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Conservation Laws
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Permanent Link: http://ufdc.ufl.edu/AA00011740/00001
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Title: Conservation Laws
Physical Description: Book
Language: en-US
Creator: Crowell, Benjamin
Publication Date: 12/9/2008
 Subjects
Subjects / Keywords: Science, Physics, Perpetual Motion, Energy, Numerical Scale of Energy, Kinetic Energy, Potential Energy, Conservation of Angular Momentum, Angular Momentum in Planetary Motion, Torque, Momentum Transfer, Proof of Kepler’s Elliptical Orbit Law, Pressure and Temperature, Entropy
Physics, Scientific Concepts
Science / Physics
 Notes
Abstract: This is a text for an introductory college physics class, covering conservation of energy, momentum, and angular momentum. The treatment is algebra-based, with applications of calculus discussed in optional sections. Contents: 1) Conservation of Energy. 2) Simplifying the Energy Zoo. 3) Work: The Transfer of Mechanical Energy. 4) Conservation of Momentum. 5) Conservation of Angular Momentum. 6) Thermodynamics. This is book 2 in the Light and Matter series of free introductory physics textbooks.
General Note: Expositive
General Note: Community College, Higher Education
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General Note: Benjamin Crowell, Fullerton College, CA
General Note: Textbook
General Note: www.lightandmatter.com
General Note: http://florida.theorangegrove.org/og/file/e279dcdf-28d6-e6dd-34c8-f3125fbc2d7f/1/ConservationLaws.pdf
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Source Institution: University of Florida
Rights Management: Copyright 1998-2008 Benjamin Crowell. This book is licensed under the Creative Commons Attribution-ShareAlike license, version 1.0, http://creativecommons.org/licenses/by-sa/1.0/, except for those photographs and drawings of which I am not the author, as listed in the photo credits. If you agree to the license, it grants you certain privileges that you …
Resource Identifier: isbn - 9780970467027
System ID: AA00011740:00001

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Book2intheLightandMatterseriesoffreeintroductoryphysicstextbooks www.lightandmatter.com

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The LightandMatter seriesof introductoryphysicstextbooks: 1NewtonianPhysics 2ConservationLaws 3VibrationsandWaves 4ElectricityandMagnetism 5Optics 6TheModernRevolutioninPhysics

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BenjaminCrowell www.lightandmatter.com

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Fullerton,California www.lightandmatter.com copyright1998-2008BenjaminCrowell rev.December9,2008 ThisbookislicensedundertheCreativeCommonsAttribution-ShareAlikelicense,version1.0, http://creativecommons.org/licenses/by-sa/1.0/,except forthosephotographsanddrawingsofwhichIamnot theauthor,aslistedinthephotocredits.Ifyouagree tothelicense,itgrantsyoucertainprivilegesthatyou wouldnototherwisehave,suchastherighttocopythe book,ordownloadthedigitalversionfreeofchargefrom www.lightandmatter.com.Atyouroption,youmayalso copythisbookundertheGNUFreeDocumentation Licenseversion1.2,http://www.gnu.org/licenses/fdl.txt, withnoinvariantsections,nofront-covertexts,andno back-covertexts. ISBN0-9704670-2-8

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ToUriHaber-Schaim,JohnDodge,RobertGardner,andEdwardShore.

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BriefContents 1ConservationofEnergy13 2SimplifyingtheEnergyZoo35 3Work:TheTransferofMechanicalEnergy49 4ConservationofMomentum75 5ConservationofAngularMomentum107 AThermodynamics143

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Contents 1ConservationofEnergy 1.1TheSearchforaPerpetualMotion Machine..............13 1.2Energy............14 1.3ANumericalScaleofEnergy...18 Hownewformsofenergyarediscovered, 21. 1.4KineticEnergy.........23 Energyandrelativemotion,24. 1.5Power.............25 Summary.............28 Problems.............30 2SimplifyingtheEnergyZoo 2.1HeatisKineticEnergy......36 2.2PotentialEnergy:EnergyofDistance orCloseness............38 Anequationforgravitationalpotential energy,39. 2.3AllEnergyisPotentialorKinetic..42 Summary.............44 Problems.............45 3Work:TheTransferofMechanicalEnergy 3.1Work:TheTransferofMechanical Energy..............49 Theconceptofwork,49.|Calculating workasforcemultipliedbydistance,50.| Machinescanincreaseforce,butnotwork., 52.|Noworkisdonewithoutmotion., 52.|Positiveandnegativework,53. 3.2WorkinThreeDimensions....55 Aforceperpendiculartothemotiondoes nowork.,55.|Forcesatotherangles,56. 3.3VaryingForce.........57 3.4 R ApplicationsofCalculus....60 3.5WorkandPotentialEnergy....62 3.6 ? WhenDoesWorkEqualForce TimesDistance?..........64 3.7 ? TheDotProduct........66 Summary.............68 10

