO} and \protect\iinn{Enrique Morgado}).}
%
%
\newpage % is needed here
\subsection{Why do clocks go clockwise?}
\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 the athletic stadium} 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{sundials};
obviously,\index{shadows 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 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.
%
\subsection{Does time flow?}
% \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{UKenglish}%
%
\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{UKenglish}%
%
\footnote{`If time is a river, what is his bed?'}\\
\end{quote}
\np The\label{tfno} expression `the \iin{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 expression `flow of time',
propagated first by some
flawed Greek %\serge
thinkers\cite{aristo} and then again by \iname[Newton, Isaac]{Newton},
continues. \iname{Aristotle}, careful to think logically, pointed out its
misconception, and many did so after him. Nevertheless, expressions such as
`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. Time does \emph{not} flow.
In the same manner, 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'.
% 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.
%
\subsection{What is space?}
\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}
\np
% Why can we distinguish one tree from another? We see that they are in
% different positions.
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, hearing and proprioperception. 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 coherent 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\se, and solid angle
are all deduced with their help. Geometers, surveyors, architects,
astronomers, carpet salesmen\se, and producers of metre sticks base their
trade on distance measurements. Space is a concept formed to summarize all
the distance relations between objects for a precise description of
observations.
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. 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}. Length and surface 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; it
will be answered only 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?
% \subsubsubsubsubsubsubsubsection{Properties of Galilean space}
{\small
\begin{table}[t]
\small
\caption{Properties of 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
%
\tabheadf{Points}
& \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}, \seepagefive{vector} \\ %
Can have vanishing distance &\iin{continuity} &
%\iin{completeness} &
\iin{denseness}, \iin{completeness} &
\leavevmode\seepagefive{topocont} \\
%
Define distances &\iin{measurability} & \iin{metricity} &
\leavevmode\seepagefive{mespde} \\
% not so good:
Allow adding translations&\iin{additivity} & \iin{metricity}&
\leavevmode\seepagefive{mespde} \\
%
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{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{Galilean
space} experimentally: space, the container of objects, is continuous,
threedimensional, isotropic, homogeneous, infinite, Euclidean\se, and unique
or `absolute'. In mathematics, a structure or mathematical concept with all
the properties just mentioned is called a threedimensional \ii{Euclidean
space}. 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
below.\seepageone{eulc}.) % 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 \ii{natural} or \ii{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, 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.
%
% \subsubsubsubsubsubsubsubsection{Table of lengths}
%
% fill in more, if possible
{\small
\begin{table}[t]
\small
\centering
\caption{Some measured 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, with 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 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{Leica Geosystems}
% SENT EMAIL FEB 2008  Alessandra.Doell@leicageosystems.com
and \protect\iname{Keyence}).}
% SENT EMAIL FEB 2008  keyence@keyence.com
%
% \subsubsubsubsubsubsubsubsection{Table of length measurement methods}
%
%
{\small
\begin{table}[t]
\small
\centering
\caption{Length measurement 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}
}
%
\subsection{Are space and time absolute or relative?}
% \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 \iin{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.
%
\subsection{Size  why area exists, but volume does not}
\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 coastlines} In fact, mathematicians have described many
such curves; 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 that it is only possible using %(countably)
infinitely many pieces?\challengedif{vitali}
To sum up, length is well defined for lines that are straight or nicely
curved, but not for intricate lines, or for lines made of infinitely many
pieces. 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 first five volumes of this
adventure, and we should never forget them. We will come back to these
assumptions in the last volume 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 have to be the sum of the areas and volumes of each element of
the set; and they must be \emph{rigid}, i.e.,{} if one cuts an area or a
volume
into pieces and then rearranges the pieces, the value remains the same. Are
such definitions possible? In other words, do such concepts of volume and
area exist?
For areas in a plane, one proceeds in the following standard way: one defines
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} one can then define an area value for all polygons.
Subsequently, one 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, one finds area values
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.}%
\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[2] 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, one \emph{can} define a rigid and
additive concept of volume for polyhedra, since for all polyhedra and, in
general, for all `nicely curved' shapes, one can again use integration for the
definition of their volume. %\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), 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}}
%
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 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.
%
\subsection{What is straight?}
\label{strrl}
%
When you see a solid object with a straight edge,\index{straight lines in
nature} it is a 99\,\%safe bet that it is manmade. Of course, there are
exceptions,\index{crystals} as shown in \figureref{icrystals}.%
%
\footnote{Another famous exception, 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 nothing
is straight or flat, in the city most objects are. 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.
%
\subsection{A hollow Earth?}
\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
(usually) call this the \ii{hollow Earth theory}.\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 this view 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 coherent 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.
%
\subsection{Curiosities and fun challenges about everyday space and time}
% Space and time lead to many thoughtprovoking questions.
\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{http://htwins.net/scale2/}.
% Mar 2008
\item What is faster: an arrow or a motorbike?\challengenor{arrowbike}
% 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
\item You have two \iin{hourglasses}: one needs 4 minutes and one needs 3
minutes. How can you use them to determine when 5 minutes are over?\challengn
% Oct 2009
\item You have two water containers: one of 3 litres, another of 5 litres.
How can you use them to bring 4 litres of water?\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{pencils} are needed to draw a line as long as the Equator
of the Earth?\challengenor{penc}
% Feb 2007
\item Everybody knows the puzzle about the bear: 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}\index{bear, colour of}
The house could be on {several} \emph{additional} spots 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, 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 horse?\challengenor{snailhorse}
% 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
it!\challengenor{breadfrac}
% 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}
\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 minute?\challengenor{minu}
\item What is the highest speed\index{throwing speed, record} achieved by throwing
(with and without a racket)? What was the projectile
used?\challengenor{speedthrow}
% 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 never asked, not even
in
other languages. Why?\challengenor{wwfi}
\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 parking issue:\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
%
% \subsubsubsubsubsubsubsubsection{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}
\caption{The exponential notation: 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.}
% 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} 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} by an unconscious apparent
distance change induced by the human brain. 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
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{twodimensional universe}\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[ropes!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 be?\challengenor{time2d}
\item Other researchers are investigating whether space could have more than
three dimensions. Can this be?\challengenor{moret3dime}
% 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,
aiplanes, cheetahs, falcons, crabs, and all other motorized systems are much
slower.\cite{oly}
% 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 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 much did the two ends of the rope come
closer together?\challengeres{knottightshort}
\end{curiosity}
%
\subsection{Summary about everyday space and time}
% Nov 2008, Impr April 2010
% Index ok
Motion defines speed, time and length. Observations of everyday life and
precision experiments are conveniently and precisely described by modelling
velocity as a Euclidean vector, space as a threedimensional Euclidean space,
and time as a onedimensional real line. These three definitions form the
everyday, or \emph{Galilean}, description of our environment.\indexs{Galilean
spacetime}\indexs{spacetime, Galilean}
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 experiments can check distances
larger than \csd{10^{25}}{m} or smaller than \csd{10^{25}}{m}; the continuuum
model is likely to be incorrect there.\index{continuity, limits of} We will
find out in the last part of our mountain ascent that this is indeed the case.
\vignette{classical}
%
%
%\newpage
% \chapter{}
\chapter{How to describe motion  kinematics}
% \chapter{}
\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{UKenglish}\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 Galilean 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}. 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' 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{leaning tower in Pisa} confirmed the statement. (It is
a \emph{false} urban legend that Galileo never performed the
experiment.)\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}
\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} also 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. It plays
an important role in selforganization. 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 diagram}. It plays an essential role in
thermodynamics.
%
\subsection{Throwing, jumping and shooting}
%
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{running speed
record, human} 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{long jump} 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[horses, 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.
\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} jumps in the air, how many times can he or she
rotate
around his or her vertical axis before arriving back on earth?\challenge % !!!5
\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{shit}  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}
% Improved in April 2012
\item What is the maximum numbers of balls that could be juggled at the same
time?\challengenor{jug} This is unclear. In 2012, the human record is eleven
balls. % see https://www.youtube.com/watch?v=UWX4QSmnw6g
For robots, \index{robot
juggling}\index{juggling!record}\index{juggling!robot} the present{\present}
record is three balls, as performed by the \iin{Sarcoman robot}. The internet
is full of material and videos on the topic.
% on
% \url{www.physio.northwestern.edu/Secondlevel/Miller/FirstLevel/histresearch.html}.
It is a challenge for people and robots to reach the maximum possible number of balls.
