pendulum notes

8/10/2019 pendulum notes

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LCP 2: MOTION AND THE PENDULUM


March 23

Without the pendulum there would have been no Principia. (R.S. Westfall)

All of physics comes from Galileo’s inclined plane. (Anon)

Fig. 1 Dome of the cathedra of Pi!a "ith the #Lam$ of %aieo#.

While watching a chandelier swing back and forth at the Cathedral of Pisa in 1!"# Galileonoticed something curious. Galileo noticed that the time period to swing through one complete

cycle is independent of the amplitude through which it swings$..%e timed the swings with his

pulse# the only timing device at hand. (IL 1 )

IL 1 &&&

IL 2 &&&

The story of Galileo and the pendl!

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Fig. 2. Fo'ca't(! Pe)d''m* the Pa)theo)* Pari!.

&hat was when ' saw the pendulum.

&he sphere# hanging from a long wire set into the ceiling of the choir# swayed back and forth with isochronal ma(esty. ' knew that the period was governed by the s)uare root of the

length of the wire and by *# that number $which binds the circumference and diameter of all

possible circles$ ("!#erto $co% ta&en fro! his #oo& 'ocalts *endl!).

IL + , 1- ++++ ,escription of or pendl! #oo& -st p#lished

IL ,2- +++ Galileos oriinal description of his inclined plane e/peri!ent

IL / ,+- ++++ S!!ary of the #oo& on the *endl! (scroll do0n to

2

http://ihpst.arts.unsw.edu.au/publications.htmlhttp://galileo.rice.edu/lib/student_work/experiment95/inclined_plane.htmlhttp://www.sced.nnov.ru/PENDULUM.HTMhttp://ihpst.arts.unsw.edu.au/publications.htmlhttp://galileo.rice.edu/lib/student_work/experiment95/inclined_plane.htmlhttp://www.sced.nnov.ru/PENDULUM.HTM


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'The "#iitos *endl!)

IL 0 ,- ++++ A ood #rief history of the pendl! 0ith nice pictres

IL ,/- ++++ An e/cellent ideo replicatin Galileos e/peri!ent

IL ,0- ++ A #rief #ioraphy of Galileo and so!e of his 0or& IL 3 ,- ++ A s!!ary of the International *endl! *ro-ect (I**)

Fig. +. The $e)d''m at differe)t !tage! of it! o!ciatio) I.

THE MAIN IDEA

This L4* is ided #y the history of the pendl!% #ased on the 0or& of si/ philosophers and

scientists on !otion5 Aristotle% 6icole 7res!e% Galileo% 4hristian 8yens% 6e0ton% and Leon

ocalt% fro! the forth centry 94 to the !iddle of the 1: th centry. We 0ill add a

dra!ati;ation inolin a n!#er of short conersations 0ith the follo0in natral philosophers

and physicists of the past5 Aristotle% 7res!e% Galileo% 8yens% 6e0ton and Leon ocalt.

$ach of the sections #eins 0ith a #rief introdction% follo0ed #y a short dra!ati;ation

for each philosopher


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presentation. These e/ercises 0ill force yo to try to thin& li&e the scientist that yo are

pretendin to #e. (The 'conersations are also fond in the Appendices).

=o 0ill notice that in follo0in the history of the concepts a#ot forces and !otion #y

sin the pendl! as the central idea% 0e 0ill recapitlate (re>discoer) the intitie physics

discssed in L4* 1. 8o0eer% these ideas 0ill #e discssed in a richer conte/t and on a !ore

sophisticated leel. Rich concepts% sch as those fond in 6e0tons dyna!ics% !st #e !astered

and lti!ately nderstood in a deep sense% and not -st sperficially !e!ori;ed fro! te/t#oo&s.

If these i!portant concepts and ideas are later presented in the sa!e 0ay as it 0as done the first

ti!e% learnin also #eco!es #orin and !any of ideas and concepts 0ill #eco!e inert.

The pendl! did not only play a central role in the deelop!ent of the &ine!atics and

dyna!ics in the seenteenth centry% #t 0as also a research instr!ent in the 1?

th

and 1:

th

centries and is still sed today in the stdy of chaos theory.

It is reco!!ended that #efore the estions% pro#le! and e/peri!ental sections are

atte!pted% for each case% stdents present the #rief conersation to the 0hole class% follo0ed #y a

discssion.

The #iitos (loo& p the !eanin of the 0ord) pendl!% toether 0ith the ersatile

inclined plane% played a central role in the deelop!ent of the &ine!atics and dyna!ics in the

seenteenth centry. In !any of the &ey pro#le!s of Galileo these si!ple deices 0ere

connected in creatie 0ays to stdy !otion% 0ithot considerin the forces inoled

(&ine!atics). Te/t#oo&s tell stdents that Galileo 'dilted raity and e/trapolated to free fall in

an atte!pt to nderstand free fall% or 0hat Aristotle called 'natral !otion. 8o0eer% te/t#oo&s

enerally dont !ention that Galileo also stdied free fall directly% sin an inenios !ethod of

ti!in the fall 0ith a pendl!. The 'interrpted s0in of the pendl! anticipated the

principle of the conseration of !echanical enery. Stdyin the pendl!% Galileo thoht that

an arc of a circle represented the 'least ti!e path of an o#-ect in a ertical plane. 8e 0as al!ost

riht.

The history of the ses of the pendl! in the stdy of &ine!atics and dyna!ics contains

eerythin reired to teach the fnda!entals of &ine!atics and dyna!ics on the ele!entary

leel.

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A #rief history of the pendl! shold reeal the nfoldin of !ost of the ideas and

!ethods of the discipline of &ine!atics and dyna!ics that is central to ele!entary physics. The

i!portance of the pendl!% ho0eer% oes #eyond the 1 th centryB 0e find the pendl! as a

research instr!ent% riht into the 2Cth centry.

THE DE4C5IPTION OF THE CONTE6T

Fig. %aieo7! i)ci)ed $a)e

A 8rief Hi!tor9 of the Pe)d''m

The stdy of !otion !st #ein 0ith Aristotles ideas. Aristotle (@th centry 9.4.) 0as ara#ly

the first to loo& at the physical 0orld in a syste!atic% 0hat 0e !iht today call a 'rational 0ay.

8e ared for certain fnda!ental principles% 0hich 0ere #ased on na&ed>eye o#serations and

partly on loic. 9t his e/planations of !otion cannot #e considered scientific in the !odern

sense #ecase he did not consider it necessary to chec& his principles #y controlled e/peri!ents.

8e insisted% ho0eer% that these principles 0ere #ased on carefl o#seration of !oin

o#-ects in eeryday life. 8e o#sered o#-ects fallin% s!o&e risin% carts !oin as they 0ere

#ein plled #y o/en and spears and arro0s !oin throh the air. In the heaens% he tried to

e/plain the irrelar !otion of the planets. 8o0eer% he thoht that the physics of the heaens

(celestial) !otion 0as alitatiely different fro! the !otion of o#-ects on earth (terrestrial

!otion). 8e thoht that his co!!on sense accont of !otion in eneral 0as sfficient to

e/plain all !otion arond s.

D


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%aieo a)d the $e)d''m

The Italian physicist and !athe!atician Galileo Galilei (1D@>1@2) laid the fondations of

&ine!atics% the stdy of !otion 0ithot considerin the forces that prodce that !otion. 8e 0as

particlarly interested in nifor!ly accelerated !otion. Leend has it that he dropped t0o

o#-ects% one !ch heaier than the other fro! the Leanin To0er of *isa 0hen he 0as a yon

!an. 8e !ost li&ely neer chec&ed his ideas a#ot free fall #y droppin o#-ects fro! the

Leanin To0er of *isa. 8e apparently did not thin& this &ind of de!onstration 0as necessary.

Galileo recorded his thohts and e/peri!ents in his #oo& &he &wo +ew ,ciences that

0as 0ritten in the last decade of his life. 8e 0rote the te/t in Italian rather than in Latin as 0as

enerally done in those days% 0hile nder hose arrest. 8ere he clearly descri#ed ho0% sin an

inclined plane% he 'dilted raity and ideali;in and e/trapolatin to free fall calclated(esti!ated) the distance a freely fallin o#-ect falls in a ien ti!e. Today% 0e are !ore interested

in the ale of 0hat 0e today call % the acceleration de to raity.

8e 0as the first to sho0 that tra-ectory !otion is para#olic and that the period of the

pendl! is fairly constant. 8e proed% and later confir!ed #y e/peri!ent that the period is

proportional to sare root of the lenth. 8e also ared that the arc of a circle represented the

least ti!e (#rachistochrone) path of an o#-ect descendin alon a frictionless path in a ertical

plane.

Galileo then trned to the estion of 0hat happens 0hen a #all rolls hori;ontally% hain

rolled do0n an inclined plane. 8e ared that it shold roll p another inclined plane and !oe

to the sa!e heiht. Later% he sed an 'interrpted pendl! to de!onstrate 0hat 0e no0 call

'conseration of !echanical enery. A#ot DC years later this idea 0as e/pressed and discssed

#y the Ger!an natral philosopher and !athe!atician Gottfried on Lei#ni;% in the for! of

&inetic enery% 0hat he called 'is ia. In fact% the ,tch natral philosopher 8yens entered

into a dispte 0ith the Ger!an natral philosopher and !athe!atician Lei#ni; oer 0hether

&inetic enery or linear !o!ent! 0ere consered in #illiard #all collisions. The dispte 0as

eentally resoled #y reali;in that in any elastic collision #oth !st #e consered.

To illstrate this% Galileo o#sered that the !otion of a pendl! 0ill rise (al!ost) to its

startin leel een if a pe is sed to the chane the path. 8o0eer% he 0ent frther and i!ained


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that if a #all 0ere to roll foreer 0ithot resistance (a thoht e/peri!ent) then the #all shold

circ!naiate the earth. This thought e-periment is i!portant in the deelop!ent or

nderstandin of !otion. In fact% it anticipates 6e0tons first la0 of !otion% enerally referred to

as the 'la0 of inertia.

The 1th ce)t'r9

H'9ge)!

The ,tch physicist 4hristian 8yens (12:>1:D) 0as second only to 6e0ton as a natral

philosopher and !athe!atician. 8e 0ent #eyond Galileo and sed the pendl! to find the

e/pression for 'centrifal force as 0ell as the !odern for!la for the period of a pendl!

for s!all anles% na!ely that T H 2 (l


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IL 1 ,12- ++ 8yens #ioraphy

IL 1/ ,1+- +++ 8yens #ioraphy

It 0as left to 6e0ton (1@2>12)% Lei#ni; (1@>11) and Johannes 9ernolli (1>1@?) to

lay the fondation of a ne0 #ranch of the calcls% in order to sole pro#le!s sch as the

#rachistochrone % or 'least ti!e of descent #et0een t0o points in a ertical plane. In the

capa#le hands of the reat S0iss !athe!atician $ler their approach then #eca!e a po0erfl

!ethod to sole !ini!! and !a/i!! pro#le!s% called 'ariational calcls.