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Problems.............70 4ConservationofMomentum 4.1Momentum...........76 Aconservedquantityofmotion,76.| Momentum,77.|Generalizationofthe momentumconcept,79.|Momentum comparedtokineticenergy,81. 4.2CollisionsinOneDimension...83 Thediscoveryoftheneutron,85. 4.3 ? RelationshipofMomentumtothe CenterofMass...........88 Momentumindierentframesofreference, 89.|Thecenterofmassframeofreference, 90. 4.4MomentumTransfer.......91 Therateofchangeofmomentum,91.| Theareaundertheforce-timegraph,93. 4.5MomentuminThreeDimensions..94 Thecenterofmass,95.|Countingequationsandunknowns,96.|Calculations withthemomentumvector,97. 4.6 R ApplicationsofCalculus....98 Summary.............100 Problems.............102 5ConservationofAngular Momentum 5.1ConservationofAngularMomentum109 Restrictiontorotationinaplane,113. 5.2AngularMomentuminPlanetary Motion..............113 5.3TwoTheoremsAboutAngular Momentum............115 5.4Torque:theRateofTransferofAngularMomentum...........120 Torquedistinguishedfromforce,120.| Relationshipbetweenforceandtorque, 121.|Thetorqueduetogravity,123. 5.5Statics.............127 Equilibrium,127.|Stableandunstable equilibria,130. 5.6SimpleMachines:TheLever...131 5.7 ? ProofofKepler'sEllipticalOrbitLaw133 Summary.............135 Problems.............137 AThermodynamics A.1PressureandTemperature....144 Pressure,144.|Temperature,148. A.2MicroscopicDescriptionofanIdeal Gas...............151 Evidenceforthekinetictheory,151.| Pressure,volume,andtemperature,151. A.3Entropy............155 Eciencyandgradesofenergy,155.| Heatengines,155.|Entropy,157. Problems.............161 Appendix1:Exercises 164 Appendix2:PhotoCredits 165 Appendix3:HintsandSolutions 166 11

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InJulyof1994,CometShoemaker-LevystrucktheplanetJupiter,depositing7 10 22 joulesofenergy,andincidentallygivingrisetoaseries ofHollywoodmoviesinwhichourownplanetisthreatenedbyanimpact byacometorasteroid.Thereisevidencethatsuchanimpactcaused theextinctionofthedinosaurs.Left:Jupiter'sgravitationalforceonthe nearsideofthecometwasgreaterthanonthefarside,andthisdifferenceinforcetoreupthecometintoastringoffragments.Twoseparate telescopeimageshavebeencombinedtocreatetheillusionofapointof viewjustbehindthecomet.ThecoloredfringesattheedgesofJupiter areartifactsoftheimagingsystem.Top:Aseriesofimagesoftheplume ofsuperheatedgaskickedupbytheimpactofoneofthefragments.The plumeisaboutthesizeofNorthAmerica.Bottom:Animageafterallthe impactswereover,showingthedamagedone. Chapter1 ConservationofEnergy 1.1TheSearchforaPerpetualMotionMachine Don'tunderestimategreedandlazinessasforcesforprogress.Modernchemistrywasbornfromthecollisionoflustforgoldwithdistasteforthehardworkofndingitanddiggingitup.Failedeorts bygenerationsofalchemiststoturnleadintogoldlednallytothe conclusionthatitcouldnotbedone:certainsubstances,thechemicalelements,arefundamental,andchemicalreactionscanneither 13