% (NO) robot? fig? Wrote to Prof Miller in Sep 2007
\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
\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{jumping
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}; 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}
%
\subsection{Enjoying vectors}
%
% 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]{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, they 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}, 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 in science the concept of \emph{length}
(specifying the `magnitude') 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 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
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
italic letters, whereas vectors are written in bold letters. A vector space
with a scalar product is called an \ii[vector space, Euclidean]{Euclidean}
vector space.
The scalar product is also useful for specifying directions. Indeed, the
scalar product between two vectors encodes the angle between them. Can you
deduce this important relation?\challengenor{scalprod}
%
\subsection{What is rest? What is velocity?}
\cssmallepsf{islope}{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{slope}{\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 at a point} is the limit of the
secant {slopes} in the neighbourhood of the point, as shown in
\figureref{islope}. 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.
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\se, 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{sets, 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.
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 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.
%
% \subsubsubsubsubsubsubsubsection{Table of accelerations}
%
%
{\small
\begin{table}[t]
\small
\caption{Some measured acceleration 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 asfalt &
$\csd{5.5}{m/s^{2}}$\\
%
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}
}
%
\subsection{Acceleration}
Continuing 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}. Like velocity, acceleration has both a magnitude and a
direction, properties indicated by the use of \textbf{bold} letters for their
abbreviations. In short, acceleration, like velocity, is a vector quantity.
% May 2005
Acceleration is felt. Our body is deformed and the sensors in our
semicircular canals in the ear feel it. 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.
% 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.) Acceleration
sensors, such are those listed in \tableref{accelsensors} or those shown in
\figureref{iaccelerationmeasurement}, all work in this way. Can you think
of a situation where one is accelerated but does \emph{not} feel
it?\challengenor{notfeelacc}
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 system.
%
% \subsubsubsubsubsubsubsubsection{Table of acceleration sensors}
%
%
{\small
\begin{table}[t]
\small
\caption{Some acceleration 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 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
%
\subsection{Objects and point particles}
\begin{quote}
\selectlanguage{german}Wenn ich den Gegenstand kenne, so kenne ich auch sämtliche
Möglichkeiten seines Vorkommens in Sachverhalten.\selectlanguage{UKenglish}%
%
\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} or a \ii{point mass} or a \ii{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).}
As an exception to the general rule, the size of a few large and nearby stars,
of red giant type, can be measured
with special instruments.%
%
\footnote{The website \url{www.astro.uiuc.edu/~kaler/sow/sowlist.html}
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[2] K.R. Lang,
C.A. Whitney/ \bt Vagabonds de l'espace  Exploration et découverte dans le
système solaire/ Springer Verlag, \yrend 1993/ The most beautiful pictures of
the stars can be found in \asi D. 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\se, and \iin{Sirius} in
Canis Major are examples of stars whose size has been measured; they are all
only a few light years from Earth.\cite{a17} Of course, like the Sun, also all
other stars have a finite size, but one cannot prove this by measuring
dimensions in photographs. (True?)\challengenor{starsize}
% Oct 2011
\cssmallepsfnb{itwinkle}{scale=1}{Regulus and Mars, photographed with 10
second exposure time on 4 June 2010 with a wobbling camera, show the
difference between a pointlike star 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 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} 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{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, 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{points}\index{space points} 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\se, or photons are
pointlike for all practical purposes. Once one knows how to describe the
motion of point particles, one can also describe the motion of extended
bodies,
rigid or deformable, by assuming 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{continuum
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.
%
A more precise description than the continuum approximation is given
below.\seepagefour{facsal} Nevertheless, all observations so far have
confirmed that the motion of large bodies can be described to high precision
as the result of the motion of their parts. This approach will guide us
through the first five volumes of our mountain ascent. Only in the final
volume will we discover that, at a fundamental scale, this decomposition is
impossible.
%
\subsection{Legs and wheels}
% Index ok
The parts 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 wheels\index{wheels in living beings} or \iin[propellers in
nature]{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[connected
bodies]{connected}.\seepagefive{manidef}
This is often assumed to be obvious, and it
is often mentioned that\cite{fakew} the \iin{blood supply}, the nerves\se, and
the lymphatic connections to a rotating part would get tangled up. However,
this argument is not so simple, as \figureref{ifakewheel3} shows. It shows
that it is indeed possible to rotate a body continuously against a second one,
without tangling up the connections. Can you find an example for this kind of
motion 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}{How an object can rotate continuously
without tangling up the connection to a second object.}
Despite the possibility of animals having rotating parts, the method of
\figureref{ifakewheel3} still cannot be used to make a practical wheel or
propeller. Can you see why?\challengenor{wheel} Evolution had no choice: it
had to avoid animals with 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
spiders,%
%
% Oct 2009
\footnote{Rolling is known for desert spiders of the \iie{Cebrennus} and the
\iie{Carparachne} genus; films can be found on
\url{www.youtube.com/watch?v=5XwIXFFVOSA} and
\url{www.youtube.com/watch?v=ozn31QBOHtk}. Cebrennus seems even to be able to
accelerate its rolling motion with its legs.} %
%
certain other animals, children\se, 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: the red
millipede \protect\iie{Aphistogoniulus erythrocephalus} (15$\,$cm body
length), a gekko 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},
\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
).}
% (no, not locomotion in general) one day, maybe add photo of snail at water
% air interface, and of snake on sand
\emph{Single} bodies, and thus all living beings, can only move through
\ii[shape deformation and motion]{deformation} of their shape:\cite{jgrayloc}
therefore they are limited to walking, running, rolling, crawling or flapping
wings or fins. Extreme examples of leg use\cite{leguse} in nature are shown
in \figureref{ilegwheel}. The most extreme example
% (OK) find fig or movie and ref by Ingo Rechenberg, Berlin gattung/genus
% Cebrennus
are rolling spiders living in the sand in Morocco;\cite{rechenberg} they use
their legs to accelerate and steer the rolling direction. Walking on water is
shown in
\figureref{iwaterstrider} on \cspageref{iwaterstrider}; % this vol I
examples of wings are given later on,\seepagefive{iwings}
% Apr 2010
as are the various types of deformations that allow swimming in
water.\seepagefive{extmot}
%
In contrast,
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\se, or 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 few?\challengenor{bodyship}}
In summary, 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} 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{wheels
in nature} Organisms such as \iie{Escherichia coli}, the wellknown
\iin[bacteria]{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} A
% !!!2 add flagellar motor image
bacterium is able to turn its 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.) Therefore wheels actually do
exist in living beings, albeit only tiny ones. But let us 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.
\csepsfnb{icometmcnaught}{scale=1}{Are comets, such as the beautiful comet
McNaught seen in 2007, images or bodies?
% Improved Apr 2013
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
%
\subsection{Curiosities and fun challenges about kinematics}
\begin{curiosity}
% Aug 2007
\item[] What is the biggest wheel ever made?\challengenor{biggestwheel}
% Aug 2007
\item A soccer ball is shot, by a goalkeeper, with around \csd{30}{m/s}.
Calculate the distance it should fly and compare it with the distances found
in a soccer match. Where does the difference come from?\challengenor{soccerz}
% Sep 2005
\item A train starts 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
% June 2005
\item Balance a 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 plane  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 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 ({\textcopyright}~\protect\iinn{Detlev Lohse}).}
% EMAILED FEB 2008  D.Lohse@tnw.utwente.nl
% add a diagram of the experimental setup !!!2 for sonoluminescence
% physics.open.ac.uk/~swebb/ach.htm ???
\item The highest \emph{microscopic} accelerations are observed in particle
collisions,
where one gets values up to \csd{10^{35}}{m/s^{2}}.
% 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\index{accelerations, highest} 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 bu 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. 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}
%
\subsection{Summary of kinematics}
The description of everyday motion of mass points with three coordinates as
$(x(t), y(t), z(t))$ is simple, precise and complete. It assumes that objects
can be followed along their paths. Therefore, the description often does not
work for an important case: the motion of images.
\vignette{classical}
%
\newpage
% \chapter{}
\chapter{From objects and images to conservation}
% \chapter{}
\markboth{\thesmallchapter\ from objects and images to conservation}%
{\thesmallchapter\ from objects and images to conservation}
\label{oim}
%
\csini{W}{alking} through a forest
%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{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 Fall, in nature only certain crystals, the octopus, the chameleon\se,
and a few other animals achieve this. Of manmade objects, television,
computer displays, heated objects\se, 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\se, 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, if we are
able to touch what we see  or more precisely, if we are able to move it 
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.} %
%
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. This 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}. 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\se, 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?}
%
\subsection{Motion and contact}
\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\se, and loving, something pushes
something else. Thus our first challenge is to describe this transfer of
motion in more precise terms.