Fig. A !i=tee)th ce)t'r9 demo)!tratio): the >?'ic@e!t de!ce)t B a di!cre$a)t e;e)t

IL 10 &&& *ictre ta&en fro! 'Galileo and the *endl!.

IL 1 ,1- +++ 8istory of the #rachistochrone pro#le!

IL 1 ,1/- +++ IA of the #rachistochrone pro#le!

IL 13 ,10- +++ IA of the #rachistochrone pro#le!

A si!ple apparats can #e #ilt% sin t0o 0ires% one straiht and the other rohly

shaped as a cycloid% 0ith t0o #all #earins slidin do0n the 0ires. This is an e/a!ple of a

discrepant eent that is sre to enerate !ch discssion. The eent is discrepant #ecase the

:

http://www-history.mcs.st-and.ac.uk/Biographies/Huygens.htmlhttp://inventors.about.com/library/inventors/bl_huygens.htmhttp://cnx.org/content/m11929/latest/http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Brachistochrone.htmlhttp://home.ural.ru/~iagsoft/BrachJ2.htmlhttp://www.kw.igs.net/~jackord/bp/i0.html/http://www-history.mcs.st-and.ac.uk/Biographies/Huygens.htmlhttp://inventors.about.com/library/inventors/bl_huygens.htmhttp://cnx.org/content/m11929/latest/http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Brachistochrone.htmlhttp://home.ural.ru/~iagsoft/BrachJ2.htmlhttp://www.kw.igs.net/~jackord/bp/i0.html/


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ti!e it ta&es for the #ead to slide do0n the loner path is actally shorterK

The 0or& of Ro#ert 8oo&e (1@C>1C)% a conte!porary of 6e0ton% shold #e inclded

in this historical presentation. Te/t#oo&s enerally !ention 8oo&e only in connection 0ith his

la0 of sprins. 8oo& is often called 'the 9ritish Leonardo. 8e 0as a poly!ath (a person 0ho

has e/tensie &no0lede in !any disciplines)5 scientist% inentor and ara#ly the reatest

e/peri!enter of the seenteenth centry. 8e 0as the crator of the Royal Society and so!eti!e

friend of 6e0ton. 8e sed his la0 ( H >&/) to sho0 that si!ple har!onic !otion (S8M)% li&e

that of the pendl!% or an oscillatin !ass attached to a sprin% arises 0hen this la0 holds.

8is scientific #attles 0ith 6e0ton 0ere leendary. When 6e0ton #eca!e the president of

the Royal Society in 1CD% he e/pned all esties of 8oo&e. Ths 0e hae no li&eness of hi!.

We identify Ro#ert 8oo&e #y the fa!os dra0in he !ade of a lose in his reoltionary'Microraphia that he p#lished at the ae of 3C. ,iscssin the confrontation #et0een 6e0ton

and 8oo&e stdents ic&ly co!e to reali;e that science is a h!an endeaor and cannot #e

captred #y a specifia#le !ethod. After 8oo&es death in 1C% 6e0ton 0as elected the

presidency of the Royal Society. "nfortnately% 6e0ton destroyed all esties of 8oo&e in the

#ildin% sch as pictres and #sts% so that today 0e only &no0 0hat he loo&ed li&e throh

descriptions.

I!aac Ne"to)

Fig. Ne"to) co)tem$ati)g gra;it9

1C


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IL 2< ,1- +++ A lon #ioraphy of 6e0ton

IL 21 ,1- ++ A short #ioraphy of 6e0ton

Isaac 6e0ton (1@2>12) sho0ed that the la0s of !otion applied to 'thins terrestrial

as 0ell as to 'thins celestial. These la0s and the inerse sare la0 of raity

e/plained at once planetary !otion% the !otion of the tides% free fall% and the !otion of

the pendl!. We 0ill see later ho0 6e0ton sed lare 0ooden spheres as pendla for

collision e/peri!ents% for testin 'centrifal forces% and to find the acceleration de to

free fall. 8e also tested the ratio of the raitational and inertial !asses of different

!aterials% sin a pendl!. 4o!#inin his second la0 ( H !a) 0ith 8oo&es la0

( H >&/) it follo0s that the period of a pendl! for s!all s0ins is ien #y

T H 29 < l % -st as 8yens fond earlier sin eo!etric reasonin and the idea of

'centrifal force.

What e/peri!ents did 6e0ton perfor! that sested and confir!ed his three la0s of

!otionE Most physics teachers do not &no0 the ans0er to this estion and te/t#oo&s

seldo! discss the e/peri!ental 0or& of 6e0ton #eyond his optical e/peri!ents. It is

not enerally &no0n that in his stdy of dyna!ics 6e0ton sed pendla to test his

second and third la0s of !otion% as 0ell as centripetal acceleration. Inertia% or his first

la0 of !otion% 0as seen as the conseence of a thoht e/peri!ent that cold not #e

e/peri!entally tested directly. 6e0ton% ho0eer% 0ent #eyond Galileos idea of inertia as

'the circ!naiation of an o#-ect on a perfectly s!ooth $arth to the idea of 'straiht

line !otion 0ith a constant speed in deep space 0hen there are no forces actin on the

o#-ect. 8is second la0% H !a% can #e applied to a pendl! to de!onstrate that if

8oo&es la0 holds (restorin force is proportional to the displace!ent of the !ass of the

pendl! fro! the ertical) then 0e hae si!ple har!onic !otion. This part of the story

is often told in te/t#oo&s% #t 6e0tons e/peri!ents to test his third la0 is seldo!

!entioned.

The third la0% 'action is eal to reaction% 0as de!onstrates #y 6e0ton sin

t0o lon (1C >1 feet) pendla and hain the! collide. 8e sed a reslt of Galileo (that

the speed of a pendl! at its lo0est point is proportional to the chord of its arc) and

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IL 2 ,21- ++ Short #ioraphy of 9en-a!in Ro#ins

IL 2/ ,22- +++ A detailed discssion of the #allistic pendl! > also featres a calclator

Later (1:C)% the $nlish physicist Geore At0ood% sed the pendl! and incorporated

it in his fa!os !achine% na!ed after hi!% as a research apparats. 7ne of the e/peri!ents he

perfor!ed 0as to test 6e0tons second la0 of !otion. At0oods !achine is foreer enshrined in

physics te/t#oo&s pro#le!s% #t it is seldo! !entioned that At0oods approach 0as the first

direct 'test of 6e0tons second la0 of !otion. The pendl! in this e/peri!ent is part of the

apparats. A si!ple plley can #e sed 0ith t0o dissi!ilar 0eihts and a pendl! to calclate

the ale of acceleration de to raity.

Fig. 1< At"ood7! Machi)e

In 1?D1% the rench *hysicist Jean ocalt desined a ery lon and heay pendl! to

de!onstrate for the first ti!e directly that the $arth reoles arond its a/is. We can offer a ood

discssion of this dra!atic and cele#rated de!onstration. Replication in the classroo! is difficlt

#t !any science !se!s and centers hae a ocalt pendl! de!onstration.

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Fig. 11 Fo'ca't demo)!trati)g hi! $e)d''m i) Pari!* 1/1.

IL 20 ,2+- ++++ ,escription of the ocalt pendl!

IL 2 ,2- +++ Lin&s to seeral ocalt pendl! descriptions ( a ariety of choices)

IL 2 ,2/- ++ 9rief description of the ocalt pendl!

IL 23 ,20- +++ ocalt pendl! applet sho0in forces present rench only

IL +< ,2- ++++ An applet of the ocalt *endl! discssin 4oriolis acceleration

(adanced)

THE P5E4ENTATION OF THE CONTE6T4

The actiity for the discssion of each of the si/ scientists shold #e introdced #y the

presentation of the short dra!ati;ations fond at the end of L4* 2% in the Appendices. =o can

also clic& on the '4onersations for each scientist% as sho0n in the preios section.

Co);er!atio)! "ith great !cie)ti!t! ao't motio) B a dramatization in six short scenes.

The presentation cold #e done #y the stdents (0ith or 0ithot the assistance of the instrctor)

This presentation is desined to set the scene for the actiities sested for each scientist

presented. The instrctor cold #ein #y hain stdents (0ith or 0ithot the instrctor) present.

After the presentation and the discssion that acco!panies sch a rop actiity the class cold

o ahead 0ith the actiities #elo0.

1@

http://en.wikipedia.org/wiki/Foucault_pendulumhttp://www.google.ca/search?hl=en&q=The+Foucault+pendulum&metahttp://mats.gmd.de/~skaley/vpa/foucault/foucault.htmlhttp://www.sciences.univ-nantes.fr/physique/perso/gtulloue/Meca/RefTerre/Foucault.htmlhttp://www.phys.unsw.edu.au/PHYSICS_!/FOUCAULT_PENDULUM/foucault_pendulum.htmlhttp://en.wikipedia.org/wiki/Foucault_pendulumhttp://www.google.ca/search?hl=en&q=The+Foucault+pendulum&metahttp://mats.gmd.de/~skaley/vpa/foucault/foucault.htmlhttp://www.sciences.univ-nantes.fr/physique/perso/gtulloue/Meca/RefTerre/Foucault.htmlhttp://www.phys.unsw.edu.au/PHYSICS_!/FOUCAULT_PENDULUM/foucault_pendulum.html


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Ari!tote a)d motio)

It is sested that stdents #ein 0ith the presentation of the Co);er!atio) "ith Ari!tote.

7ne often hears scientists or te/t#oo&s refer to the personal nderstandin of !otion #y stdents

as 'Aristotelian% sestin that these ie0s are si!plistic and 'nscientific. We !st

re!e!#er% ho0eer% that althoh his ideas a#ot !otion see! 'nscientific to s% AristotleNs

ideas a#ot !otion 0ere a part of a co!ple/ philosophical syste!. $en today% these ideas see!

ery sond to the intitie thin&in of !ost people.

8ac@gro')d to ca!! acti;itie! a)d f'rther di!c'!!io)!

IL +1 ,2- +++ A ery co!prehensie discssion of Aristotles thin&in in eneral

IL +2 ,23- ++ A s!!ary of Aristotles ideas a#ot !otion on $arthIL ++ ,+eident and cold not #e estioned.