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a / Themagnetdrawsthe balltothetopoftheramp,where itfallsthroughtheholeandrolls backtothebottom. b / Asthewheelspinsclockwise,theexiblearmssweep aroundandbendandunbend.By droppingoffitsballontheramp, thearmissupposedtomake itselflighterandeasiertoliftover thetop.Pickingitsownballback upagainontheright,ithelpsto pulltherightsidedown. increasenordecreasetheamountofanelementsuchasgold. Nowashforwardtotheearlyindustrialage.Greedandlaziness havecreatedthefactory,thetrain,andtheoceanliner,butineach oftheseisaboilerroomwheresomeonegetssweatyshovelingthe coaltofuelthesteamengine.Generationsofinventorshavetriedto createamachine,calledaperpetualmotionmachine,thatwouldrun foreverwithoutfuel.SuchamachineisnotforbiddenbyNewton's lawsofmotion,whicharebuiltaroundtheconceptsofforceand inertia.Forceisfree,andcanbemultipliedindenitelywithpulleys, gears,orlevers.Theprincipleofinertiaseemseventoencourage thebeliefthatacleverlyconstructedmachinemightnoteverrun down. Figuresaandbshowtwooftheinnumerableperpetualmotion machinesthathavebeenproposed.Thereasonthesetwoexamples don'tworkisnotmuchdierentfromthereasonalltheothershave failed.Considermachinea.Evenifweassumethataproperly shapedrampwouldkeeptheballrollingsmoothlythrougheach cycle,frictionwouldalwaysbeatwork.Thedesignerimaginedthat themachinewouldrepeatthesamemotionoverandoveragain,so thateverytimeitreachedagivenpointitsspeedwouldbeexactly thesameasthelasttime.Butbecauseoffriction,thespeedwould actuallybereducedalittlewitheachcycle,untilnallytheball wouldnolongerbeabletomakeitoverthetop. Frictionhasawayofcreepingintoallmovingsystems.The rotatingearthmightseemlikeaperfectperpetualmotionmachine, sinceitisisolatedinthevacuumofouterspacewithnothingtoexert frictionalforcesonit.Butinfactourplanet'srotationhasslowed drasticallysinceitrstformed,andtheearthcontinuestoslow itsrotation,makingtodayjustalittlelongerthanyesterday.The verysubtlesourceoffrictionisthetides.Themoon'sgravityraises bulgesintheearth'soceans,andastheearthrotatesthebulges progressaroundtheplanet.Wherethebulgesencounterland,there isfriction,whichslowstheearth'srotationverygradually. 1.2Energy Theanalysisbasedonfrictionissomewhatsupercial,however.One couldunderstandfrictionperfectlywellandyetimaginethefollowingsituation.Astronautsbringbackapieceofmagneticorefrom themoonwhichdoesnotbehavelikeordinarymagnets.Anormal barmagnet,c/1,attractsapieceofironessentiallydirectlytoward it,andhasnoleft-orright-handedness.Themoonrock,however, exertsforcesthatformawhirlpoolpatternaroundit,2.NASA goestoamachineshopandhasthemoonrockputinalatheand machineddowntoasmoothcylinder,3.Ifwenowreleaseaball bearingonthesurfaceofthecylinder,themagneticforcewhipsit aroundandaroundateverhigherspeeds.Ofcoursethereissome 14 Chapter1ConservationofEnergy