Contact is not the only way to put something into motion; a counterexample is
an \iin[apples]{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 forces are not easily avoided. Taking this choice we will
make a similar experience to that of \iin[bicycle riding]{cyclists}. (See
\figureref{isteer}.) If you ride a bicycle at a {sustained} speed and try to
turn left by pushing the righthand steering bar, you 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.
%
\subsection{What is mass?}
\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[movable property]{movable} and \iin[immovable property]{immovable}
property 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 $v_{2}+\Delta v_{2}$}%
]
% !!!1 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. The high precision version of the
experiment is shown in \figureref{ibilliard}. Repeating the experiment with
various pairs of objects, we find  as in everyday life  that 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:mdef1}
\end{equation}
where $\Delta v$ is the velocity change produced by the collision. The more
difficult it is to move an object, the higher the number. The number $m_{i}$
is called the \ii{mass} of the object $i$.
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 the difficulty of getting
something moving. High 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 \lived(17431794), French chemist and a
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. There is a good, but most probably false story
about him: When he was (unjustly) sentenced to the guillotine during the
French revolution, he decided to use the situations for a scientific
experiment. 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, if a decapitated has no pain
or shock, he 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{Galilean mass is a measure
for the \iin{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}
\cssmallepsfnb{ihuygens}{scale=0.13}{Christiaan Huygens
\livedfig(16291695).}
%
\subsection{Momentum and mass}
The definition of mass can also be given in another way. We 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
\ii{momentum} of the body. The sum, or \ii{total momentum} of the system, is
the same before and after the collision; momentum is a \emph{conserved}
quantity.
\begin{quotation}
\noindent \csrhd \emph{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
Momentum and mass are conserved in everyday motion of objects. 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 it is \emph{not} valid in
special relativity.
%
% \subsubsubsubsubsubsubsubsection{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 galaxy collision& up to $\csd{10^{46}}{Ns}$ \\
%
%
\bottomrule
\end{tabular}%
}% end of sbox
%
\centering
\begin{minipage}{\wd\cshelpbox}%
\caption{Some measured 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 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[aeroplanes]{aeroplane} speed as a
multiple of the \iin{speed of sound} in air (about \csd{0.3}{km/s}) is named
after him. He 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. 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. This
definition has been studied 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
conservation implies momentum conservation} because mass is defined in such a
way that the principle holds.\index{momentum conservation follows from mass
conservation}
\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]{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 origin of the accelerations is 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\se, 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 has no
real meaning; this becomes especially clear when one takes an observation
point that is 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}
%
% \subsubsubsubsubsubsubsubsection{Table of masses}
{\small
\begin{table}[t]
\small
\centering
\caption{Some measured 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 devices is given in \tableref{masssensors}
and \figureref{imassmeasurement}. Some measurement results are listed in
\tableref{massmetab}. We also discover the main properties of mass. It 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}; 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\se, and all other processes.
%
% \subsubsubsubsubsubsubsubsection{Properties of mass}
%
\begin{table}[t]
\small
\caption{Properties of Galilean mass.}
\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}&
\seepagefive{mespde} \\
Can change gradually &\iin{continuity} & \iin{completeness} &
\seepagefive{topocont} \\
Can be added & \iin{quantity of matter} &\iin{additivity} & \seepageone{eulc}
\\
Beat any limit & \iin{infinity}& \iin{unboundedness, 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}
Later we will find that in the case of mass all these properties, summarized
in \tableref{massprop}, are only approximate. Precise experiments show that
none of them are correct.%
%
\footnote{In particular, 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.
The definition of mass through momentum conservation implies that when an
object falls, the Earth is accelerated upwards by a tiny amount. If one could
measure this tiny amount, one 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}
\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}
%
% \subsubsubsubsubsubsubsubsection{Table of mass sensors}
{\small
\begin{table}[t]
\small
\caption{Some mass 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 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 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
%
\subsection{Is motion eternal?  Conservation of momentum}
\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 motion during
interactions.\index{disappearance of motion}\index{conservation of
momentum}\index{creation of motion}
% Feb 2010, added solutions of drawn puzzles
\csepsf{iselfprop}{scale=1}{What happens in these
four\protect\challengenor{foursit} situations?}
Momentum is an \ii{extensive quantity}.\indexs{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 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}
(`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 fluid. In an interaction, the invisible fluid is transferred from
one object to another. In such transfers, the sum 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 the answer lies in the microscopic aspects of these systems.
A muscle only \ii{transforms} one type of motion, namely that of the electrons
in certain chemical compounds\footnote{Usually adenosine triphosphate
(\csaciin{ATP}), the fuel of most processes in animals.\cite{ATPbook}} 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 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. 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 these 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{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} 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 \iin{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.
%
\subsection{More conservation  energy}
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{collisions} 
collisions without \iin{friction}  the following quantity, called the
\ii{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
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.
\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 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 concept of work is just the precise version of what is
meant by work in everyday life. As usual, (physical) \ii[work]{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.
% 
Another, equivalent definition of energy will become clear later:
\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{god!and energy} implied that energy was the same as momentum.
Leibniz, instead, knew that energy increases with the square of the mass and
proved Newton wrong.\index{Newton!his energy mistake} In 1722, Willem Jacob 's
Gravesande\indname{Gravesande@'s Gravesande, Willem Jacob} even showed this
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.
%
% \subsubsubsubsubsubsubsubsection{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 & $\csd{10^{44}}{J}$ \\
%
Gamma ray 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}%
\caption{Some measured 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.
Another distinction 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
damage?\challengenor{cardamage} What changes if the parked car has its brakes
on?
To get a better feeling for energy, here is an additional approach. 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.) 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. (See \tableref{powmetab} for some power values
found in nature, and \tableref{powersensors} for some measurement devices.)
In particular, if we look at the \iin{energy consumption in First World
countries}, the average inhabitant there has machines working for them
equivalent to several hundred `servants'. Can you point out some of these
machines?\challengenor{mach}
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 in
fact a process transforming kinetic energy, i.e.,{} the energy connected with
the
motion of a body, into heat. On a microscopic scale, energy is conserved.%
%
\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}, the concept of time would not be definable. We will
show this 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.
%
% \subsubsubsubsubsubsubsubsection{Table of powers}
%
{\small
\begin{table}[t]
\small
\caption{Some measured 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} 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}
}
%
% \subsubsubsubsubsubsubsubsection{Table of power sensors}
%
%
{\small
\begin{table}[t]
\small
\caption{Some power 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
\csepsfnb{ipowermeasurement}{scale=1}{%
Some power measurement devices: a bicycle power meter, electical power meter,
laser power meter ({\textcopyright}~\protect\iname{XXX})}
%
\subsection{The cross product, or vector product}
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} ${\bm a}\times{\bm b}$.
The result of the vector product is another 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 that is orthogonal to both vectors to be
multiplied, whose orientation is given by the \ii{righthand rule}, and whose
length is given by $a b \sin \sphericalangle ({\bm a}, {\bm b})$, i.e.,{} by
the surface area of the parallelogram spanned by the two vectors.
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 properties\challengn
%
\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}) \;,
%
{\bm a} \times ({\bm b}\times {\bm c}) = {\bm b} ({\bm a} {\bm c})  {\bm c}
({\bm a} {\bm b}) \;,
%
\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 c} (({\bm a}
\times {\bm b} ) {\bm d} )  {\bm d} (({\bm a} \times {\bm b} ) {\bm c} )
\;,
%
\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}) = 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
The vector product is useful to describe systems that \emph{rotate}  and
(thus) also systems with magnetic forces. The main reason for the usefulness
is that the motion of an orbiting body is always perpendicular both to the
axis and to the shortest line that connects the body with the axis.
%  because we live in \emph{three}
% spatial dimensions
% 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 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}
This is 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, it is 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 by three arbitrary vectors ${\bm
a}$, ${\bm b}$ and ${\bm c}$ has the\challengn volume $V= {\bm c}\,({\bm
a}\times {\bm b})$. Show that the \ii{pyramid} or \ii{tetrahedron} formed
by the same three vectors has one sixth of that volume.
%
\subsection{Rotation and angular momentum}
\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; similarly, a body also has a
\iin{reluctance to turn}. This quantity is called its \ii{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{angular velocity}, usually abbreviated $\omega$  pronounced `omega'. A
few values found in nature are given in \tableref{rotmetab}.
%
% \subsubsubsubsubsubsubsubsection{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}\\
%
Ultracentrifuge & $>2 \pi\;\cdot$ \csd{2 \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}%
\caption{Some measured 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.