1D

Ari!tote7 Fo')datio)a A!!'m$tio)! ao't Ph9!ic!:

1. The "ord i! ratio)a a)d @)o"ae.

2. Phe)ome)a ca) e i);e!tigated 9 "a9 of a ded'cti;e mode of rea!o)i)g.

+. There are first principles of $h9!ic! that ca) e di!co;ered* 9

o!er;atio)! a)d i)t'itio).

. e ca) o)9 !t'd9 terre!tria $he)ome)a "ith !'cce!!. Cee!tia

$he)ome)a "i remai) a m9!ter9.

/. There are )ece!!ar9* ')i;er!a !cie)tific fact! that ca) e ded'ced from

@)o"ae $ri)ci$e!.

http://en.wikipedia.org/wiki/Aristotlehttp://csep10.phys.utk.edu/astr161/lect/history/aristotle_dynamics.htmlhttp://aether.lbl.gov/www/classes/p10/aristotle-physics.htmlhttp://en.wikipedia.org/wiki/Aristotlehttp://csep10.phys.utk.edu/astr161/lect/history/aristotle_dynamics.htmlhttp://aether.lbl.gov/www/classes/p10/aristotle-physics.html


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The follo0in are Aristotles irst Principles of Physics/

The follo0in are 0hat Aristotle called the 0niversal Propositions for Physics/

It is interestin that after Galileo these #eca!e !easra#le antities and sed to

esta#lish la0s of !otion% e/pressed in !athe!atical ter!s. 8o0eer% as 0e 0ill see% this

&ind of physics 0old hae #een alien to Aristotle.

The follo0in are 0hat Aristotle thoht 0ere fnda!ental estions in physics5

1

Ari!tote7! Fir!t Pri)ci$e! of Ph9!ic!

1. A motio) i! either )at'ra or ;ioe)t.

2. A )at'ra motio) i! to"ard a )at'ra $ace.

+. ioe)t motio) i! ca'!ed 9 the co)ti)'i)g actio) of a) age)t.

. A ;ac''m i! im$o!!ie.

Ari!tote7! U)i;er!a Pro$o!itio)! for Ph9!ic!:1 Light tra;e! i) a !traight i)e.

2. A hea;9 oect! fa to"ard the ce)tre of the earth.

+. A o$a?'e oect! ca!t a !hado".

Predicates Proper to Physics/

. Po!itio)* !$eed* re!i!ta)ce.

Ari!tote7! F')dame)ta G'e!tio)! of Ph9!ic!:

1. hat are the ')i;er!a $ro$o!itio)! a)d defi)itio)! that are $ro$er to

$h9!ic!

2. hat @i)d of mathematica rea!o)i)g ca) "e tr'!t "he) de!crii)g the

"ord

+. h9 do thi)g! mo;e


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The follo0in 0ere the pro#le!s of physics for Aristotle5

Oestions #ased on Aristotles physics of !otion5

4o!!ent on Aristotles scientific thin&in #y ans0erin the follo0in estions5

1. What do yo thin Aristotle !eant #y 'rational thin&in in descri#in the 0orldE

*erhaps today 0e 0old call it 'o#-ectie% or 'scientific thin&in. ,iscss andco!!ent.

2. Aristotle thoht that celestial !otion (essentially the !otion of the planets and the

stars) !oed in response to la0s that astrono!ers coldneer nderstand. 8o0eer%

Gree& thin&ers and astrono!ers% sch as *ythaoras% $do/s #efore Aristotle% and

8ipparchs% Aristarchs and *tole!y after hi!% proposed ery sccessfl !odels to

e/plain planetary !otion. Aristarchs een sested a sn> centered !odel that 0as

reisited #y 4opernics si/teen hndred years later. Loo& p the !odels proposed #y

these Gree& natral philosophers and then set p discssion rops% and hae each

rop report to the 0hole class.

IL + ,+1- ++ Pisal representation of planetary !otion and ood applets

IL +/ ,+2- +++ 9asic !odels of planetary !otion

IL +0 ,++- +++ 9est sorce for the estion a#oe

3. Aristotle ared that the irst Principles of Physics cold #e discoered #y

o#seration and intition. We 0old call this &ind of reasonin today 'indctie and

'intitie. Then% to e/plain eeryday pheno!ena sch as free fall% 0e 0old hae to

are dedctiely fro! these principles to 0hat he called a 'scientific fact.

1

Ph9!ic! Proem! for Ari!tote:

1. Ho" ca) I de!crie the motio) of a free9 fai)g oect

2. Ho" ca) I de!crie the motio) of $roectie* a a;ei)

+. hat i! a $ro$er !cie)tific arg'me)t

http://csep10.phys.utk.edu/astr161/lect/retrograde/aristotle.htmlhttp://library.thinkquest.org/C005921/prevmodel.htmhttp://en.wikipedia.org/wiki/Ptolemaic_systemhttp://csep10.phys.utk.edu/astr161/lect/retrograde/aristotle.htmlhttp://library.thinkquest.org/C005921/prevmodel.htmhttp://en.wikipedia.org/wiki/Ptolemaic_system


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Fig. 12

Ari!tote7! $h9!ic!.

Oestions and Actiities for the stdents5

1. What is an indctie ar!ent and 0hat is a dedctie ar!entE Gie an e/a!ple

for each in5

a. $eryday discorse

#. Mathe!atics< eo!etry

c. Loic

d. *hysics

2. What is considered an eidential ar!ent in the follo0in areasE

a. $eryday discorse

#. Mathe!atics< eo!etry

c. Loic

d. *hysics

IL + ,+- +++ ,iscssion of indctie


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The follo0in are typical ar!ents that Aristotle considered scientifically alid5

Another 0ay to present this ar!ent is as follo0s5

Why is there a round shadow on the oon2

A rond shado0 is prodced 0hen a rond opae #ody (the $arth) is in the path of

the sns liht that traels to the !oon in straiht lines.

'7h% I see. The sns liht traelin to0ard the !oon is intercepted #y the earth and

that is 0hy there is a rond shado0 on the !oon.

,iscss each of these. ,o yo find these ar!ents persasieE

3. ,iscss critically each of the for irst Principles of Physics that Aristotle proposed.

"se yor o0n e/perience and &no0lede of physics. ,o yo aree 0ith Aristotle% or

disaree 0ith hi!E

@. Aristotle said that 'it is not enoh -st to na!e a scientific fact. =o !st also &no0

0hy it is a scientific fact. That is% 0e !st #ac& p a scientific clai! 0ith ood

eidence. 4onsider the follo0in e/a!ples of 'scientific facts5

a. The $arth !oes arond the sn.

1:

Arg'me)t 1:

A hea;9 oect! fa to"ard the ce)ter of the Earth.Thi! i! a hea;9 oect.

Therefore it "i fa do") ;ertica9.

Arg'me)t 2:

Light i) the hea;e)! come! from the !').

The moo) a)d the earth are !ometime! i) i)e.

Light tra;e! i) a !traight i)e.

A o$a?'e oect! ca!t a !hado".

QQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQ

Therefore: the ro')d !hado" o) the Moo) i! the !hado" of the Earth.


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#. The rond shado0 on the !oon drin a lnar eclipse is prodced #y the earth

#ein in line 0ith the sn.

c. There is heli! in the sn.

d. The s!allest part of an ele!ent is an ato!.

e. The !oon has no at!osphere.

f. The distance to the nearest star is a#ot @ liht years.

. =or o0n e/a!ples.

or each try to ie an eidential ar!ent for #eliein these cont as 'scientific

facts. 7f corse% Aristotle cold hae only ans0ered the first t0o.

D. 8o0% do yo thin&% Aristotle 0old hae e/plained the !otion of an oscillatin

pendl!E Re!e!#er% he classified !otion into t0o cateories5 natural motion andviolent motion.

Oestions for the !odern physics stdent5

1. =o are thro0in a s!all !etallic sphere straiht p in to the air. What force acts on

the #all5

a. -st after #ein released%

#. at the !idpoint oin p%

c. at the hihest point of the tra-ectory% and

d. on the 0ay #ac& to0ard the rondE

8o0 0old Aristotle e/plain this !otionE

Fig. 1+ Three $o!itio)! for a a thro") ;ertica9. I

2C


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2. A roc&et is in deep space% leain the solar syste!. We &no0 that to acco!plish that%

the roc&et !st leae the earth 0ith a elocity of at least 33 &!tho'ght e=$erime)t.

21

http://en.wikipedia.org/wiki/Jean_Buridanhttp://en.wikipedia.org/wiki/Nicolas_Oresmehttp://www-history.mcs.st-and.ac.uk/Biographies/Oresme.htmlhttp://en.wikipedia.org/wiki/Jean_Buridanhttp://en.wikipedia.org/wiki/Nicolas_Oresmehttp://www-history.mcs.st-and.ac.uk/Biographies/Oresme.html


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'Thoht $/peri!ents for the stdent. ('Thoht e/peri!ents 0ill #e defined and

discssed a little later)

1. I!aine a tnnel throh the earth% as sho0n in i. 1. If yo dropped a heay

o#-ect fro! the srface of the earth do0n this tnnel% descri#e the !otion. 6elect

frictional effects.

2. 6o0 i!aine a pendl!% 0ith the lenth of the radis of the earth. 4o!pare the

!otion of the pendl! to that of an o#-ect fallin throh the earth.

Fig. 1: Motio) i) a t'))e thro'gh the Earth

6ote5 This pro#le! 0ill #e considered aain% after discssin 6e0tons physics.

%aieo de!crie! the motio) of a $e)d''m

IL 2 ,+3- +++ A 0ell 0ritten short history of Galileo and the pendl!

IL + ,


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9ac&rond to class actiities and frther discssions

Galileo chaned the approach to stdyin !otion and !oed fro! a alitatie to a antitatie

description. 8e stdied Aristotle and 0as acainted 0ith the ideas of 9ridan and 7res!e. In

fact% one of the reslts of !edieal physics of !otion 0as the !ean speed theore! that Galileo

sed as the startin point for his !athe!atical description of free fall.

%aieo7! !cie)tific thi)@i)g:

ollo0in Aristotle% Galileo declared that the 0orld is rational and cold #e nderstood. 9t he

0ent frther. 8e #elieed that5

&he world must be studied not through secondary )ualities but through primary )ualities

imbedded in the language of mathematics.$/a!ples of secondary alities are color% shape% s!ell and the feelin of hot and cold.

*ri!ary alities are lenth% speed% acceleration and te!peratre. Galileo ared that secondary

alities are not !easra#le #t the pri!ary ones are.