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c / Amysteriousmoonrock makesaperpetualmotion machine. d / Example1. friction,butthereisanetgaininspeedwitheachrevolution. Physicistswouldlaylongoddsagainstthediscoveryofsucha moonrock,notjustbecauseitbreakstherulesthatmagnetsnormallyobeybutbecause,likethealchemists,theyhavediscovered averydeepandfundamentalprincipleofnaturewhichforbidscertainthingsfromhappening.Therstalchemistwhodeservedto becalledachemistwastheonewhorealizedoneday,Inallthese attemptstocreategoldwheretherewasnonebefore,allI'vebeen doingisshuingthesameatomsbackandforthamongdierent testtubes.Theonlywaytoincreasetheamountofgoldinmylaboratoryistobringsomeinthroughthedoor."Itwaslikehaving someofyourmoneyinacheckingaccountandsomeinasavingsaccount.Transferringmoneyfromoneaccountintotheotherdoesn't changethetotalamount. Wesaythatthenumberofgramsofgoldisa conserved quantity.Inthiscontext,thewordconserve"doesnothaveitsusual meaningoftryingnottowastesomething.Inphysics,aconserved quantityissomethingthatyouwouldn'tbeabletogetridofeven ifyouwantedto.Conservationlawsinphysicsalwaysrefertoa closedsystem ,meaningaregionofspacewithboundariesthrough whichthequantityinquestionisnotpassing.Inourexample,the alchemist'slaboratoryisaclosedsystembecausenogoldiscoming inoroutthroughthedoors. Conservationofmassexample1 Ingured,thestreamofwaterisfatternearthemouthofthe faucet,andskinnierlowerdown.Thisisbecausethewaterspeeds upasitfalls.Ifthecross-sectionalareaofthestreamwasequal allalongitslength,thentherateofowthroughalowercrosssectionwouldbegreaterthantherateofowthroughacrosssectionhigherup.Sincetheowissteady,theamountofwaterbetweenthetwocross-sectionsstaysconstant.Thecrosssectionalareaofthestreammustthereforeshrinkininverseproportiontotheincreasingspeedofthefallingwater.Thisisan exampleofconservationofmass. Ingeneral,theamountofanyparticularsubstanceisnotconserved.Chemicalreactionscanchangeonesubstanceintoanother, andnuclearreactionscanevenchangeoneelementintoanother. Thetotalmassofallsubstancesishoweverconserved: thelawofconservationofmass Thetotalmassofaclosedsystemalwaysremainsconstant.Energy cannotbecreatedordestroyed,butonlytransferredfromonesystem toanother. Asimilarlightbulbeventuallylitupintheheadsofthepeople whohadbeenfrustratedtryingtobuildaperpetualmotionmachine. Inperpetualmotionmachinea,considerthemotionofoneofthe Section1.2Energy 15

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balls.Itperformsacycleofrisingandfalling.Onthewaydownit gainsspeed,andcomingupitslowsbackdown.Havingagreater speedislikehavingmoremoneyinyourcheckingaccount,andbeing highupislikehavingmoreinyoursavingsaccount.Thedeviceis simplyshuingfundsbackandforthbetweenthetwo.Havingmore ballsdoesn'tchangeanythingfundamentally.Notonlythat,but frictionisalwaysdrainingomoneyintoathirdbankaccount:" heat.Thereasonwerubourhandstogetherwhenwe'recoldisthat kineticfrictionheatsthingsup.Thecontinualbuildupintheheat account"leaveslessandlessforthemotionaccount"andheight account,"causingthemachineeventuallytorundown. Theseinsightscanbedistilledintothefollowingbasicprinciple ofphysics: thelawofconservationofenergy Itispossibletogiveanumericalrating,calledenergy,tothestate ofaphysicalsystem.Thetotalenergyisfoundbyaddingupcontributionsfromcharacteristicsofthesystemsuchasmotionofobjects init,heatingoftheobjects,andtherelativepositionsofobjects thatinteractviaforces.Thetotalenergyofaclosedsystemalways remainsconstant.Energycannotbecreatedordestroyed,butonly transferredfromonesystemtoanother. Themoonrockstoryviolatesconservationofenergybecausethe rock-cylinderandtheballtogetherconstituteaclosedsystem.Once theballhasmadeonerevolutionaroundthecylinder,itsposition relativetothecylinderisexactlythesameasbefore,sothenumericalenergyratingassociatedwithitspositionisthesameasbefore. Sincethetotalamountofenergymustremainconstant,itisimpossiblefortheballtohaveagreaterspeedafteronerevolution.If ithadpickedupspeed,itwouldhavemoreenergyassociatedwith motion,thesameamountofenergyassociatedwithposition,anda littlemoreenergyassociatedwithheatingthroughfriction.There cannotbeanetincreaseinenergy. Convertingoneformofenergytoanotherexample2 Droppingarock: Therocklosesenergybecauseofitschanging positionwithrespecttotheearth.Nearlyallthatenergyistransformedintoenergyofmotion,exceptforasmallamountlostto heatcreatedbyairfriction. Slidingintohomebase: Therunner'senergyofmotionisnearly allconvertedintoheatviafrictionwiththeground. Acceleratingacar: Thegasolinehasenergystoredinit,which isreleasedasheatbyburningitinsidetheengine.Perhaps10% ofthisheatenergyisconvertedintothecar'senergyofmotion. Therestremainsintheformofheat,whichiscarriedawaybythe exhaust. 16 Chapter1ConservationofEnergy

