The pendulum : a case study in physics

301 Pages·2005·11.675 MB·English

by  Baker, Gregory L., Blackburn, James A

THEPENDULUM
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The Pendulum
A Case Study in Physics
GREGORY L. BAKER
Bryn Athyn College of the NewChurch,
Pennsylvania,USA
and
JAMES A. BLACKBURN
Wilfrid Laurier University,
Ontario, Canada
AC
AC
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Preface
Tolookatathingisquitedifferentfromseeingathing
(OscarWilde,fromAnIdealHusband)
Thependulum:acasestudyinphysicsisanunusualbookinseveralways.
Most distinctively, it is organized around a single physical system, the
pendulum,incontrasttoconventionaltextsthatremainconfinedtosingle
fieldssuchaselectromagnetismorclassicalmechanics.Inotherwords,the
pendulumisthecentralfocus,butfromthismainpathwebranchtomany
importantareasofphysics,technology,andthehistoryofscience.
Everyone is familiar with the basic behavior of a simple pendulum—a
pivoted rod with a mass attached to the free end. The grandfather clock
comestomind.Itmightseemthatthereisnotmuchtobesaidaboutsuch
an elemental system, or that its dynamical possibilities would be limited.
But,inreality,thisisaverycomplexsystemmasqueradingasasimpleone.
On closer examination, the pendulum exhibitsa remarkable variety of
motions. By considering pendulum dynamics, with and without external
forcing,wearedrawntotheessentialideasoflinearityandnonlinearityin
driven systems, including chaos. Coupled pendulums can become syn-
chronized, a behavior noted by Christiaan Huygens in the seventeenth
century.Evenquantummechanicscanbebroughttobearonthissimple
type of oscillator. The pendulum has intriguing connections to super-
conducting devices. Looking at applications of pendulums we are led to
measurements of the gravitational constant, viscosity, the attraction of
chargedparticles,theequivalenceprinciple,andtime.
Whilethestudyofphysicsistypicallymotivatedbythewishtounder-
stand physical laws, to understand how the physical world works, and,
throughresearch,toexplorethedetailsofthoselaws,thissciencecontinues
tobeenormouslyimportantinthehumaneconomyandpolity.Thepen-
dulum,initsownway,isalsopartofthisdevelopment.Notjustadeviceof
purephysics,thependulumisfascinatingbecauseofitsintriguinghistory
andtherangeofitstechnicalapplicationsspanningmanyfieldsandseveral
centuries.Thusweencounter,inthisbook,Galileo,Cavendish,Coulomb,
Foucault,KamerlinghOnnes,Josephson,andothers.
Wecontemplatedarangeofpossibilitiesforthestructureandflavorof
ourbook.Thewidecoverageandhistoricalconnectionssuggestedabroad
approach suited to a fairly general audience. However, a book without
equations would mean using words to try to convey the beauty of the
theoretical(mathematical)basisforthephysicsofthependulum.Graphs
andequationsgivephysicsitspredictivepowerandpreeminentplaceinour
understandingofthephysicalworld.Withthisinmind,weoptedinstead
vi Preface
forathoroughtechnicaltreatment.Inplaceswehavesuppliedbackground
material for the nonexpert reader; for example, in the chapter on the
quantumpendulum,weincludeashortintroductiontothemainideasof
quantumphysics.
Thereisanothersignificantdifferencebetweenthisbookandstandard
physicstexts.Asnoted,thisworkfocusesonasingletopic,thependulum.
Yet,inconventionalphysicsbooks,thependulumusuallyappearsonlyas
an illustration of a particular theory or phenomenon. A classical
mechanicstextmighttreatthependulumwithinacertaincontext,whereas
a book on chaotic dynamics might describe the pendulum with a very
differentemphasis.Intheeventthatabookonquantummechanicswereto
considerthependulum,itwoulddosofromyetanotherpointofview.In
contrast,herewehavegatheredtogetherthesemanythreadsandmadethe
pendulumtheunifyingconcept.
Finally,webelievethatThePendulum:ACaseStudyinPhysicsmaywell
serveasamodelforanewkindofcourseinphysics,onethatwouldtakea
thematic approach, thereby conveying something of the interrelation of
disciplines in the real progress of science. To gain a full measure of
understanding, the requisite mathematics would include calculus up to
ordinary differential equations. Exposure to an introductory physics
coursewouldalsobehelpful.Anumberofexercisesareincludedforthose
who do wish to use this as a text. For the more casual reader, a natural
curiosity and some ability to understand graphs are probably sufficient
togainasenseoftherichnessofthescienceassociatedwiththiscomplex
device.
Webeganthisprojectthinkingtocreateabookthatwouldbesomething
of an encyclopedia on the topic, one volume holding all the facts about
pendulums. But the list of potential topics proved to be astonishingly
extensive and varied—too long, as it turned out, for this text. So from
manypossibilities,wehavemadethechoicesfoundinthesepages.
Thebook,then,isathemeandvariations.Wehopethereaderwillfind