Galileo 0ent #eyond Aristotles approach of findin niersal principles of loic and

physics and arin dedctiely. 8e 0as loo&in for !athe!atical (eo!etric) description of

pheno!ena (sch as free fall and the period of a pendl!) that cold #e confir!ed #y

syste!atic e/peri!entation.

Li&e Aristotle% Galileo !ade ass!ptions a#ot the 0orld. 8is fondational ass!ptions

0ere5

,iscssion5

1. Galileo follo0ed Aristotles 'rational !ethod in tryin to nderstand the pheno!ena of

the 0orld. 9t he 0ent #eyond Aristotle. or hi! 'rational !eant that pheno!ena

2D

%aieo7! 7 Fo')datio)a A!!'m$tio)! ao't Ph9!ic!:

1. The "ord ca) e ')der!tood 9 ratio)a tho'ght.

2. The a)g'age of Nat're i! the a)g'age of mathematic! ,geometr9-.

+. The "ord m'!t e !t'died o)9 thro'gh $rimar9 ,mea!'rae- ?'aitie!

of e)gth* time* area* etc.


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hori;ontal. The ertical !otion is si!ply a free fall !otion and the hori;ontal !otion is a

constant elocity !otion. 7f corse% the frictional effects of the air are nelected here.

IL +++ Tra-ectory !otion as seen #y i!pets theorists and as nderstood #y Galileo.

Fig. 13 PreB%aiea) thi)@i)g ao't $roectie motio).

These estions enerated the follo0in pro#le!s and e/peri!ents that Galileo tried to sole

and perfor!.

3. Galileo tried to nderstand !otion throh a !edi! (sch as a sphere fallin in

0ater) #t 0as na#le to descri#e this !otion. It 0as 6e0ton 0ho 0as first a#le to descri#e this

co!ple/ !otion. We 0ill see in L4* 3 that !otion the force on a car is proportional to the sare

of the elocity.

Galileo and thoht e/peri!ents (T$)..

We hae already encontered a T$ 0hen 0e discssed the 7res!es i!ainary o#-ect fallin

throh the $arth.

T$s hae a lon history and a natral connection to the deelop!ent of physical

concepts. They can #e traced #ac& to enos parado/es of !otion and Aristotles prora! ofe/plainin pheno!ena ided only #y na&ed eye o#seration and rational thoht. We 0ill se

shortly ho0 Galileo cha!pioned T$s to sho0 that !otion% as o#sered on a ship !oin 0ith a

constant speed in a straiht line% cannot #e distinished fro! !otion o#sered at rest. 8e also

presented T$s to are that the !otion of heay o#-ects in free fall are identical and constrcted

2

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ship that 0as s!oothly !oin 0ith a constant elocity% the cannon #all 0old fall

directly #elo0 hi!. Whateer the sailor 0old do% ass!in a s!ooth constant

elocity in a straiht line% he 0old not #e a#le to tell that the ship 0as in !otion if he

confined his attention to the ship. "sin !odern ter!inoloy% Galileo ths clai!ed

that there is no distinction between constant motion and rest. All of s hae

e/perienced this 0hen traelin in train on a leel trac& !oin 0ith a constant

speed.

Fig. 2< A Jtho'ght e=$erime)t7: 5e!t a)d co)!ta)t motio) are e?'i;ae)t.

IL / +++ An e/cellent applet to illstrate Galilean inertia.

3% To illstrate the idea of inertia% Galileo first i!ained a #all rollin do0n an inclined

plane and then rollin p another one . See IL . 8e ared that 0hen the #all finally rolls

on a leel srface% in the ideal case 0hen there is no resistance% it 0old roll arond the

earth and contine to do so foreer. See i. 21.

2:

http://id.mind.net/~zona/mstm/physics/mechanics/forces/galileo/galileoInertia.htmlhttp://id.mind.net/~zona/mstm/physics/mechanics/forces/galileo/galileoInertia.html


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Fig. 21 %aiea) I)ertia

IL 0 &&& An adanced discssion of inertia and !otion% 0ith history.

Oestions Galileo as&ed5

1. 8o0 does the speed of a freely fallin o#-ect aryE 4an I e/press this as a

proportionality state!entE

2. 8o0 does the period of a pendl! ary 0ith the lenthE

3. 8o0 can I descri#e the !otion of a hori;ontally pro-ected o#-ectE

8o0 did Galileo ans0er these estionsE 8o0 did he are and 0hat e/peri!ent did

he perfor!E Ans0er these estions the 0ay Galileo 0old hae ans0ered the!.

Fig. 22 %aieo determi)e! the acceeratio) of free fa.

3C

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Research pro#le!s for Galileo5

1. Galileo proed that the ti!e of descent of an o#-ect (on a frictionless srface) alon

an incline represented #y any chord (see fire 1?) is the sa!e and is eal to the

ti!e it 0old ta&e for an o#-ect to fall throh a distance of t0ice the radis.

Fig. 2+ %aieo7! geometric arg'me)t

2. Galileo sho0ed that the period is proportional to the sare root of the lenth and

#elieed that the arc represented the least ti!e (#rachistochrone) of an o#-ect

descendin alon a frictionless path in a ertical plane.

"sin a s&etch% sho0 Galileos ar!ent.

Oestions for the stdent5

The follo0in estions are #ased on an the e/cellent internet lin&% IL @@. . Stdy and listen to

the Pideo in 0hich a replication of Galileos fa!os inclined plane e/peri!ent can #e seen.

IL ,1- ++ Srey of the inclined plane

IL ,- ++++ An e/cellent ideo replicatin Galileos inclined plane e/peri!ent

1. 8o0 did Galileo slo0 do0n !otion so he cold !easre the !otion of a fallin

o#-ectE What 0as this ar!ent to sho0 that the !otion alon any inclined plane is

al0ays a !otion descri#ed #y constant accelerationE

2. 8o0 is the ti!e of descent !easred in the ideoE

3. 8o0 did Galileo actally !easre the ti!e interalsE

31

http://en.wikipedia.org/wiki/Inclined_planehttp://www.anselm.edu/homepage/dbanach/h-galileo-experiments-instructions.htmhttp://en.wikipedia.org/wiki/Inclined_planehttp://www.anselm.edu/homepage/dbanach/h-galileo-experiments-instructions.htm


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2. Sho0 that the acceleration for 3CU 0old #e of % or a#ot D !


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Test this state!ent #y Mersenne sin a pendl! of a#ot 1! lenth and deter!ine

the period for a s!all anle (a#ot D derees) and a lare anle (a#ot @D derees)

and co!!ent.

A special research e/peri!ent for Galileo It

is not 0ell &no0n% that Galileo sed the pendl! to !easre free fall directly . Galileo &ne0

that the period of the pendl! 0as nearly constant (for s!all anles of displace!ent) and sed

this to !easre the actal ti!e it too& for a !etallic sphere to fall fro! a heiht of a#ot 3 !eters

(of corse% he sed cubits% not !eters). 8e fond the ratio of the distance fallen fro! rest to the

lenth of the pendl! that 0as ti!in that fall #y s0inin the ertical throh a s!all arc.

(See fire 2C.)

A pendulum was held out from the vertical board and then released simultaneouslywith the dropping of a small metallic sphere. Galileo ad(usted the height from

which the metallic sphere fell until the 5thud5 of the sphere hitting the floor

coincided with that of the pendulum hitting the wall. Galileo found that the ratio

was 1.16!.

Research e/peri!ent for the stdent5

Try to replicate this e/peri!ent 0ith another stdent. (This e/peri!ent shold pro#a#ly #e done

for the 0hole class #y the instrctor% assisted #y stdents). =o can easily replicate this cleer

and si!ple e/peri!ent #y Galileo. "se a lenth of the pendl! of #et0een 2 and 3 !eters. ind

the ratio of the heiht fro! 0hich the sphere fell to the lenth of the pendl!. =o !st% of

corse% re!e!#er that Galileo did not hae a 0atch to ti!e the period of the pendl!. 8o0 did

he find the period of the pendl!E

3@


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H'9ge)! '!e! the $e)d''m a! a re!earch i)!tr'me)t

IL /+ ,/- +++ Infor!ation on !otion of the pendl! and the !otion alon cres% li&e the

cycloid

IL / ,0- +++ $/peri!ental setp to sho0 different paths yield different ti!es throh the

sa!e chane in heiht

IL // &&& A ery ood discssion of the !otion alon a cycloid

It is sested that stdents #ein 0ith the presentation of the Co);er!atio) "ith H'9ge)!.

The ,tch natral philosopher 8yens 0as the first to find the !odern for!la% na!ely that

T H 29 < l for the si!ple pendl! and also the first to 0rite the !athe!atical state!ent for

'centrifal acceleration as a H 2 < R% arond 1C. 8e sed lon and heay pendla to

deter!ine the ale of raitational acceleration. 8e later correlated latitde and the local ale

of to test his ideas. 8yens 0as also the first sho0 (eo!etrically) that the path alon 0hich a

pendl! 0old sho0 isochronos !otion 0as a cycloid. Later (a#ot 1CC)% Johannes

9ernolli% 6e0ton and others sho0ed that not only is this path isochronos #t it is also the least

ti!e descent of an o#-ect in a ertical plane #et0een t0o points.

IL /0 ,- && A #rief #ioraphy of 8yens

Fig. 20 The co)ica $e)d''m

IL / ,3- ++++ Applet of the conical pendl!

IL / ,/


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*ro#le!s for the stdent5

1. ind the lenth of the pendl! that has a period of 1 second.

2. Sho0 that a 1 !eter pendl! has a period of close to e/actly 2 seconds.

3. 7n a certain planet the period of a 1 ! pendl! 1.DC seconds. What is the

raitational attraction on the srface !easred in in !fire accracy in the period.

3. =o hae landed on a planet and yo 0ant to find ot 0hat the local raity is

!easred in !


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5oert Hoo@e di!co;er! the force a" that $rod'ce! !im$e harmo)ic motio)

Fig. 2: Image Hoo@e dre" '!i)g hi! micro!co$e

Fig. 2: Hoo@e7! micro!co$e

IL / , /2- +++ Sorce of i. 22

IL /3 ,/+- + Sorce of i. 23

IL 0< ,/- +++ 9rief #ioraphy of 8oo&e

IL 01 ,//- +++ A ery co!prehensie discssion of 8oo&e5 #ioraphy and his 0or&)

Ro#ert 8oo&e sed his la0 ( H >&/) to sho0 that si!ple har!onic !otion (S8M)% li&e that of

the pendl!% or an oscillatin !ass attached to a sprin% arises 0hen this la0 holds. 8is

scientific #attles 0ith 6e0ton 0ere leendary. When 6e0ton #eca!e the president of the Royal

Society in 1CD% he e/pned all esties of 8oo&e. Ths 0e hae no li&eness of hi!. We

identify Ro#ert 8oo&e #y the fa!os dra0in he !ade in his reoltionary 'Microraphia that

he p#lished at the ae of 3C.