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e / Example3. Cruisinginacar: Asyoucruiseatconstantspeedinyourcar,all theenergyoftheburninggasisbeingconvertedintoheat.The tiresandenginegethot,andheatisalsodissipatedintotheair throughtheradiatorandtheexhaust. Steppingonthebrakes: Alltheenergyofthecar'smotionisconvertedintoheatinthebrakeshoes. Stevin'smachineexample3 TheDutchmathematicianandengineerSimonStevinproposed theimaginarymachineshowninguree,whichhehadinscribed onhistombstone.Thisisaninterestingexample,becauseit showsalinkbetweentheforceconceptusedearlierinthiscourse, andtheenergyconceptbeingdevelopednow. Thepointoftheimaginarymachineistoshowthemechanical advantageofaninclinedplane.Inthisexample,thetrianglehas theproportions3-4-5,buttheargumentworksforanyrighttriangle.Weimaginethatthechainofballsslideswithoutfriction,so thatnoenergyiseverconvertedintoheat.Ifweweretoslide thechainclockwisebyonestep,theneachballwouldtakethe placeoftheoneinfrontofit,andtheoverallcongurationwould beexactlythesame.Sinceenergyissomethingthatonlydependsonthestateofthesystem,theenergywouldhavetobe thesame.Similarlyforacounterclockwiserotation,noenergyof positionwouldbereleasedbygravity.Thismeansthatifweplace thechainonthetriangle,andreleaseitatrest,itcan'tstartmoving,becausetherewouldbenowayforittoconvertenergyof positionintoenergyofmotion.Thusthechainmustbeperfectly balanced.Nowbysymmetry,thearcofthechainhangingunderneaththetrianglehasequaltensionatbothends,soremoving thisarcwouldn'taffectthebalanceoftherestofthechain.This meansthataweightofthreeunitshangingverticallybalancesa weightofveunitshangingdiagonallyalongthehypotenuse. Themechanicaladvantageoftheinclinedplaneistherefore5 = 3, whichisexactlythesameastheresult,1 = sin ,thatwegotbeforebyanalyzingforcevectors.WhatthisshowsisthatNewton'slawsandconservationlawsarenotlogicallyseparate,but ratherareverycloselyrelateddescriptionsofnature.Inthecases whereNewton'slawsaretrue,theygivethesameanswersas theconservationlaws.Thisisanexampleofamoregeneral idea,calledthecorrespondenceprinciple,abouthowscienceprogressesovertime.Whenanewer,moregeneraltheoryisproposedtoreplaceanoldertheory,thenewtheorymustagreewith theoldoneintherealmofapplicabilityoftheoldtheory,since theoldtheoryonlybecameaacceptedasavalidtheorybybeingveriedexperimentallyinavarietyofexperiments.Inother words,thenewtheorymustbebackward-compatiblewiththeold one.EventhoughconservationlawscanprovethingsthatNewSection1.2Energy 17