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 so for a brick?\challengenor{brick}
%
% \subsubsubsubsubsubsubsubsection{Table of angular momenta}
% New in Oct 2011
{\small
\begin{table}[p]
\small
\centering
\caption{Some measured angular momentum values.}
\label{angmomvaltab}
\dirrtabularstar
\begin{tabular*}{\textwidth}{@{}>{\columncolor{hks152}[0pt][2cm]} p{80mm}
@{\extracolsep{\fill}} p{40mm} @{}}
%
\toprule
%
\tabheadf{Observation} & \tabhead{Angular momentum} \\[0.5mm]
%
\midrule
%
{Smallest observed} in nature ($\hbar/2$)  applies to the zcomponent of
most 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 disk) playing& \csd{0.029}{Js} \\
%
\iin[walking man!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]{Atmosphere} & 1 to \csd{1\cstimes
10^{26}}{kg\,m^2/s}\\ % from an el Nino paper
%
\iin[Earth!angular momentum]{Earth around its axis} & \csd{7.1 \cstimes
10^{33}}{kg\,m^2/s}\\
%
\iin[Moon!angular momentum]{Moon around Earth} & \csd{2.9 \cstimes
10^{34}}{kg\,m^2/s}\\
%
\iin[Earth!angular momentum]{Earth around Sun} & \csd{2.7 \cstimes
10^{40}}{kg\,m^2/s}\\ % two sources
%
\iin[Sun!angular momentum]{Sun} around its axis
& \csd{1.1 \cstimes 10^{42}}{kg\,m^2/s} \\
%
\iin[Jupiter!angular momentum]{Jupiter around Sun} & \csd{1.9 \cstimes
10^{43}}{kg\,m^2/s} \\
%
\iin[Solar system!angular momentum]{Solar system around Sun} & \csd{3.2
\cstimes 10^{43}}{kg\,m^2/s} \\
%
\iin[Milky Way!angular momentum]{Milky Way} & \csd{10^{68}}{kg\, m^2/s} \\
%
\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 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} & $\bm {L}=\Theta \bm \omega$
\\
& energy & $mv^2/2$ & energy & $\Theta \omega^2/2$ \\
%
%
Motion & velocity & $\bm v$ & {angular velocity}\index{angular velocity} &
$\bm \omega$ \\
%
& acceleration & $\bm a$ &
{angular acceleration}\index{angular acceleration} & $\bm \alpha$ \\
%
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 no conservation of the moment of
inertia. The value of the moment of inertia depends 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, one distinguishes intrinsic and extrinsic angular
momenta.\indexs{angular momentum, intrinsic}\indexs{angular momentum,
extrinsic}\indexs{momentum, angular} (By the way, the\label{masscentre}
\ii{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{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. 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 this energy, you would become
famous. For undistorted rotated objects, rotational energy is conserved.
Every object that has an orientation also has an intrinsic angular momentum.
(What about a sphere?)\challengenor{sphereori} Therefore, point particles do
not have intrinsic angular momenta  at least in first approximation. (This
conclusion will change in quantum theory.) The \ii[angular momentum,
extrinsic]{extrinsic} angular momentum $\bm L$ of a point particle is given by
\begin{equation}
{\bm L} = {\bm r} \times {\bm p} = \frac{2 {\bm a}(T) m }{ T }
\qhbox{so that} L= r\, p = \frac{2 A(T) m }{ T }
\end{equation}
where $\bm p$ is the momentum of the particle, ${\bm a}(T)$ is the surface
swept by the position vector $\bm r$ of the
particle during time $T$. %
%
%
%
The angular momentum thus points along the rotation axis, following the
righthand rule, as shown in \figureref{iangmomentum}.
% Oct 2011
A few observed values are given in \tableref{angmomvaltab}.
% Feb 2012
A body can rotate simultaneously about 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 vector
sum of these two rotations. Rotations thus are vectors.\index{rotation!as
vector}
% corrected mistake in Apr 2006
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). However, for closed (undisturbed) systems both quantities are
always conserved. In particular, on a microscopic scale, most objects are
undisturbed, so that conservation of rotational energy and angular momentum is
especially obvious there.
% 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}
assuming that the plate on which the ape rests can turn around the axis
without friction?\challengenor{aperot}
% May 2007
We note that the effects of rotation are the same as for acceleration.
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 not prevent bodies from
changing their orientation. Cats learn this in their youth. After they have
learned the trick, if they are dropped legs up, they can turn themselves in
such a way that they always land feet first.\cite{catfall} Snakes also know
how to rotate themselves, as \figureref{isnake} shows. During the Olympic
Games one can watch \iin{board divers} and \iin{gymnasts} perform similar
tricks. Rotation is thus different from translation in this aspect.
(Why?)\challengedif{whyrotdif}
%
\subsection{Rolling wheels}
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}
\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\se, 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}
%
\subsection{How do we walk?}
\begin{quote}
Golf is a good walk spoiled.\\
Mark Twain\indname{Twain, Mark}
\end{quote}
\np Why do we move our arms when walking or running? To save energy or to be
graceful?\index{walking} In fact, whenever a body movement is performed with
as little energy as possible, it is natural and graceful. This correspondence
can indeed be taken as the actual definition of grace.\indexs{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, 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
no arms, such as \iin{penguins} or \iin{pigeons}, have more difficulty
walking; they have to move their whole torso with every step.
Which muscles do most of the work when walking, the motion that experts call
\ii{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
oder 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 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]{run}. In 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.
%
\subsection{Curiosities and fun challenges about conservation and rotation}
\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.
\begin{curiosity}
% 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}
% Sep 2011
\item Death\label{deatmetabolism} is a physical process; let us explore
it.\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. 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
\cssmallepsf{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 (\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
flled 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{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.
\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}
\item Housewives know how to extract a \iin{cork} of a \iin[wine bottle]{wine}
\iin{bottle} using a cloth. 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?\challenge % !!!5
\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 vapour is $g/7$?\challengedif{fallingdrop}
\item You have two hollow spheres: they have the same weight, the same
size\se, 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 a tilted plane?\challengenor{hollsphe}
\item What is the shape of a rope when rope jumping?\challenge % !!!5
\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, 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
\csepsf[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 measure the mass of a ship?\index{ship!mass of}
% 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
\csepsf{iceltic}{scale=1}{The famous Celtic wobble stone  above, right 
and a version made
with a spoon  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}
\end{curiosity}
%
% Aug 2007
\subsection{Summary on conservation}
% Aug 2007
\begin{quote}
The gods\index{gods} 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
closed systems in everyday life:
\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
laws applies to motion of images.
% Aug 2007
\np These 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 above quote, 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
% \chapter{}
\chapter{From the rotation of the earth to the relativity of motion}
% \chapter{}
\markboth{\thesmallchapter\ from the rotation of the earth}%
{to the relativity of motion}
\begin{quote}
\selectlanguage{italian}Eppur si muove!\selectlanguage{UKenglish}\\
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}
\cssmallepsf{iparallaxis}{scale=1}{The parallax  not drawn to scale.}
\csini{I}{s} the Earth rotating? 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} maintained 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}
\cssmallepsf{iflattening}{scale=1}{Earth's deviation from spherical shape due
to its rotation (exaggerated).}
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}{{}^\prime}, changing sign every 12 hours, to the aberration due to
the motion of the Earth around the Sun, about \csd{20.5}{{}^\prime}. In modern
times, astronomers have found a number of additional proofs, but none is
accessible to the man on the street.
Furthermore, 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 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. He was one of the key figures in the quest for
the \iin{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 this \bt Histoire du Docteur
Akakia et du natif de SaintMalo/ \yrend 1753/ Maupertuis
(\url{www.voltaireintegral.com/Html/23/08DIAL.htm}) 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.
% !!!2 psfrag
\cssmallepsf{ifallhit}{scale=1}{The deviations of free fall towards the east
and towards the Equator due to the rotation of the Earth.}
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 Earth,
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 thus cannot
be drawn on a 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 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}
${\bm a}$ 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}
% Nov 2006
On Earth, the {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, the general wind patterns on Earth and the
deflection of ocean currents and \iin{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. (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 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, 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 be downloaded at
\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
%
% !!!1 add figure of Zach espiritu (email 23. Mai 2013)
Foucault was also the inventor and namer of the
\ii{gyroscope}.\label{gyroscope} He built the device, shown in
\figureref{igyro4}, 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.
(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. In the most
modern versions, one uses 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 simple 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.