3:

http://en.wikipedia.org/wiki/Image:Flea-Hooke.gifhttp://www.ucmp.berkeley.edu/history/images/hookemicro.gifhttp://www.ucmp.berkeley.edu/history/hooke.htmlhttp://en.wikipedia.org/wiki/Robert_Hookehttp://en.wikipedia.org/wiki/Image:Flea-Hooke.gifhttp://www.ucmp.berkeley.edu/history/images/hookemicro.gifhttp://www.ucmp.berkeley.edu/history/hooke.htmlhttp://en.wikipedia.org/wiki/Robert_Hooke


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Fig. 23: A !$ri)g a)d a $e)d''m o!ciati)g i) ')i!o)

An e/peri!ent 8oo&e perfor!ed5

1. 8oo&e had to sho0 that 0hen a sprin 0as e/tended the lenth of e/tension 0as

directly proportional to the force applied. 8e soon discoered that this la0 only holds

0ithin the li!it of 0hat 0e no0 call the 'elastic li!it.

Research e/peri!ent for stdents:

1. Attach a sprin scale to a heay pendl! and pll the pendl! to a#ot 1


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pendl! that s0ins 0ith the sa!e period.

Fig. +< E=am$e! of 4HM: !$ri)g* $e)d''m* i?'id i) a UBt'e*

"edge oc@* roc@i)g oat

IL 02 ,/0- +++ 8oo&es la0 appletIL 0+ ,/- +++ Illstrations of 8oo&es la0

IL 0 ,/- +++ Adanced discssion of S8M

IL 0/ ,/3- +++ IA for the pendl! and S8M

IL 00 ,0


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Tae I

,iscssion5

1. Li&e Galileo% 6e0ton #elieed that the lanae of science (physics) is !athe!atics.We !st% ho0eer% #e carefl here and re!e!#er that 0hen Galileo says

'!athe!atics he !eans eo!etry and $clidean ratios. As 0e hae seen% in his

ar!ents a#ot !otion he ses eo!etry and ratios only% and not ale#raic

eations% as 0e no0 do. 6e0ton in his Principia also ses co!plicated eo!etry

!ost of ti!e% #t occasionally does present ale#raic e/pressions and eations.

Today 0e se predo!inantly ale#raic e/pressions to descri#e !otion.

2. The raitational force #et0een !asses% say the !oon and the earth% act

3. Space can #e descri#ed #y $clidean eo!etry. That !eans that if yo !easred the

anles in a trianle that connected three points in space% ho0eer far apart 0old

al0ays contain three anles that added p to 3CU. 7f corse% if yo !easred

the anles of a trianle connectin Toronto% 6e0 =or&% and 4hicao% the s! of the

anles 0old #e !ore than 3CU. (We 0ill discss this later in !ore detail).

@. 6e0ton #elieed that5

Absolute# true# and mathematical time # from its own nature# passes e)uably withoutrelation the anything e-ternal# and thus without reference to any change or way of

measuring of time 3e.g.# the hour# day# month# or year4. (Ta&en fro! the Principia).

That !eans that if a !easre a ti!e interal in !y la#oratory% in a satellite or#itin the

earth 0e 0old !easre e/actly the sa!e ti!e interal. $instein sho0ed in 1:CD that

@3

Ne"to)7! Fo')datio)a A!!'m$tio)! for $h9!ic!:

1. Mathematic! i! the core of de!cri$tio) a)d e=$a)atio) i) $h9!ic!

2. Ma!! $oi)t! i)teract 9 "a9 of a ce)tra force.

+ 4$ace i! E'cidia).

Time i! a!o'te.

/. Ma!! $oi)t! i)teract i)!ta)ta)eo'!9.


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ti!e is not a#solte. We 0ill discss the ar!ents of $instein in L4* 1C.

D. Accordin to 6e0ton% a planet or#itin the sn 0old feel the effect of the

raitational pll instantaneosly% that is% as if the raitational force traeled 0ith an

infinite speed. 6e0ton did not li&e this ass!ption #t 0as forced to !a&e it. 8e

called it an 'occlt pheno!enon. 6e0ton said5

&hat one body may act upon another at a distance through a vacuum without the

mediation of anything else# by and through which their action and force may be conveyed

from one another# is to me so great an absurdity that# ' believe# no man who has in

philosophic matters a competent faculty of thinking could ever fall into it.@

We 0ill discss this pheno!enon in L4* 1C.

Tae II

,iscssion5

1. The fnda!ental antities 6e0ton are an e/tension of those Galileo sed to

descri#e &ine!atics% that is% the stdy of !otion 0ithot considerin the

forces inoled. Galileo introdced the definition of speed% nifor! acceleration

and distance% sho0ed ho0 these 0ere related in free fall !otion.

The ne0 antities that 6e0ton needed to introdce 0ere the ideas of force

!o!ent! and i!plse.

@@

Fo')datio)a G'e!tio)! for Ne"to):

1. hat are the f')dame)ta $h9!ica ?'a)titie! i) term! of "hich "e ca)

de!crie the d9)amic! of free fa* coi!io) a)d ce)tri$eta force

2. Ho" !ho'd "e defi)e acceerati)g force A! m ;* or F m;2r% or

F ma

+. Ho" ca) "e de!crie the gra;itatio)a force a)d the motio) of a $a)et

aro')d the !'). Do the a"! of motio) o) earth ,terre!tria a"!- a!o go;er) the motio)

i) of the $a)et! ,cee!tia motio)-

http://en.wikipedia.org/wiki/Newton's_law_of_universal_gravitation#_note-3%23_note-3http://en.wikipedia.org/wiki/Newton's_law_of_universal_gravitation#_note-3%23_note-3


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2. 6e0ton strled 0ith these different 0ays of e/pressin the idea of force% as 0e 0ill

see.

3. 7ne of 6e0tons reat achiee!ents 0as that% sin his three la0s of !otion and the

inerse sare la0 of raitational attraction% he 0as a#le to calclate the

!otion of the planets. We 0ill discss this a little later.

@. Aristotle #elieed that the la0s of !otion on earth (terrestrial physics) and the la0s of

!otion idin the planets (celestial physics) are different. 6e0ton% ho0eer% sho0ed

that the sa!e la0s apply to the earth as 0ell as to celestial !otion.

The central pro#le!s he soled 0ere5

Tae III:

@D

Ne"to)7! Fo')datio)a Ph9!ic! Proem!:

1. To !ho" that %aieo7! a" of free fa i! '!t a !$ecia ca!e ge)erated 9 the !eco)d

a" of motio).

2. 4ho" that the motio) of a $e)d''m ,for !ma a)ge!- ca) e de!cried 9 '!i)g

Hoo@e7! a" a)d the !eco)d a" of motio).

+. To fi)d the mathematica e=$re!!io) for force i) coi!io)* !traight i)e motio)* a)d

i) )o)Bi)ear motio)* a! i) a circe or a) ei$!e.

. To fi)d the )at're of the $ath of a $a)et oe9i)g the i);er!e !?'are force a".

/. To !ho" that e$er7! a"! are deri;ae from the a"! of motio) a)d the

')i;er!a gra;itatio)a a".

0. To !ho" that a !$herica9 homoge)eo'!9 di!tri'ted ma!! ha! the !ame

gra;itatio)a effect a! a $oi)t e?'i;ae)t ma!!.

. To !ho" that the gra;itatio)a ma!! of a) oect i! the !ame a! it! i)ertia ma!!.


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,iscssion5

1. $/a!inin free fall% sin 6e0tons second e/pressed as H !a% 0e find that the

!otion of the freely fallin heay o#-ect !st #e sch that a H . Sho0 this.

1. "se a force diara! to sho0 that the restorin force (prodced #y raity) actin

on an oscillatin pendl! can #e considered linear% p to a#ot ?>1C derees

(See i. ). That !eans that 0e can consider the !otion prodced #y a constant

n#alanced force on an oscillatin pendl! can #e considered linear% p to a#ot

?>1C derees (See i. ). That !eans that 0e can consider the !otion prodced

#y a constant n#alanced force.

2. 6e0tons ar!ent here can #e nderstood if 0e considerer circlar !otion. See

IL . A ery co!prehensie discssion is ien 9. 4ohens article

(see references)

3. 6e0ton sho0ed in the Principa that an inerse sare la0 for raitational

attraction and sin the sn as the sorce of a central force 0ill necessarily

prodce an ellipse. =o can conslt te/t #oo&s that deal 0ith classical !echanics.

Also see IL .

@. We hae already discssed this. See.

D. 6e0ton soled this pro#le! sin his discoery of calcls (0hat he called

'fl/ions). The !ain idea here is that% if 0e consider the earth as a perfect

Sphere% 0ith constant density% it can #e sho0n that the raitational effect on an

o#-ect otside the earth (or on the srface of the earth) 0old #e the sa!e

Yas if all the !ass 0ere concentrated at the center. This effect is called 6e0tons

shell theore!.

See i. and conslt IL and IL

. We sho0ed (see ,iscssions 1.) that in free fall a heay o#-ect falls 0ith a

constant acceleration of a H . Sho0 that this can only #e tre if the

raitational !ass ! is eal to the inertial !ass !i .

@


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IL < + + An adanced discssion of 6e0tons Yshell theore!.

IL 1 ++ A #rief history of 6e0tons !otiation to 0rite the Principia.

Tae I:

Fo')datio)a E=$erime)t! for Ne"to)ia) D9)amic!:

1. I);e!tigati)g the d9)amic! of the !im$e $e)d''m* '!i)g Hoo@e7! a".

2. I);e!tigati)g coi!io) ,headBo)- "ith $e)d'a.

+. I);e!tigati)g the eha;ior of a co)ica $e)d''m.

. U!i)g At"ood7! machi)e to te!t the !eco)d a" ,F ma- , 1


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Tae I:

Ne"to)7! G'e!tio)! for the F't're:

1. h9 are i)ertia a)d gra;itatio)a ma!!e! e?'i;ae)t

2. Ca) force a)d ma!! e e=$re!!ed i) a )o)Bcirc'ar "a9

+. >Ho" ca) "e ?'a)titati;e9 demo)!trate the "a9 $artice! of matter i) motio)

e)do"ed "ith force! $rod'ce the o!er;ed $he)ome)a i) )at're* for oth !ma

a)d argeB!cae $he)ome)a

,iscssion.