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DiscussionquestionA.The waterbehindtheHooverDam hasenergybecauseofitspositionrelativetotheplanetearth, whichisattractingitwithagravitationalforce.Lettingwaterdown tothebottomofthedamconverts thatenergyintoenergyofmotion. Whenthewaterreachesthe bottomofthedam,ithitsturbine bladesthatdrivegenerators,and itsenergyofmotionisconverted intoelectricalenergy. ton'slawscan'tthatperpetualmotionisimpossible,forexample, theyaren'tgoingto disprove Newton'slawswhenappliedtomechanicalsystemswherewealreadyknewNewton'slawswere valid. DiscussionQuestion A Hydroelectricpowerwaterowingoveradamtospinturbines appearstobecompletelyfree.Doesthisviolateconservationofenergy? Ifnot,thenwhatistheultimatesourceoftheelectricalenergyproduced byahydroelectricplant? B Howdoestheproofinexample3failiftheassumptionofafrictionless surfacedoesn'thold? 1.3ANumericalScaleofEnergy Energycomesinavarietyofforms,andphysicistsdidn'tdiscoverall ofthemrightaway.Theyhadtostartsomewhere,sotheypicked oneformofenergytouseasastandardforcreatinganumerical energyscale.Infactthehistoryiscomplicated,andseveraldierent energyunitsweredenedbeforeitwasrealizedthattherewasa singlegeneralenergyconceptthatdeservedasingleconsistentunit ofmeasurement.Onepracticalapproachistodeneanenergy unitbasedonheatingwater.TheSIunitofenergyisthejoule, J,rhymeswithcool",namedaftertheBritishphysicistJames Joule.OneJouleistheamountofenergyrequiredinordertoheat 0.24gofwaterby1 C.Thenumber0.24isnotworthmemorizing. Notethatheat,whichisaformofenergy,iscompletelydierentfromtemperature,whichisnot.Twiceasmuchheatenergy isrequiredtopreparetwocupsofcoeeastomakeone,buttwo cupsofcoeemixedtogetherdon'thavedoublethetemperature. Inotherwords,thetemperatureofanobjecttellsushowhotitis, buttheheatenergycontainedinanobjectalsotakesintoaccount theobject'smassandwhatitismadeof. 1 Laterwewillencounterotherquantitiesthatareconservedin physics,suchasmomentumandangularmomentum,andthemethod fordeningthemwillbesimilartotheonewehaveusedforenergy: picksomestandardformofit,andthenmeasureotherformsby comparisonwiththisstandard.Theexibleandadaptablenature ofthisprocedureispartofwhathasmadeconservationlawssucha durablebasisfortheevolutionofphysics. Heatingaswimmingpoolexample4 Ifelectricitycosts3.9centsperMJMJ=1megajoule=10 6 J,howmuchdoesitcosttoheata26000-gallonswimmingpool 1 Instandard,formalterminology,thereisanother,nerdistinction.The wordheat"isusedonlytoindicateanamountofenergythatistransferred, whereasthermalenergy"indicatesanamountofenergycontainedinanobject. I'minformalonthispoint,andrefertobothasheat,butyoushouldbeaware ofthedistinction. 18 Chapter1ConservationofEnergy

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from10 Cto18 C? Convertinggallonstocm 3 gives 26000gallons 3780cm 3 1gallon =9.8 10 7 cm 3 Waterhasadensityof1grampercubiccentimeter,sothemass ofthewateris9.8 10 7 g.Onejouleissufcienttoheat0.24g by1 C,sotheenergyneededtoheattheswimmingpoolis 1J 9.8 10 7 g 0.24g 8 C 1 C =3.3 10 9 J =3.3 10 3 MJ. Thecostoftheelectricityis.3 10 3 MJ$0.039/MJ=$130. Irishcoffeeexample5 YoumakeacupofIrishcoffeeoutof300gofcoffeeat100 C and30gofpureethylalcoholat20 C.OneJouleisenough energytoproduceachangeof1 Cin0.42gofethylalcoholi.e., alcoholiseasiertoheatthanwater.Whattemperatureisthe nalmixture? Addingupalltheenergyaftermixinghastogivethesameresult asthetotalbeforemixing.Weletthesubscript i standforthe initialsituation,beforemixing,and f forthenalsituation,anduse subscripts c forthecoffeeand a forthealcohol.Inthisnotation, wehave totalinitialenergy=totalnalenergy E ci + E ai = E cf + E af Weassumecoffeehasthesameheat-carryingpropertiesaswater.Ourinformationabouttheheat-carryingpropertiesofthetwo substancesisstatedintermsofthe change inenergyrequiredfor acertain change intemperature,sowerearrangetheequationto expresseverythingintermsofenergydifferences: E af )]TJ/F102 10.9091 Tf 10.909 0 Td [(E ai = E ci )]TJ/F102 10.9091 Tf 10.909 0 Td [(E cf Usingthegivenratiosoftemperaturechangetoenergychange, wehave E ci )]TJ/F102 10.9091 Tf 10.909 0 Td [(E cf = T ci )]TJ/F102 10.9091 Tf 10.909 0 Td [(T cf m c = .24g E af )]TJ/F102 10.9091 Tf 10.909 0 Td [(E ai = T af )]TJ/F102 10.9091 Tf 10.909 0 Td [(T ai m a = .42g Settingthesetwoquantitiestobeequal,wehave T af )]TJ/F102 10.9091 Tf 10.909 0 Td [(T ai m a = .42g= T ci )]TJ/F102 10.9091 Tf 10.909 0 Td [(T cf m c = .24g. Section1.3ANumericalScaleofEnergy 19