\cstftlepsf{ihagen}{scale=1}{Showing the rotation of the Earth through the
rotation of an axis.}[12mm]
{icomptonwheelnew}{scale=1}{Demonstrating the rotation of the Earth
with water.} % !!!2 psfrag both, increase the frames
% 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
of the experiment, which uses lasers instead of lamps, is shown in
\figureref{ilasergyrointerf}.\cite{wettzellref}
%
% !!!2 find literature of fringes and interference
% Fringes are \ldots
%
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\se, 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 \lived(18511940) was a British physicist 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 one 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, observations\index{Earth rotation!speed of} show that the Earth
surface rotates at \csd{463}{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 also implies an acceleration, at the Equator, of
\csd{0.034}{m/s^2}. We are in fact \emph{whirling} through the universe.
%
\subsection{How does the Earth rotate?}
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's 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.
But why does the Earth rotate at all? The rotation derives from 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 one, turn around
themselves in the same direction, and that they also all turn around 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. 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 flat.
% Sep 2007
\csepsfnb{ipolarmotion}{scale=1}{The motion of the North Pole 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 of the Earth, 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{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%
%
\footnote{The circular motion, a wobble, was predicted by the
great Swiss mathematician \iinn{Leonhard Euler} \lived(17071783). In an
incredible 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
Only with help of the exact position of the Earth's axis is the high precision
of the \csac{GPS} system possible; 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.
\cssmallepsfnb{iplate}{scale=0.4}{The continental plates are the objects of
tectonic motion.} % !!!2 add source and copyright, did not find it in 2012
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[continental motion]{continents}
\emph{move}, and that they are all fragments of a single continent that broke
up 200 million years ago.%
% Jun 2005
\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 flowed
back. Today, the river still flows eastwards and is called the
\iin{Amazon} River.}
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{10}{mm} every year. 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{magentization 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 motion.\cite{platetect}
%
% Sep 2007
% \subsubsubsubsubsubsubsubsection{Earth data}
%
{\small
\begin{table}[tp]
\small
\caption{Modern measurement 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{21.5mm} @{\extracolsep{\fill}} % 21 is minimum
>{\PBS\raggedright} p{45.5mm} @{}} % 45 was too small by .9 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
%
\subsection{Does the Earth move?}
% May 2005
\cssmallepsfnb{ibessel}{scale=0.4}{Friedrich~Wilhelm Bessel
\livedfig(17841846).}
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 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, and 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}
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 and that the Earth indeed revolves around the Sun. The
motion itself was not a surprise. It confirmed the result of the mentioned
\iin{aberration} of light, discovered in
1728 by\label{bradleybio} \iinns{James Bradley}%
%
\footnote{James Bradley \lived(16931762), English astronomer. He was one of
the first astronomers to understand the value of precise measurement, and
thoroughly modernized Greenwich. He discovered the aberration of light, a
discovery that showed that the Earth moves and also allowed him to measure the
speed of light; he also discovered the nutation of the Earth.}
%
and to be discussed below;\seepagetwo{aberrrr} the Earth moves around the Sun.
\csepsf{ieaorbit}{scale=1}{Changes in the Earth's motion around the Sun.}
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'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} This motion is due 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} is due in large part to the influence of the
other planets; a small remaining part will be important in the chapter on
general relativity. It was the first piece of data confirming the 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, 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, 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.
We turn around the galaxy centre because the formation of galaxies, like that
of solar systems, always happens in a \iin{whirl}. By the way, can you
confirm from your own observation that 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's 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}, to be studied later, produces no motion.}
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} 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
\subsection{Is velocity absolute?  The theory of everyday relativity}
% \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 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{ships 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, there is no
difference between constant, undisturbed motion, however rapid it may be, and
rest. This is now called \ii[relativity, Galileo's principle of]{Galileo's
principle of relativity}. In everyday 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.%
%
% Plagiarism or not? I cannot remember
\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.'} %
%
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.
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 not to?\challengenor{wrongwitt}
%
\subsection{Is rotation relative?}
% Jul 2005
When we turn 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
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}
As usual for philosophical issues, Newton's answer was guided by the mysticism
triggered by his father's early death.\index{gods} Newton saw absolute space
as a religious concept and was not even able to conceive an alternative.
Newton thus sees rotation as an absolute concept. Most modern scientist have
fewer 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: rotation is relative.\index{rotation!absolute or relative} Also the
theory of general relativity confirms this conclusion.
%
\subsection{Curiosities and fun challenges about relativity}
\begin{curiosity}
% Dec 2005
\item[] When travelling in the train, you can test Galileo's statement about
everyday relativity of motion. Close your eyes and ask somebody to turn you
around many 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 a bullet vertically,
where does the bullet fall?\challengenor{verticalcannon}
\item Why are most \iin{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{465}{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
First of all, the speed determines the size of typical weather phenomena.
This size, the socalled \ii{Rosby length}, is given by the speed of sound
divided by the local rotation speed. At moderate latitudes, the Rosby length
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 Rosby length 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
\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 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.
\item Would travelling through interplanetary space be healthy?\index{space
travel} 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} lean 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
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 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 Rotation holds a surprise for anybody who studies it carefully. Angular
momentum is a quantity with a magnitude and a direction. However, it is 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 easily
check this for yourself.\challengn For this reason, \iin{angular momentum} is
called a \ii{pseudovector}. The fact has no important consequences in
classical physics; but we have to keep it in mind for later occasions.
\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. 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{earthquakes} 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 The\label{angmomten} moment of inertia of a body depends on the shape of
the body; usually, angular momentum and the angular velocity do not point in
the same direction. Can you confirm this with an example?\challengenor{angt}
\item Can it happen that a satellite dish for geostationary TV \iin{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 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}
\item What is the moment of inertia of a homogeneous sphere?\challengenor{sphere}
\item The moment of inertia is determined by the values of its three principal
axes. These 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}
% 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 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}
\cssmallepsfnb{icircumpolar}{scale=0.8}{A long exposure of the stars at night
when facing north  above the Gemini telescope in Hawaii
({\textcopyright}~\protect\iname{Gemini Observatory/AURA}).}
% this is the figure www.ausgo.unsw.edu.au/gallery/telescopes01.html
% Gemini north telescope Mauna Kea
\item An impressive confirmation that the Earth is round can be seen at sunset, if
one turns, against usual habits, one's back on the Sun. On the eastern sky
one
can see the impressive rise of the \iin{Earth's shadow}. (In fact, 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} One can admire a vast shadow rising
over the whole horizon, clearly having the shape of a segment of a huge
circle.
\item How would \figureref{icircumpolar} look if taken at the
Equator?\challengenor{circpol}
\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 has interesting consequences
for volley balls and for \iin{girlwatching}. 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} Men have a real interest that
this mechanism is strictly followed; if not, looking at girls on the beach
could cause the muscles moving the eyes to get knotted up.\index{women,
dangers of looking after}
\end{curiosity}
%
\subsection{Legs or wheels?  Again}
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{wheels!vs legs} 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 these downwards forces are
transformed into grip; modern racing tyres allow forward, backward\se, and
sideways accelerations (necessary for speed increase, for braking\se, 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 all these efforts 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 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. In
short, legs \emph{outperform} wheels.
\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}).}
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.
% 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}
%
\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), 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{waterwalking 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. No robot can walk or even run as fast
as the animal system it tries to copy. For twolegged robots, the most
difficult ones, 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}. 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 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
% 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{sprinting} 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
\subsection{Summary on Galilean relativity}
% Feb 2012
The Earth rotates. The acceleration is so small that we do not feel it. The
speed is large, but we do not feel it, because there is no way to so.
Undisturbed or inertial motion cannot be felt or measured. It is thus
impossible to distinguish motion from rest; the distinction 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.
Only later on will we discover that one example of motion in nature 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{}
\chapter{Motion due to gravitation}
% \chapter{}
\markboth{\thesmallchapter\ motion due to gravitation}%
{\thesmallchapter\ motion due to gravitation}
\begin{quote}\selectlanguage{italian}%
Caddi come\indname{Dante Alighieri}\indname{Alighieri, Dante} corpo
morto cade.\selectlanguage{UKenglish}\\
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 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\se, 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.%
%
\footnote{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}?
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?
By the way, if the Earth is round, the top of two buildings is further
apart than their base. Can this effect be\challengenor{builddist}
measured?} %
%
How does gravitation change when two bodies are far apart? The experts on
distant objects are the astronomers. Over the years they 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\se,
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 of planetary
motion.\seepagethree{exisdefi} This is not an easy task, as the observation
of \figureref{imarsretrograde} shows.