1. The fact that heay o#-ects fall 0ith the sa!e acceleration in the earths raity !eans

sests that inertial !ass of an o#-ect (!i) and its raitational !ass (!i) are eialent.

6e0ton tested this sin a pendl! 0ith different !asses and different !aterials

(0ooden and arios !etallic spheres) and fond that the period of the pendl! for a

ien settin does not chane. In L4* 1C 0e 0ill discss ho0 this eialence #eca!ethe fondation principle for $insteins General Theory of Graity.

2. 6e0ton nderstood that% sin his second la0% H !a (or% as he e/pressed it

H X (!) < Xt ) can #e sed to e/press !ass as ! H < a. 8o0 can yo escape this

circlar definition% connectin !ass and forceE 8e did not &no0 the ans0er.

3. This estion 0as as&ed #y 6e0ton to0ard the end of his life. 8e 0as only acainted

0ith raitational force. Static electric and !anetic forces 0ere &no0n% #t not

nderstood. In !odern physics 0e hae for forces that hae #een identified. What are

these forcesE Loo& p the ans0er to this estion and discss it 0ith yor instrctor and

other stdents.

Oestions for the stdent% in reference to the a#oe.

The follo0in estions shold #e discssed in s!all rops% and then later a report shold #e

@?


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'f a bucket is hanging from a very long cord and is continually turned around

until the cord becomes twisted tight# and if the bucket is thereupon filled with

water and is at rest along with the water and then# by some sudden force# is made

to turn around in the opposite direction and# as the cord unwinds# perseveres for a

while in this motion7 (6e0ton 12 1:::% @12>@13).

Fig. +2: Ne"to)7! 8'c@et

Fig. ++ 8eha;ior of ater i) the Ne"to)7! 8'c@et

IL 2 ,0- &&+ Sorce of i. 32.

IL + ,0/- ++++ An e/cellent sorce for thoht e/peri!ents F sorce of i. 3CIL ,00- +++ Adanced discssion of the thoht e/peri!ent

IL / ,0- ++ Adanced discssion of 6e0tons thoht e/peri!ents and Machs principle

DC

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6e0ton contines to descri#e for staes of the #c&et e/peri!ent. Initially% the #c&et is filled

0ith 0ater% the cord has not #een released% and the srface of the 0ater is leel (ire @.1). In

the second stae the cord #eins to n0ind% there is a relatie !otion #et0een the 0ater and the

#c&et% and the 0ater is o#sered to #e leel (ire @.2). 6e/t% the 0ater catches p 0ith the

sides of the #c&et and there is no relatie !otion #et0een the 0ater and the #c&et% and the

0ater is o#sered to #e concae (ire @.3). inally% the #c&et stops% the 0ater is spinnin

relatie to the #c&et and the 0ater is o#sered to #e concae (ire @.@). At stae 2 and at

stae @% the #c&et and the 0ater are in !otion relatie to each other. 8o0eer% in the first case

the 0ater has a leel srface and in the second case the 0ater has a concae srface. It appeared

that the shape of the srface of the 0ater 0as not dependent on the relatie !otion of the 0ater

and the sides of the #c&et. 6e0ton conclded that it 0as the spin of the 0ater 0ith respect toa#solte space that !attered. If the 0ater 0as not spinnin 0ith respect to a#solte space then its

srface 0as leel% #t 0hen it 0as spinnin 0ith respect to a#solte space% the srface of the

0ater 0as concae.

6e0ton e/peri!ented 0ith pendla5

1. 6e0ton sho0ed that the periodic s0in of a pendl!% o#eyin 8oo&es la0

illstrated the second la0 H !a% 0here H >&/. ree fall itself 0as rearded as a

special case of the second la0% 0here inertial and raitational !asses are considered

eialent. ,iscss 0ith yor fello0 stdents and yor instrctor.

2. 6e0ton tested the eialence of inertial and raitational !ass #y sin pendla of

the sa!e 0eiht #t !ade fro! different !aterials. "sin the eation of the period

of the pendl! for s!all anles% sho0 ho0 this can #e done.

3. 6e0ton also tested his third la0B co!!only 0ritten as 'for eery action there is eal

and opposite reaction. An e/cerpt fro! the Principia is ien #elo0% as 0ell as the

pictre of the e/peri!ent that he perfor!ed. ,iscss 6e0tons e/peri!ent and his

!athe!atical and eo!etric ar!ent. (See fire 3@).

IL 0 ,0- & Sorce of i. 3@

D1

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Fig. +: Ne"to)7! !@etch of coidi)g $e)d'a from the Principia

3. What forces act on a pendl!% at the !o!ent it is released% at the !idpoint% at

the #otto!E This estion 0as posed earlier% #t &no0 0e shold #e a#le to ans0er it

ided #y an nderstandin of 6e0tons la0s of !otion. 6ote5 this is a difficlt estion

and the stdents shold discss it a!on the!seles as 0ell as 0ith the the instrctor.

IL ,03- & Sorce of i. 32

D2

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Fig. +/: Force acti)g o) a $e)d''m II

(6ote5 This is a difficlt pro#le!. Stdents shold #e a#le to ans0er this at the

end of this L4*).

6e0tons third la0 of !otion and the conseration of linear !o!ent! principle.

6e0ton did not se the idea of conseration of linear !o!ent!% or the idea of &inetic enery%

6or the idea of raitational potential enery in the Principia% as 0e do today. 8is third la0 as

de!onstrated #y the collision of t0o 0ooden spheres% as he analy;ed this% a!onted to the sa!e

thin. Today 0e 0old say5 In an elastic collision the !e!ento #efore i!pact are eal to the

!o!enta after i!pact. Mo!ent! is a ector antity and &inetic enery is a scalar antity.

The follo0in pro#le! 0ill illstrate ho0 these antities% !o!ent! (* H !) and &inetic

enery (1on) elastic collision. The 1& #all is plled p hih

enoh so that it collides 0ith the 2 & #all 0ith a elocity of 3!


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IL ,


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cold sho0 that an inerse sare la0 descri#es the force of raity #et0een the sn and

the planets.

( eplers third la05 T2 < R 2 H a constant% 0here T is the period of the planets and R

the distance to the center of the earth and the e/pression for centripetal acceleration for

circlar !otion is ac H 2 < R% 0here is the elocity of the satellite and R the distance

#et0een the planet and the sn. Re!e!#er% eplers la0s apply to the elliptical !otion

of planets% #t it also applies to circlar !otion% #ecase a circle is -st a special case of

an ellipse).

3. After findin his forth la0% (na!ely the inerse sare la0 of raity)% 6e0ton applied

it to the !otion of the !oon arond the earth.Try to follo0 the ar!ent in IL to find ot ho0 6e0ton calclated the period of the

!oon. Re!e!#er that this is a !odern ersion of ho0 6e0ton !ade this calclation.

6ote5 We 0ill se these ideas aain in reater detail in in L4* ?% 0here 0e 0ill discss the

!otion of Mars and ho0 to et to there.

*ro#le!s for the stdent

1. Sho0 that the period of a satellite -st a#oe the earths srface 0old #e a#ot ?@

!intes.

2. Sho0 that a pendl! 0old hae a period of ?@ !intes if its lenth 0ere the radis

of the earth. Why

3. 6o0 loo& at the !otion of the pendl! throh the eyes of 6e0ton5 "se his physics

to e/plain this !otion. Specifically% sho0 the n#alanced force actin on the

pendl! at A% 9% 4% ,% and $. A is the startin point and 4 the lo0est point. See

fire 3@ #elo0

@.. inally% co!pare the e/planations for the !otion of the pendl!% as seen #y

Aristotle% 7res!e% Galileo% and 6e0ton.

(This is a ood e/ercise to s ho0 that 0e see pheno!ena arond s throh or ideas

and concepts.

D


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Fig. + Force! acti)g o) a) o!ciati)g $e)d''m. III.

IL ,+- ++++ Applet on pendl! !otion

%eorge At"ood i);e)t! hi! machi)e to e '!ed a! a re!earch i)!tr'me)t

IL / ,- +++ IA for At0oods !achine

Geore At0ood (1@D>1?C) 0as a cele#rated lectrer at Trinity 4ollee% $nland. 8e 0as #est

&no0n for his #oo&% A &reatise of ectilinear otion% a conte!porary te/t#oo& on 6e0tonian

dyna!ics. 8e 0as the inentor of the fa!os At0oods Machine% sed ntil the #einnin of the

2Cth centry for e/peri!ents in 6e0tonian dyna!ics. It 0as sed for #oth research and teachin.

To find the ti!e of descent of the larer !ass he sed a pendl! cloc&. See pictre #elo0.

D?

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Fig. +3: At"ood7! Machi)e re;i!ited

IL 0 ,/- && 9ioraphy of At0ood

IL ,0- +++ ,etails and pictres of At0oods !achine

Research e/peri!ent for the stdent

1.4hec& 6e0tons second la0 of !otion #y sin a lon pendl! (a#ot 2 !) and

han it aainst a 0all so that yo can deter!ine the ti!e for 1


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Fig. < 4et'$ for !t'de)t re!earch e=$erime)t ,ao;e-


1. "sin 6e0tons second la0% sho0 that 0hen t0o !asses !1 and !2 are sspended%

0here !1 is larer than !2% then the acceleration of the syste! is ien #y

a H (!1 > !2) < (!1 Z !2).

2. What !st #e the ratio #et0een the !asses so that the acceleration of C.D E

An e/peri!ent for the stdent5

1. "se IL @% 0hich is an interactie prora! to stdy At0oods !achine% to find the

acceleration for arios !asse attached to the strins. 4o!pare so!e of these ales 0ith

those yo o#tain sin the for!la a#oe.

2. "sin a si!ple plley% set p an e/peri!ent% as sho0n In i. a#oe. *lace t0o

0eihts% one larer the other% and release the!% at least 1 ! a#oe the floor.

*redict the acceleration of the syste! and then test yor prediction. 4o!!ent.

A thoht e/peri!ent for the stdent5

4o!pare the reslt of the At0ood !achine e/peri!ent on $arth 0ith a si!ilar

e/peri!ent on the Moon. Wold yo find that the for!la for the acceleration of the

C


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syste! a H (!1 > !2) < (!1 Z !2) yo fond in pro#le! 1 a#oe 0old still 0or&E

,iscss.

Leo) Fo'ca't '!e! a $e)d''m to co)firm the rotatio) of the Earth

IL ,- +++ ,iscssion of ocalts pendl!

IL 3 ,- ++++ An e/cellent lin& 0ith a ood ideo of the ocalt pendl!