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Inthenalmixturethetwosubstancesmustbeatthesametemperature,sowecanuseasinglesymbol T f = T cf = T af forthe twoquantitiespreviouslyrepresentedbytwodifferentsymbols, T f )]TJ/F102 10.9091 Tf 10.909 0 Td [(T ai m a = .42g= T ci )]TJ/F102 10.9091 Tf 10.91 0 Td [(T f m c = .24g. Solvingfor T f gives T f = T ci m c 0.24 + T ai m a 0.42 m c 0.24 + m a 0.42 =96 C. Onceanumericalscaleofenergyhasbeenestablishedforsome formofenergysuchasheat,itcaneasilybeextendedtoothertypes ofenergy.Forinstance,theenergystoredinonegallonofgasoline canbedeterminedbyputtingsomegasolineandsomewaterinan insulatedchamber,ignitingthegas,andmeasuringtheriseinthe water'stemperature.Thefactthattheapparatusisknownasa bombcalorimeter"willgiveyousomeideaofhowdangerousthese experimentsareifyoudon'ttaketherightsafetyprecautions.Here aresomeexamplesofothertypesofenergythatcanbemeasured usingthesameunitsofjoules: typeofenergyexample chemicalenergy releasedbyburning About50MJarereleasedbyburning akgofgasoline. energyrequiredto breakanobject Whenapersonsuersaspiralfractureofthethighboneacommon typeinskiingaccidents,about2J ofenergygointobreakingthebone. energyrequiredto meltasolidsubstance 7MJarerequiredtomelt1kgoftin. chemicalenergy releasedbydigesting food AbowlofCheerieswithmilkprovides uswithabout800kJofusableenergy. raisingamassagainst theforceofgravity Lifting1.0kgthroughaheightof1.0 mrequires9.8J. nuclearenergy releasedinssion 1kgofuraniumoxidefuelconsumed byareactorreleases2 10 12 Jof storednuclearenergy. Itisinterestingtonotethedisproportionbetweenthemegajoule energiesweconsumeasfoodandthejoule-sizedenergiesweexpend inphysicalactivities.Ifwecouldperceivetheowofenergyaround usthewayweperceivetheowofwater,eatingabowlofcereal wouldbelikeswallowingabathtub'sworthofenergy,thecontinual lossofbodyheattoone'senvironmentwouldbelikeanenergy-hose leftonallday,andliftingabagofcementwouldbelikeicking itwithafewtinyenergy-drops.Thehumanbodyistremendously 20 Chapter1ConservationofEnergy