% Mar 2012
\csepsf{imarsretrograde}{scale=1}{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 example
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)
In 1684, all observations 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.} %
%
%
% May 2004
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. This is called
\ii{universal gravitation}, or the \ii{universal `law' of gravitation},
because it is valid in general, both on Earth and in the sky. The
proportionality constant $G$ is called the \ii{gravitational constant}; it is
one of the fundamental constants of nature, like the speed of light or the
quantum of action. More about $G$ will be said
shortly.\seepageone{unigrpropzz} The effect of gravity thus decreases with
increasing distance; gravity 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.
\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
This inverse square dependence is often called Newton's `law' of gravitation,
because the English physicist \iinn{Isaac Newton} % \lived(16421727)
proved more elegantly than Hooke that it 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 this description of astronomical
motion 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%
%
\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.\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 distance $h$ of 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 a 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 of the Earth during a 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}' 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 describes motion due to gravity 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 the 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
\iin{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.
\cssmallepsfnb{ifloatingastronaut}{scale=1}{The man in orbit feels no
weight, the blue atmosphere, which is not, does (NASA).}
% 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.
%
\subsection{Properties of gravitation}
Gravitation\label{unigrpropzz} implies 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}
\caption{Some measured 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 thus 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
it?\challengenor{atwood}
Note that 9.8 is roughly $\pi^{2}$. 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}}.\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}. 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 for objects in everyday life. Gravity
must also work \emph{sideways}.\index{gravity, sideways action of} This is
indeed the case, even though the effects are so small that they were measured
only long after universal gravity had predicted them. In fact, measuring this
effect allows the gravitational constant $G$ to be determined. Let us see
how.
We note that measuring the gravitational constant $G$ is also the only way to
determine the mass of the \ii[Earth, mass of]{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
\csepsfnb{ifourmilab}{scale=1}{An experiment that allows weighing the Earth
and proving that gravity also works sideways and 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 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
\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 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{gravyty} 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. At his time, this was a surprising result, because rock only has 2.8
times the density of water.
% Impr. Jun 2010
Gravitation 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} will be studied
later\seepagethree{edynstzys} in our walk: it is called \ii{electromagnetism}.
\cssmallepsf{ipotedef}{scale=1}{The potential and the gradient.}
%
\subsection{The gravitational potential} % added Aug 2009
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 ${\bm{\nabla}} \varphi= (\partial
\varphi/\partial x, \partial \varphi/\partial y, \partial \varphi/\partial z)
= {\rm grad\,} \varphi$. The minus sign in (\ref{minussigndef}) is introduced
by convention, in order to have higher
potential values at larger heights.%
%
\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.} %
%
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
{\frac{ 1 }{ 2}}
m_{\rm 1} {\bm v_{\rm 1}}^{2}

{\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 accelerations can be derived from a potential; systems with this
property are called \ii[systems,
conservative]{conservative}.\seepageone{enconszz} Observation shoes 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. When the spherical 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}
%
\subsection{The shape of the Earth} % added Aug 2009
For a spherical 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, one has
\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 \iin{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{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
$\Delta \varphi \sim \bar\phi \phi(x)$?)\challenge % !!!5
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{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
of]{shape} of the Earth. 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{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 some
philosophers pretended.\seepageone{wittgsonne}
%
\subsection{Dynamics  how do things move in various dimensions?}
% Apr 2005, impr. Jul 2005
The concept of potential is a powerful tool. 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 short,
motion in \emph{one dimension} follows directly from energy
conservation.\index{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.}
%
\subsection{Gravitation in the sky}
% improved May 2005, Oct 2009, Mar 2012
The expression $\smash{a=GM/r^2}$ for the acceleration 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
\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 orbit an ellipse?\cite{heckman} The simplest argument is
given in \figureref{iellipsebook}. We know that\indexs{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 Moon.
%
\subsection{The Moon}
% Jun 2005
How long 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.
\csmovfilmrepeat{lunation3}{scale=0.25}{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{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, 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
\cssmallepsfnb{fhiddenmoonside}{scale=0.8}{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 area 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.\footnote{The web
pages \url{cfawww.harvard.edu/iau/lists/Closest.html} and
\href{cfawww.harvard.edu/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{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{tectonic
activity} of the Earth, and maybe also the drift of the
continents.\index{earthquakes}
The Moon has many effects on animal life. A famous example is the midge
\iie{Clunio}, which lives on coasts with pronounced \iin{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 spring tide. Spring tides are the
especially strong \iin{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[astronomers, smallest
known]{astronomers}.
If \iin{insects} can have \iin{circalunar} cycles, it should come as no
surprise that \iin{women} also have such a cycle;\index{menstrual cycle}
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
ages.\seepageone{moonprecsn}
% % 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
%
\subsection{Orbits  and conic sections}
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}s or
\ii{hyperbola}s, 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 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}).}
% !!!1 add a figure with the irregular orbits of a planetoid or comet
% http://ifa.hawaii.edu/~barnes
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, 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 of
the energy and the angular momentum needed for these paths to
appear?\challenge % !!!5
In practice, parabolic paths do not exist in nature. (Though some comets seem
to approach this case when moving around the Sun; almost all comets follow
elliptical paths). 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
\end{equation}
and therefore the motion lies in a plane. Also the energy $E$ is a constant
\begin{equation}
E = \frac{1}{2} m \left (\frac{\diffd r}{\diffd t} \right )^2 +
\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\challenge % !!!5
\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
(ellipsis, 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 still a
topic of research, and the results are mathematically fascinating. Let us
look at a few examples.
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}, as shown in \figureref{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 no \ii{antigravity}. Can
gravity be used for \ii{levitation} nevertheless, maybe 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.
% % http://apod.nasa.gov/apod/ap120411.html I have permission!
% % 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
% % http://apod.nasa.gov/apod/ap120411.html I have permission!
% % 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
% % 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{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.
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} 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}, changes 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 \iinn{Albert
Einstein}.\seepagetwo{perishift}
\csepsfnb{i4tides}{scale=1}{Tides at SaintValéry en Caux on September 20,
2005 ({\textcopyright}~\protect\iinn{Gilles Régnier}).}
% EMAILED FEB 2008  gilles@gillesregnier.com
%
\subsection{Tides}
\label{tides}%
%
Why do physics texts always talk about \iin{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}
% !!!2 psfrag
\cstftlepsf{itidea}{scale=1}{Tidal deformations due to
gravity.}[20mm]{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 deformation of four pieces. We can study it in free fall,
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 \ii{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\se, 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{tides 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}\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.
% !!!2 both need psfrag
\cstftlepsf{itidecur}{scale=1}{Particles falling sidebyside approach over
time.}[30mm]{ilightgravbend}{scale=1}{Masses bend light.}
%
\subsection{Can light fall?}
\begin{quote}\selectlanguage{german}%
Die Maxime, jederzeit selbst zu denken, ist die Aufklärung.\\
\iinn{Immanuel Kant}\selectlanguage{UKenglish}%
% 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 has a finite velocity  a story which we will tell in
detail 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}
One sees that 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}{{}^{\prime\prime}}$=\;$\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}{{}^{\prime\prime}}, 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.
%
\subsection{What is mass?  Again}
Mass describes how an object interacts with others. In our walk, we have
encountered two of its aspects. \ii[inertial mass]{Inertial mass}
is\index{mass, inertial} the property that keeps objects moving and that
offers resistance to a change in their motion. \ii[gravitational
mass]{Gravitational mass} is\index{mass, gravitational} 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 values shown by a balance for a person of \csd{85}{kg}
juggling three balls of \csd{0.3}{kg}\challenge each?} % !!!5
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}
%
But 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 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, as far
as we can tell, gravitational mass and inertial mass are
\emph{identical}. What is the origin of this mysterious equality?
This socalled `mystery' is a typical 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 origins of the accelerations do 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
electromagnetic interaction, and gravitational mass\index{mass, identity of
gravitational and inertial} are identical \emph{by definition}.
We also note that we have never defined a separate concept of `passive
gravitational mass'. The mass being accelerated by gravitation is the
inertial mass. Worse, there is no way to define a `passive gravitational
mass'. Try it!\challengenor{passgrmass} All methods, such as weighing an
object, cannot be distinguished from those that determine inertial mass from
its reaction to acceleration. Indeed, all methods of measuring mass use
nongravitational mechanisms. Scales are a good example.
If the `passive gravitational mass' were different from the inertial mass, we
would have strange consequences. For those bodies for which it were different
we would get into trouble with energy conservation. 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? Each of these
possibilities contradicts either energy or momentum conservation.