IL 3< ,3- ++ ocalt pendl! on the Soth *oleK

IL 31 ,


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and had to #e restarted eery hors. ,rin this ti!e the pendl! eered the e/pected C to

C derees cloc&0ise. In 1?D1 he ths p#lically proided the first direct proof that the earth

rotates. ocalt sho0ed that it 0as the $arth that 0as reolin and not the pendl! that

prodced the apparent rotationK See fires 3D and 3.

Fig. 1: Fo'ca't7! demo)!trate! hi! $e)d''m i) 1/1.

Fig. 2: Fo'ca't7! $e)d''m o) the North Poe

IL 32,1- ++ Sorce of i. @1.

4lass actiities and discssions

2

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1. A hndred and fifty tears #efore ocalts fa!os de!onstration of the rotation of

the $arth% 6e0ton pointed ot that if the earth rotated then it shold #e possi#le to

directly test this !otion the follo0in 0ay5 ,rop a heay o#-ect do0n a deep 0ell

and chec& the point of i!pact and see 0ere it hits the #otto! of the 0ell. (See fire

@C.) It is interestin that he failed to thin& of sin a lon pendl! to de!onstrate

the rotation of the $arth.

Fig. +: A !@etch of a fai)g oect i) a ;er9 dee$ "e.

IL 3+ ,2- An applet for the *

IL 3 ,+- A ideo of the *


1. Where 0old a heay o#-ect fall% relatie to the ertical% if the 0ell is 1CC ! deepE

8o0 cold 6e0ton hae tested thisE

2. What 0as the period of ocalts pendl! (*)E

3. irst% ie an ar!ent to sho0 that the period of rotation at any latitde % ien #y T

H 2@ h < sin is a plasi#le ans0er.

@. 8o0 lon 0old it ta&e for a * to rotate 3C % on the 6orth *oleE 7n the eatorE

D. 4alclate the period of rotation of a * in Winnipe% Toronto% on the eator% Sydney%

or the city or to0n yo lie in.

Research estions for the stdent5

3

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1. We hae tal&ed a#ot t0o !ethods (0ays) to sho0 that the $arth rotates. There are

others. ,escri#e so!e of these.

Research $/peri!ent for the stdent5

1. 4onstrct a ocalt pendl!% sin a heay #o# and a lon stron 0ire% and

conince others that the $arth rotates.

5e;i!iti)g Ore!me7! >t'))eBthro'ghBtheBearth tho'ght e=$erieme)tQ

7ne day Isaac 6e0ton receied a letter fro! Ro#ert 8oo&e. In this letter% 8oo&e otlined the

!athe!atics oernin ho0 o#-ects !iht fall if dropped throh hypothetical tnnels drilled

throh the $arth at aryin anles. Thoh it see!s that 8oo&e 0as !ostly interested in the

physics of the thoht e/peri!ent% an i!pro#a#le yet intriin idea fell ot of the data5 adi;;yinly fast transportation syste!.

See the ILs #elo05

IL 3/ +++

IL 30 +++. The co!plete history and the soltion to the pro#le! can #e fond in these lin&s.

Thoht e/peri!ents for the stdent5

1. I!aine a tnnel throh the earth% as sho0n #elo0. ,rop a heay o#-ect at A an

descri#e the !otion.

2. 6o0 i!aine a pendl!% 0ith the lenth of the radis of the earth. ,escri#e the

!otion of the pendl!. What is the period of the pendl!E

3. inally% find ot% or calclate the period of a satellite in circlar !otion% -st

a#oe the srface of the earth. 4o!pare the ti!es for the three !otions

and co!!ent.

@

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ro#otics. It is hard to i!aine a !ore !otiatin lare conte/t in 0hich to teach the fondations

of statics and dyna!ics 0ith a stron lin& to the !odern 0orld arond s.

More IL!:

IL 3 31 ++ ,iscssion of raity po0ered transportation F throh earth tnnels

IL 3 32 +++ ,etailed discssion of raity train F earth tnnel trael

IL 33 3+ ++ 9rief discssion of raity train F earth tnnel trae

C'rric''m Co))ectio)!: To e com$etedR

http://www.damninteresting.com/?p=696http://en.wikipedia.org/wiki/Gravity_trainhttp://www.math.purdue.edu/~eremenko/train.htmlhttp://www.damninteresting.com/?p=696http://en.wikipedia.org/wiki/Gravity_trainhttp://www.math.purdue.edu/~eremenko/train.html


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Co);er!atio) "ith Ari!tote: ,8ac@ to Hi!tor9 of the Pe)d''m-

,8ac@ to Pre!e)tatio) of the Co)te=t!-

I!aine that yo cold o #ac& in ti!e and hae a conersation 0ith reat physicists.7r first scientist is Aristotle. We are in Greece% in the forth centry% 94.

Aristotle appears. 8e 0al&s #ac& and forth in deep thoht. The stdent approaches hi!.

Aristotle5

Greetins !y yon friend. In anticipation of yor isit I hae #een thin&in a#ot!otion arond s. The !otion of a leaf fallin% a -aelin #ein thro0n% the !otion of acart% etcMotion in eneral is an ine/hasti#ly deep s#-ect. I nderstand that yo 0ish

to discss !otion 0ith !e.

Stdent5

=es% I do. It is #e0ilderin% I &no0. 9t yo hae clearly cateori;ed !otion. $specially%

yo hae separated the la0s of !otion in the heaens and those that oern !otion on$arth% terrestrial !otion and celestial !otion.

Aristotle5

=es. 9t let s only tal& a#ot !otion on $arth only. Aristotle takes a stone and a feather. %e looks at the student and then asks5

Which of these 0ill fall to the rond first% the stone or the featherE

Stdent5

I thin& the stone 0ill fall faster .

Aristotle5

Alriht.

%e drops the stone and the feather. As e-pected# the stone reaches the ground first.

&he student then takes a small piece of wood. %e looks at Aristotle and says/

Stdent5I hae here a s!all piece of liht 0ood. The stone is clearly !any ti!es heaier than the 0ood.

,o yo areeE

%e gives these to Aristotle. Aristotle smiles and hands them back to the student. &he studentdrops these and they seem to fall to the ground at the same time.

?


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This si!ple de!onstration sests that all heay o#-ects fall at the sa!e rate% dont yo thin&E

Aristotle5

I a! not coninced a#ot that. I thin& if 0e 0ent to a hih to0er and dropped these o#-ects% thenthe stone 0old hit the rond first.

Stdent5That !ay #e so. 8o0eer% yo cannot say that if 0e had t0o o#-ects% say t0o spheres !ade of

different !etals% #t of the sa!e si;e% and one t0ice as heay as the other% the heaier one 0old

fall t0ice as fast.

Aristotle5

Well% !ay#e not. 9t I hae o#sered heaier than 0ater o#-ects fall throh 0ater. %e picks up two metallic spheres.

Lets place these in the 0ater and allo0 the! to fall. %e walks over to a glass container and lets the spheres fall through the water.

6o0 o#sere the! as they fall throh the 0ater.

Stdent5

It loo&s li&e the heaier falls !ch faster.

Aristotle5

=o see. 4learly% heaier o#-ects fall faster in 0ater.

Stdent5

Pery i!pressie. 9t then% i!aine an o#-ect fallin to0ard the earth if there 0ere no air.

,ont yo thin& that then a feather and a stone 0old reach the rond at the sa!e ti!eE

Aristotle5

6oK That is i!possi#leK

Stdent5

9t 0hyE

Aristotle5

9ecase if there 0ere no air% 0e 0old hae a ac! a#oe the earth. And% I #eliee that

ac! cannot e/istK

Stdent5

Aristotle% 0hy do yo #eliee a ac! cannot e/istE

:


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!otion. Therefore I 0old say that it is partly natral and partly iolent !otion.

Stdent5

Than& yo ery !ch% Aristotle. This 0as ery ill!inatin.

Aristotle5

=o are 0elco!e. =o !st co!e and -oin s in !y Acade!y. *erhaps yo can contine to stdy

of the !otion of a % 0hat did yo% the pendl!.

Stdent5

Than& yo% I 0ill do so ery soon. Good #ye.

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Co);er!atio) "ith Ore!me: ,8ac@ to Hi!tor9 of the Pe)d''m-


7r stdent !eets 7res!e in his roo! at La Saint 4hapelle% in 133% 0here he 0as

eleated to the post of dean of the 4athedral of Roen.

7res!e5I a! loo&in for0ard to or discssion% yon !an. I nderstand that yo hae stdied the

physics of Aristotle.

Stdent5

Master 7res!e% than& yo for areein to hae this discssion 0ith !e. =es% I hae stdied the

physics of Aristotle% especially his ideas of !otion.

7res!e5

Wonderfl. 8e 0as a reat thin&er in eery field of stdy% especially in loic and #ioloy. 8es!iles. 9t in physics% !any of ideas 0ere estiona#le. $specially those that had to do 0ith!otion.

Stdent5=es. I% too% 0as a little p;;led 0hen Aristotle e/plains !otion #y 0ay of his ideas of iolent and

natral !otion. or e/a!ple% his e/planation of the case of !otion of a pro-ectile is in !y

opinion% do#tfl.

7res!e5

Actally% Aristotles ideas a#ot !otion 0ere first challened a#ot eiht hndred years ao #y

John *hiloponos (ofE...) 8e re-ected the Aristotelian la0 that the speed of an o#-ect dependsdirectly on the force and inersely on the resistance. Specifically% he ared that speed is

proportional to the force !ins the resistance force.

Stdent5

That of corse neatly sidesteps the pro#le! of the ac!.

7res!e5

=es% indeed. I aree 0ith *hiloposs. That !eans% of corse% that !otion in a oid (ac!)

0here there is no resistance% is possi#le.

Stdent59t 0e still hae the pro#le! of the 'case of !otion.

7res!e5

*hiloposs 0ent frther and ared that it is not the air that proides the !otie po0er to propel

a pro-ectile li&e a -aelin% #t an i!pressed force that he called i!pets. The i!pets% ho0eer%dies ot.

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Stdent5

Is there any te/t aaila#le that 0as 0ritten #y *hilopossE

7res!e5

=es% indeed.

%e picks up a heavy book and looks for a passage.8ere 0e are.

The concldin 0ords in his Commentary on Aristotle’s Physics . s!s it p 0ell5

rom these considerations and from many others we may see how impossible it is for forced

motion to be caused in the way indicated. ather is it necessary to assume that some incorporeal

motive force is imparted by the pro(ector to the pro(ectile# and that the air set in motioncontributes either nothing at all or else very little to this motion of the pro(ectile. 'f# then# forced

motion is produced as ' have suggested# it is )uite evident that if one imparts motion 5contrary tonature5 or forced motion to an arrow or a stone the same degree of motion will be produced

much more readily in a void than in a plenum. And there will be no need of any agency e-ternalto the pro(ector. . . .