itarichandsatisfyingdiscourse.
Acknowledgments
We are indebted to many individuals for helping us bring this project to
completion.Inthesummerof2003,wevisitedseveralscientificmuseums
in order to get a first hand look at some of the famous pendulums to
which we refer in the book. In the course of these visits various curators
and other staff members were very generous in allowing us access to the
museum collections. We wish to acknowledge the hospitality of Adrian
Whicher, Assistant Curator, Classical Physics, the Science Museum of
London, Jonathan Betts, Senior Curator of Horology, Matthew Read,
Assistant Curator of Horology, and Janet Small, all of the National
Maritime Museum, Greenwich, G. A. C. Veeneman, Director, Hans
Hooijmaijers, Curator, and Robert de Bruin all of the Boerhaave
MuseumofLeyden,andLaurenceBobis,DirectricedelaBibliothe`quede
L’Observatoire de Paris. We wish also to thank William Tobin for
helpful information on Le´on Foucault and his famous pendulum, James
Yorke for some historical information, Juan Sanmartin for further
articlesonOBotafumeiro,BernieNickelforusefuldiscussionsaboutthe
Foucault pendulum at the University of Guelph, Susan Henley and
William Underwood of the Society of Exploration Geophysicists, and
RajarshiRoyandStevenStrogatzforusefuldiscussions.Otherstowhom
weowethanksareMargaretWalker,BobWhitaker,PhilipHannah,Bob
Holstro¨m, editor of the Horological Science Newsletter, and Danny
Hillis and David Munro, both associated with the Long Now clock
project. For clarifying some matters of Latin grammar, JAB thanks
ProfessorsJoannFreedandJudyFletcherofWilfridLaurierUniversity.
Finally, both of us would like to express gratitude to our colleague and
friend, John Smith, who has made significant contributions to the
experimental work described in the chapters on the chaotic pendulum
and synchronized pendulums.
Library and other media resources are important for this work. We
wouldliketothankRachelLongstaff,NancyMitzen,andCarrollOdhner
oftheSwedenborgLibraryofBrynAthynCollege,AmyGillinghamofthe
Library, University of Guelph, for providing copies of correspondence
between Christiaan Huygens and his father, Nancy Shader, Charles
Greene,andthestaffofthePrincetonManuscriptLibrary.GLBwishesto
thankCharlesLindsay,DeanofBrynAthynCollegeforhelpingtoarrange
sabbaticals that expedited this work, Jennifer Beiswenger and Charles
Ebertforcomputerhelp,andtheResearchcommitteeoftheAcademyof
theNewChurchforongoingfinancialsupport.
Financial support for JAB was provided through a Discovery Grant
fromtheNaturalSciencesandEngineeringResearchCouncilofCanada.
viii Acknowledgments
Thenatureofthisbookprovidedastrongincentivetousefiguresfroma
widevarietyofsources.Wehavemadeeveryefforttodetermineoriginal
sources and obtain permissions for the use of these illustrations. A large
number, especially of historical figures or pictures of experimental appa-
ratus, were taken from books, scientific journals, and from museum
sources. Credit for individual figures is found in the respective captions.
Manyresearchersgenerouslygaveuspermissiontousefiguresfromtheir
publications. In this connection we thank G. D’Anna, John Bird, Beryl
Clotfelter,RichardCrane,JensGundlach,JohnLindner,GabrielLuther,
RileyNeuman,JuanSanmartin,DonaldSullivan,andJamesYorke.The
book contains a few figures created by parties whom we were unable to
locate.Wethankthosepublisherswhoeitherwaivedorreducedfeesforuse
offiguresfrombooks.
IthasbeenapleasureworkingwithOUPonthisprojectandwewishto
express our special thanks to Sonke Adlung, physical science editor,
Tamsin Langrishe, assistant commissioning editor, and Anita Petrie,
productioneditor.
Finally,wewishtoexpressprofoundgratitudetoourwives,Margaret
Baker and Helena Stone, for their support and encouragement through
thecourseofthiswork.
Contents
1 Introduction 1
2 Pendulumssomewhatsimple 8
2.1 Thebeginning 8
2.2 Thesimplependulum 9
2.3 Someanalogsofthelinearizedpendulum 13
2.3.1 Thespring 13
2.3.2 Resonantelectricalcircuit 15
2.3.3 Thependulumandtheearth 16
2.3.4 Themilitarypendulum 19
2.3.5 Compoundpendulum 20
2.3.6 Kater’spendulum 21
2.4 Someconnections 23
2.5 Exercises 24
3 Pendulumslesssimple 27
3.1 OBotafumeiro 27
3.2 Thelinearizedpendulumwithcomplications 29
3.2.1 Energyloss—friction 29
3.2.2 Energygain—forcing 34
3.2.3 Parametricforcing 42
3.3 Thenonlinearizedpendulum 45
3.3.1 Amplitudedependentperiod 45
3.3.2 Phasespacerevisited 51
3.3.3 Anelectronic‘‘Pendulum’’ 53
3.3.4 Parametricforcingrevisited 56
3.4 Apendulumofhorror 63
3.5 Exercises 64
4 TheFoucaultpendulum 67
4.1 WhatisaFoucaultpendulum? 67
4.2 Framesofreference 71
4.3 Publicphysics 74
4.4 Aquantitativeapproach 75
4.4.1 Startingthependulum 78
4.5 Adarkerside 85
4.6 TowardabetterFoucaultpendulum 86
4.7 Afinalnote 89
4.8 Exercises 91