% Another beautiful proof of this statement was given by \iinn{{A.E.}
% Chubykalo},
% % valeri @ canera.reduaz.mx
% and \iinn{{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, they
% prove
% that the inertial and gravitational mass must be proportional to each
% other.
% Can you see how?\challenge
No wonder that all measurements confirm the equality of all mass types. The
issue is usually resurrected in general relativity, with no new
results.\seepagetwo{grinma} `Both' masses remain equal. \emph{Mass is a
unique property of bodies}.
%
%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.
%nor by general relativity. We will discover the answer later during our
% walk.
%We need a bit of patience.
%
\subsection{Curiosities and fun challenges about gravitation}
\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{UKenglish}\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 2009
\item Is the acceleration due to gravity constant? 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{asteroids}. 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's mass,
time variation}
\item Incidentally, discovering objects hitting the Earth is not at all easy.
Astronomers like to point out that an 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=0.15}{Brooms fall more rapidly than
stones ({\textcopyright}~\protect\iinn{Luca Gastaldi}).}
% EMAILED FEB 2008
\item Imagine that 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, 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{1}{8} \frac{m}{M} \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 weight 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
weight 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 lift a mass of a kilogram from the floor on 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
\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.
(Why?)\challengn
What is the escape velocity for 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 its escape velocity would be larger than the speed
of light?\challengenor{bh1}
\item What is the largest \iin{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
\item The \ii[shape of the Earth]{shape} of the Earth 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
\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}\se, 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\se, 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[whales]{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 \iin{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; around 80 are known today.
\csepsfnb{isedna}{scale=1}{The orbit of Sedna in comparison with the
orbits of the planets in the solar system (NASA).}
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}.
% Sometimes these bodies change
% trajectory due to the attraction of a nearby planet: that is the birth of a
% new \iin{comet}.
\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{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 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
\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 a star or the Sun 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 the star Pegasi 51, was discovered in this way, in 1992 and
1995. In the meantime, over 400{\present} planets have been discovered with
this and other methods. Some have even masses comparable to that of the
Earth.
\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
obervations of 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
Newton's
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}
% !!!2 psfrag
\cssmallepsf{ishellgrav}{scale=1}{The vanishing of gravitational force inside
a spherical shell of matter.}
\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 :
%
% 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 around 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{shadows!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 hypthetical particles are called \ii{gravitons}. On the other
hand, no such particles have never been observed. Only in the final part of
our mountain ascent will we settle the
issue of the origin of gravitation. %\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 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 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}?\challenge % !!!5
How does it compare\index{Moon!weight of} with the weight of the
\iin[Alps!weight of]{Alps}?
\item Owing to the slightly flattened shape of the Earth, 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}
\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}
\caption[An unexplained property of nature: the TitiusBode rule.]{An
unexplained property of nature: planet distances 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's moons}
% 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 a 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.
% May 2005
\item Around\label{babylo} 3000 years ago, the \iin{Babylonians} had measured the
orbital times of the seven celestial bodies. Ordered from longest to
shortest, they wrote them down in \tableref{babiggghhh}.
%
% \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}
\caption{The orbital
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} The Babylonians %(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}
% !!!2 Find a figure from the cited paper
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?\challenge % !!!5
% 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
\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 show in \figureref{iearthwire}?\challengenor{uprightwire}
% Dec 2006
\item Everybody knows that there are roughly two \iin[tides, once or twice
per day]{tides} per day. But there are places, such as on the coast of
\iin{Vietnam}, where there is only one tide per day. See
\url{www.jason.oceanobs.com/html/applications/marees/marees_m2k1_fr.html}.
Why?\challenge % !!!5
% 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
\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.
% Sep 2008
\item The observed motion due to gravity can be shown to be the simplest
possible, in the following sense. If we measure change of a falling object
with $\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?\challenge % !!!5
% 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
\subsection{Summary on gravitation}
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}, describes snowboarders, skiers, paragliders,
athletes, couch potatoes, pendula, stones, canons, rockets, tides, eclipses,
planet shapes, planet motion and much more. It is the first example of a
unified description, in this case, of \emph{how everything falls}.
\vignette{classical}
%
%
%
\newpage
% \chapter{}
\chapter{Classical mechanics and the predictability of motion}
% \chapter{}
\markboth{\thesmallchapter\ classical mechanics and the predictability of
motion}%
{\thesmallchapter\ classical mechanics and the predictability of motion}
% !!!2 a photo of a jumping water parabola in a fountain
% ESTHER MAMA Nonna Opa Oma
\csini{A}{ll} those types of motion in which the only permanent property
of\linebreak body is mass define
% the mass of a body is its only
% permanent property form
the field of \ii{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\se, and masses are
measured.
More specifically, our topic of investigation so far is called \ii[classical
mechanics]{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.
%
% ll
% 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\se, 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? Let us
start with the exploration of this issue.
We know that there is more to the world than gravity. A simple observation
makes the point: \emph{friction}. Friction cannot be due to gravity, because
friction is not observed in the skies, where motion follows
gravity rules only.%
%
\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.} %
%
Moreover, on Earth, friction is not related to gravity, as you might want to
check.\challengn There must be another interaction responsible for friction.
We shall study it shortly. But one issue merits a discussion right away.
%
\subsection{Should one use force? Power?}
\begin{quote}
The direct use of force is such a poor solution to any problem,
it is generally employed only by small children and large 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\se, 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 origin.
% all composed of only four
% \ii[fundamental interaction]{fundamental} types of
% interactions:\indexs{interaction, fundamental} the gravitational, the
% electromagnetic and the two nuclear interactions.
%
% \subsubsubsubsubsubsubsubsection{Table of forces}
{\small
\begin{table}[t]
\small
\centering
\caption{Some force 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}. Since momentum is
conserved, we can say that force measures the \emph{flow} of
momentum.\index{momentum!flow} If a force accelerates a body, momentum flows
into it.\index{momentum!as a liquid} Indeed, momentum can be imagined to be
some invisible and intangible liquid. Force measures how much of this liquid
flows into or out of a body per unit time.
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} \lived(17071783) 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, 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 muscular force. For example, the more force is used, the
further a stone can be thrown.
% 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 a
nanonewton, can be detected by measuring the deflection of small levers with
the help of a reflected laser beam.
% !!!2 add images 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} \cdot {\bm v}
\end{equation}
in which \ii[work, physical]{(physical) work} $W$ is defined as $W = {\bm F}
\cdot {\bm s} $. 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. Note that a man who walks carrying a heavy
rucksack is hardly doing any work; why then does he get
tired?\challengenor{tiredman}
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}
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{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 force and everyday effort is
the main hurdle in \iin{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).}
% Often equation (\ref{nl}) is not recognised as the definition of force.
% This
% is mainly due to the fact that there seem to be forces without any
% associated
% acceleration or momentum change, such as mechanical tension or water
% \iin{pressure}. Pushing against a tree, there is no motion, yet a force is
% applied. If force is momentum flow, where does the momentum go? In fact,
% the
% tree is slightly deformed, and the associated momentum change of the
% molecules, the atoms, or the electrons of the bodies \emph{is} observed, but
% only at the microscopic level. For this reason one never needs the concept
% of
% force in the microscopic description of nature. For the same reason, the
% concept of \ii{weight}, also a force, will not be used (much) in our walk.
Often the flow of momentum, 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, 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, one can indeed find
that 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.
As force is net momentum flow, it is only needed as a separate concept in
everyday life, where it is useful in situations where net momentum flows are
less than the total flows. At the microscopic level, momentum alone suffices
for the description of motion. For example, the concept of \ii{weight}
describes the flow of momentum due to gravity. Thus we will hardly ever use
the term `weight' in the microscopic part of our adventure.
% true, as Alpha shows
% On the other hand, at the macroscopic level the concept of force is indeed
% useful, especially when no apparent motion can be associated to it, such as
% in
% the case of \iin{pressure}.
Before we can answer the question in the section title, on the usefulness of
force and power, we need more arguments. Through its definition, the concept
of force is distinguished clearly from `mass', `momentum', `energy'\se, and
`power'. 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}
%
\subsection{Forces, surfaces and conservation}
% June 2010
We saw\label{forcesuftg} that force is the change of momentum. We also saw
that momentum is conserved. 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
flowing \emph{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} 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, when the momentum of a body changes, this happens
through a surface. Momentum change is due to momentum flow. In other words,
the
concept of force always assumes 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 are flows through surfaces.