Stdent58e 0as certainly ahead of his ti!e.

7res!e5I aree. 9t !y teacher% John 9ridan% deeloped the i!pets theory of *hiliposs een frther.

8e thoht that an i!pressed force on a pro-ectile 0as per!anent nless acted on #y resistances

or other forces. I hae #een deelopin 9ridans ideas a little frther. I #eliee that it is not possi#le to detect nifor! straiht>line !otion.

Stdent5

Actally% that !a&es sense to !e. When yo are on a ship on a cal! day% yo are not a0are of!otion.

7res!e$/actly. This is 0hy I #eliee that it !ay #e the earth that is rotatin and not the sn. 9t I hae

had pro#le!s 0ith that idea #ecase it aainst the teachins of the 4hrch. Re!e!#er% I a! a

9ishop in the 4atholic 4hrch. %e hesitates and then continues

8ere is an interestin idea% !y yon friend. I!aine a tnnel throh the earth. Let !e sho0

yo. %e draws a sketch in the sand

6o0 i!aine a heay o#-ect #ein dropped into the tnnel. Also i!aine that there is no

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resistance offered to the !otion. What &ind of !otion 0old yo e/pectE

Stdent5

Let !e thin&. If 0e ass!ed the earth as a perfect sphere% that is% the !aterial distri#tion 0ere

the sa!e throhot% the o#-ect shold arrie at the other end and stop. Then the !otion sholdrepeat itself. Why% the o#-ect 0old !oe li&e a pendl!.

7res!e5Pery ood. Actally% this idea is not oriinal 0ith !e.

>resme opens a book and shows the student a passage.

My colleae Al#ert of Sa/ony discssed this idea first. Let !e sho0 yo the te/t.

'f the earth were completely perforated# and through that hole a heavy body were descending

)uite rapidly toward the center# then when the center of gravity 3medium gravitatis4 of thedescending body was at the center of the world# that body would be moved on still further

9beyond the center< in the other direction# i.e.# toward the heavens# because of the impetus in it

not yet corrupted. And# in so ascending# when the impetus would be spent# it would converselydescend. And in such a descent# it would again ac)uire unto itself a certain small impetus by

which it would be moved again beyond the center. When this impetus was spent# it would

descend again. And so it would be moved# oscillating 3titubando4 about the center until there nolonger would be any such impetus in it# and then it would come to rest.

Stdent5Wonderfl. So no0 0e hae a #etter e/planation of the !otion of the pendl! than the one

Aristotle ae s.

7res!e55Indeed. The !otion of the i!ainary o#-ect fallin throh a tnnel then can #e de!onstrated #y

the !otion of a pendl!.

Stdent5

9t 0e still cannot !a&e predictions a#ot the ti!e of oscillation inoled for a ien pendl!.

or e/a!ple% ho0 !any s0ins 0old a pendl! of lenth of 1CC c#its !a&e in ien ati!eE

7res!e5

May#e yo 0ill #e a#le to do that 0hen yo #eco!e a natral philosopher.I !st o no0. I hear the #ells callin for espers. Good #ye.

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Co);er!atio) "ith %aieo: ,8ac@ to Hi!tor9 of the Pe)d''m-


7r ti!e traeler finds hi!self in Galileos la#oratory in the year 11C. 8e loo&s arond as

Galileo sddenly appears. There is an inclined plane% pendla and !any instr!entson the ta#le. 7ne can see a 0ater cloc& and a itar. At the ery front of the ta#le there is a s!all

telescope. Galileo is 0ritin. 8e loo&s p.

Galileo5

Welco!e to !y la#oratory% yon !an. I hae -st co!pleted !y !anscript &he ,tarry essenger % in 0hich I descri#e the !otion of the !oons of Jpiter. A ery reoltionary

astrono!ical discoery. 8o0eer% in yor letter yo said yo are interested in the physics of

terrestrial !otion.

Stdent5

Well% if yo hae ti!e 0e cold also discss yor 0or& on the !otion in the heaens.

Galileo5

Well% 0e 0ill see.

I a! -st 0or&in on an interestin pro#le!. %e shows the student a pendulum# points to a very long inclined plane and a water clock.

I hae sho0n that if a !etallic sphere rolls do0n an inclined plane li&e this

%e demonstrates the motion.

The distance coered is proportional to the sare of the elapsed ti!e.

So that if the sphere traels one c#it in one second% it 0ill trael for c#its in t0o seconds% ninec#its in three seconds% etc.

Stdent58o0 do yo !easre the elapsed ti!e% Sinor GalileoE

Galileo5I hae sed !y plse at the #einnin% #t I fond that a 0ater cloc& ies the #est reslt.

%e points to a large cylinder

Stdent5

I see. So ol!e of 0ater is eated 0ith ti!e.

Galileo5=es. >>>

Stdent5,id yo -st o ahead and !easre distance traeled aainst ti!e elapsed to arrie at this resltE

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Galileo5

Well% I cold hae done it that 0ay. 9t that reslt 0old not hae told !e ho0 speed and ti!e

are related. 4an yo see thatE

Stdent5

I thin& so. 9t ho0 !any different 0ays cold a #all roll as far as the relationship #et0een ti!e

and speed is concernedE

Galileo5

At least three different 0ays. I called these !y hypotheses a#ot free fall.Galileo stops for a moment and then continues.

Stdent59t yo are not !easrin the free fall directly.

Galileo5=o see% I ared that I cannot !easre the ti!e elapsed in free fall directly% #ecase of the s!all

ti!e interal inoled.

%e looks at the student ..

Stdent5

=es% I can see that. And then yo e/trapolate it to :C deree ertical for free fall. Pery

I!ainatie.

Galileo5

Than& yo. 8o0eer% I cold !easre the elapsed ti!e for a freely !oin sphere alonan inclined plane. I then ared that free fall !otion 0as the sa!e as the !otion alon the

inclined plane. I si!ply 'dilted raity% as it 0ere.

&hey both laugh.

Stdent5

Pery cleer% SinorK

Galileo5

Than& yo. >>>To co!e #ac& to or pro#le!. Try to ess these three hypotheses of

!otion% yon !an.

Stdent5

Lets see. We cold ess that the speed of a freely fallin o#-ect aries accordin to the

distance coered.

Galileo5

Good. "nfortnately% this hypothesis leads to an a#srdity. If yo insisted% I cold proe


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this . 9t lets inore this one.

Stdent5

Than& yo. The other hypothesis cold #e that !otion is sch that the speed ariesaccordin to ti!e.

Galileo5Pery ood. There is a third one #t it trns ot that it is really eialent to this one.

%e stops for a moment.

It is ite easy to sho0 that if the speed aries as the ti!e elapsed then the distance !st

ary as the sare of the elapsed ti!e.

Stdent5I 0ish Aristotle and 7res!e 0ere here to hear this.

Galileo5Well% 7res!e 0old #e i!pressed #t I do not thin& Aristotle 0old #e. I do not thin& he 0old

hae #een interested in a antitatie !easre!ent of !otion.

Galileo walks over to the table and picks up an orange and grape.

9t this !ay hae #affled Aristotle.

%e holds the orange about two cubits above the table and then drops then at the same time.

So% ho0 0old hae Aristotle reacted to this si!ple de!onstration% !y yon friendE

StdentBI dont &no0. 8e 0old pro#a#ly hae said that if yo 0ent to the top of a tall #ildin and

dropped these the orane 0old arrie first.

Galileo5Well% 0e 0ont do that% een thoh the Leanin To0er is close #y.>>>9t let s retrn to the

pendl!.

%e makes the pendulum oscillate and looks at this motion# lost in deep thought.

As a yon !an% sittin in a cathedral drin !ass !y thohts drifted to other thins. I noticed

that identical chandeliers in a chrch s0in 0ith the sa!e period% no !atter 0hat the a!plitdeis.

%e e-plains the idea of a period# an amplitude and demonstrates it.

I also noticed that the !ass of the chandelier doe not affect the period. I 0as coninced then andthere that neither !ass nor a!plitde affected the period.

%e smiles and then continues.


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7f corse% I sed !y plse as a ti!in deice.

Stdent5

That is ery crios and ne/pected. Actally% the discoery that !ass !a&es no difference

follo0s fro! the fact that all heay !asses fall at the sa!e rate. 9t that the a!plitde does noteffect the period I find astonishin.

Galileo5=es% so did I. 6e/t I 0anted to find ot ho0 one can calclate the period of a pendl! on the

#asis of the lenth alone.

Stdent5

I 0old hae essed that if yo do#led the lenth% the period 0old also do#le.

9t that !st #e 0ron.

Galileo5=es% it is. >>>I first ared sin eo!etry and free fall>!otion that the ti!e it ta&es for a sphereto roll do0n any inclined plane that connects fro! the #ase point of a circle is the sa!e as it

0old ta&e a freely fallin o#-ect to fall the distance eal to the dia!eter of the circle.

Galileo goes to the blackboard draws the appropriate diagram and e-plains.

Stdent5

This% too% is not o#ios% Sinor Galileo. 9t ho0 is this connected to the period of the

pendl!E

Galileo5

Well% it sests that the period of the pendl! is proportional to the sare root of thelenth. We can easily sho0 this.

Galileo counts the time it takes for ten swings for a certain length. &hen he doubles the

length of the pendulum and repeats the e-periment.

Galileo looks at the student

What lenth do 0e need to do#le the periodE

Stdent5

That is easy. or ti!es the oriinal lenthK

Galileo5

9raoK

Stdent5Than& yo. inally% I 0old li&e to hae yor opinion on so!ethin that 7res!e

sested. Al!ost three hndred years ao he said5

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A body falling through a well that has been drilled from one side of the earth# through thecenter# to the other end# would oscillate like a pendulum and eventually come to a stop at

the center.

Galileo5

=es. I% too% discssed this idea in !y #oo&

%e brings a book and opens it.

… From this it seems possible to me... to believe that if the terrestrial globe were perforated through thecenter, a cannon ball descending through the hole would have acquired at the center such animpetus from its speed that it would pass beyond the center and be driven upward through asmuch space as it had fallen, its velocity beyond the center always diminishing with losses equal tothe increments acquired in the descent; and I believe that the time consumed in this secondascending motion would be equal to its time of descent.

Stdent5

Pery interestin% Sinor Galileo. This is ery si!ilar to the state!ent !ade #y Al#ert of

Sa/ony and then lat

pendulum notes

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