%
{\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 and devices changing the motion of bodies.}
\label{motors}\index{actuators}\index{motors}
\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{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 \& 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}
}
%
%
\subsection{Friction and motion}
% !!!2 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 one that lets us choose the direction of
our gaze to the one that carries a butterfly through the landscape, can be
put into one of the two leftmost columns of \tableref{motors}. Physically,
the 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
% April 2009
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{static friction} or
\ii{sticking friction}, is common and important: without it, turning the
wheels of bicycles, trains\se, 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} would stay tightened and no \iin{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 is based on friction}
In fact not only our own motion, but all \ii{voluntary motion} of living
beings is \emph{based} on \iin{friction}. The same is the case for
selfmoving machines. Without static friction, the propellers in ships,
aeroplanes\se, 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 required whenever
we or an engine want to move relative to our environment.\index{motion,
passive}\index{motion, voluntary}
%
\subsection{Friction, sport, machines and predictability}
Once an object moves through its environment, it is hindered by another type
of friction; it is called \ii{dynamic friction} and acts between bodies in
relative motion. Without it, falling bodies would always rebound to the same
height, without ever\cite{fricake} coming to a stop; neither \iin{parachutes}
nor brakes would work; worse, we would have no memory, as we will see later.%
%
\footnote{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.} %
%
All motion examples in the second column of \tableref{motors} include
friction. In these examples, macroscopic energy is not conserved: the systems
are \ii{dissipative}. In the first column, macroscopic energy is constant:
the
systems are \ii{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 we necessarily have to use force if we
want to describe the motion of the system. For example, the \ii{friction} or
\ii{drag} force $F$ due to \iin{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} $c_{\rm
w}$ 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 low
velocities, when the fluid motion is not turbulent but laminar, drag is called
\ii{viscous} and follows an (almost) linear relation with speed.) You may
check that aerodynamic resistance cannot be derived from a\challenge % !!!5
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.%
%
\footnote{Calculating drag coefficients in 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.
The topic of aerodynamic shapes is even more interesting for fluid bodies.
They are kept together by \ii{surface tension}. For example, surface tension
keeps the hairs of a wet brush together. Surface tension also determines the
shape of \iin{rain drops}.\index{drop} Experiments show that it 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.} %
%
%
%
% to be always larger
% than 0.0168, % NOT SURE, TAKEN OUT
% which corresponds to the optimally streamlined \iin[shape,
% optimal]{tear shape}.\index{tear shape}
%
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.
Wind resistance is also of importance to humans, 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
Likewise, parachuting 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{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. The
study of what happens in that contact area is still a topic of \iin{research};
researchers are investigating the issues using instruments such as \iin{atomic
force microscopes}, \iin{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 many particles, usually millions or
more. Such systems are called \ii[systems, dissipative]{dissipative}. Both
the\indexs{dissipative systems} 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. As a result, all motion is
conservative on a microscopic scale. Therefore, on a microscopic scale it is
possible 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}: 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
microscopic details.%
%
\footnote{In the case of human relations the evaluation should be somewhat
more discerning, as the research by James Gilligan\cite{gilligan} shows.}
%
%A beautiful book on the control of force is ..
%
Secondly, friction is not an obstacle to predictability.
\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{UKenglish}\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}
%
\subsection{Complete states  initial conditions}
\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 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. In other words, 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. Can you
specify the necessary quantities 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 it for a system
consisting of $N$ point particles?\challengenor{phasespsol}
Given that we have a description of both properties and states of 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 variables to discover. Each
property is the basis of a field of physical enquiry.
% FEb 2012
Speed is the basis for mechanics, temperature is the basis fir thermodynamics,
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.
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} As a hint, recall that when a stone is thrown, the
initial conditions summarize the effects of the thrower, his history, the way
he got there etc.; in other words, initial conditions summarize the past of a
system, i.e., the effects that the environment had during the history of a
system.
\begin{quoteunder}
An optimist is somebody who thinks that the future is uncertain.\\
Anonymous
\end{quoteunder}
%
\subsection{Do surprises exist? Is the future determined?}
% subsubsection{Evolution, time\se, and determinism}
\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{UKenglish}%
%
\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 show 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[voluntary
motion]{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 equations]{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,
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}. The
unpredictability of motion can have four origins:
\begin{Strich}
\item[{1.}] an
impracticably large number of particles involved,
\item[{2.}] the mathematical complexity of the evolution equations,
\item[{3.}] insufficient information about initial conditions, and
\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
It is hard to predict the weather over periods longer than about a week or
two. (In 1942, Hitler made a fool of himself by ordering 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 deterministic and time \ii{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.\index{causal}
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 only be resolved in
the last part of 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\se, 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, \emph{the use of time implies determinism, and vice versa.} 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.
%
\subsection{Free will}
\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 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 \ii{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, free will \emph{cannot} be
observed. (Psychologists also have a lot of their own data to support this,
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]{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.
%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.
The lack of surprises in everyday life is built deep into our body: the
concept of \iin{curiosity} is based on the idea that everything discovered 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 definition of free will: it is a \emph{feeling}, in
particular of independence of others, of independence from fear\se, and of
accepting the consequences of one's actions. Free will is 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.%
%
\footnote{That free will is a feeling can also be confirmed by careful
introspection. The idea of free will always appears \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.} %
%
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\se, 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 who insists on the
existence of free will who is running away from responsibility.
%
\subsection{Summary on predictability}
Despite difficulties to predict specific cases, all motion we encountered so
far is deterministic and predictable. The future is determined. In fact,
this is the case for all motion in nature, even in the domain of quantum
theory. If motion were not predictable, we could not have introduced the
concept of `motion' in the first place.
%
%
%
%1020304050607078
%
%
% %{La fisica classica, seconda parte}
%
%
% % versione 8.762 full OED spellcheck Dec 2006
% % 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
% % exch appropriate ", and" with "\se, and" Aug 2003
% % checked [az][azAZ] Aug 2003
% % capitalized \bt downcased \ti Aug 2003
% % figure with triple excl. Aug 2003
% % 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}
\subsection{Predictability and global descriptions of motion}
\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 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 `law' of 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, and then explore the
details of each approach.
\begin{Strich}
\item[{1.}] \ii[action principle]{Action principles} or \ii[variational
principles]{Variational principles}, the first global approach to motion,
arise when we overcome a 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 all times and positions. The
global approach required by questions such as this one will lead us to a
description of motion which is simple, precise\se, and fascinating: the
socalled principle of \iin[laziness, cosmic, principle of]{cosmic laziness},
also known as the principle of 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 different 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 symmetry transformations, lead us to a global description, 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?}[15mm]{icompastraight}{scale=1}{A famous mechanism, the
\protect\ii{PeaucellierLipkin linkage}, that allows drawing a straight line
with a compass: fix point F, put a {pencil} into joint P\se, and move C with a
compass along a circle.}
\medskip
\item[{3.}] \emph{Mechanics of\indexs{mechanics} extended and rigid}
bodies\index{bodies, rigid}, rather than mass points, is required to
understand many objects, plants and animals. As an example, the
counterintuitive result of the experiment in \figureref{ihanged} shows why
this topic is worthwhile.\challenge 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. 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 a} drawing a
\emph{straight} line with the help of a \iin{compass}.
Another\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 space.\seepagetwo{riegym}
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[extended bodies, nonrigid]{nonrigid extended bodies}. For example,
\ii{fluid mechanics} studies the flow of fluids (like honey, water\se, or air)
around solid bodies (like spoons, ships, sails\se, or wings). Fluid mechanics
thus
describes how insects, birds\se, 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, full} full of wine can be emptied in the
fastest way possible.\challengenor{eggb}
\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 deformable \emph{solids}.
This area of research is called \ii{continuum mechanics}. 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 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. They 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?
{\mbox{%
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\raise 0.1ex\hbox{?} }} %
% the
% evolution of stars and galaxies, or {\mbox{%
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% \raise 0.1ex\hbox{.} }} %
All these are examples of \ii{selforganization} processes; life scientists
simply speak of \ii{growth} processes. Whatever we call them, all these
processes are characterized by the spontaneous appearance of patterns,
shapes\se, and cycles. Such 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
\np We will now explore to these six global approaches to motion. We will
begin with the first approach, namely, the global description of motion using
a variational principle. This beautiful method was the result of several
centuries of collective effort, and is the highlight of particle dynamics.
Variational principles also provide the basis for all the other global
approaches just mentioned and for all the further descriptions of motion that
we will explore afterwards.
\vignette{classical}
\clearpage\endinput
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