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f / Example6. inecient.Thecaloriesweburn"inheavyexercisearealmostall dissipateddirectlyasbodyheat. YoutakethehighroadandI'lltakethelowroad.example6 Figurefshowstworampswhichtwoballswillrolldown.Comparetheirnalspeeds,whentheyreachpointB.Assumefriction isnegligible. Eachballlosessomeenergybecauseofitsdecreasingheight abovetheearth,andconservationofenergysaysthatitmustgain anequalamountofenergyofmotionminusalittleheatcreated byfriction.Theballslosethesameamountofheight,sotheir nalspeedsmustbeequal. It'simpressivetonotethecompleteimpossibilityofsolvingthis problemusingonlyNewton'slaws.Eveniftheshapeofthetrack hadbeengivenmathematically,itwouldhavebeenaformidable tasktocomputetheballs'nalspeedbasedonvectoradditionof thenormalforceandgravitationalforceateachpointalongtheway. Hownewformsofenergyarediscovered Textbooksoftengivetheimpressionthatasophisticatedphysics conceptwascreatedbyonepersonwhohadaninspirationoneday, butinrealityitismoreinthenatureofsciencetoroughoutanidea andthengraduallyreneitovermanyyears.Theideaofenergy wastinkeredwithfromtheearly1800'son,andnewtypesofenergy keptgettingaddedtothelist. Toestablishtheexistenceofanewformofenergy,aphysicist hasto showthatitcouldbeconvertedtoandfromotherformsof energy;and showthatitrelatedtosomedenitemeasurablepropertyof theobject,forexampleitstemperature,motion,positionrelativeto anotherobject,orbeinginasolidorliquidstate. Forexample,energyisreleasedwhenapieceofironissoakedin water,soapparentlythereissomeformofenergyalreadystoredin theiron.Thereleaseofthisenergycanalsoberelatedtoadenite measurablepropertyofthechunkofmetal:itturnsreddish-orange. Therehasbeenachemicalchangeinitsphysicalstate,whichwe callrusting. Althoughthelistoftypesofenergykeptgettinglongerand longer,itwasclearthatmanyofthetypeswerejustvariationson atheme.Thereisanobvioussimilaritybetweentheenergyneeded tomelticeandtomeltbutter,orbetweentherustingofironand manyotherchemicalreactions.Thetopicofthenextchapteris howthisprocessofsimplicationreducedallthetypesofenergy toaverysmallnumberfour,accordingtothewayI'vechosento countthem. Section1.3ANumericalScaleofEnergy 21

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Itmightseemthatiftheprincipleofconservationofenergyever appearedtobeviolated,wecouldxitupsimplybyinventingsome newtypeofenergytocompensateforthediscrepancy.Thiswould belikebalancingyourcheckbookbyaddinginanimaginarydeposit orwithdrawaltomakeyourguresagreewiththebank'sstatements. Stepaboveguardsagainstthiskindofchicanery.Inthe1920s therewereexperimentsthatsuggestedenergywasnotconservedin radioactiveprocesses.Precisemeasurementsoftheenergyreleased intheradioactivedecayofagiventypeofatomshowedinconsistent results.Oneatommightdecayandrelease,say,1.1 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(10 Jof energy,whichhadpresumablybeenstoredinsomemysteriousform inthenucleus.Butinalatermeasurement,anatomofexactlythe sametypemightrelease1.2 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(10 J.Atomsofthesametypeare supposedtobeidentical,sobothatomswerethoughttohavestarted outwiththesameenergy.Iftheamountreleasedwasrandom,then apparentlythetotalamountofenergywasnotthesameafterthe decayasbefore,i.e.,energywasnotconserved. Onlylaterwasitfoundthatapreviouslyunknownparticle, whichisveryhardtodetect,wasbeingspewedoutinthedecay. Theparticle,nowcalledaneutrino,wascarryingosomeenergy, andifthispreviouslyunsuspectedformofenergywasaddedin, energywasfoundtobeconservedafterall.Thediscoveryofthe energydiscrepanciesisseenwithhindsightasbeingstepinthe establishmentofanewformofenergy,andthediscoveryoftheneutrinowasstep.Butduringthedecadeorsobetweenstep andsteptheaccumulationofevidencewasgradual,physicists hadtheadmirablehonestytoadmitthatthecherishedprincipleof conservationofenergymighthavetobediscarded. self-checkA Howwouldyoucarryoutthetwostepsgivenaboveinordertoestablishthatsomeformofenergywasstoredinastretchedorcompressed spring? Answer,p.166 MassIntoEnergy Einsteinshowedthatmassitselfcouldbeconvertedtoandfromenergy, accordingtohiscelebratedequation E = mc 2 ,inwhich c isthespeed oflight.Wethusspeakofmassassimplyanotherformofenergy,and itisvalidtomeasureitinunitsofjoules.Themassofa15-grampencilcorrespondstoabout1.3 10 15 J.Theissueislargelyacademicin thecaseofthepencil,becauseveryviolentprocessessuchasnuclear reactionsarerequiredinordertoconvertanysignicantfractionofan object'smassintoenergy.Cosmicrays,however,arecontinuallystrikingyouandyoursurroundingsandconvertingpartoftheirenergyof motionintothemassofnewlycreatedparticles.Asinglehigh-energy cosmicraycancreateashowerofmillionsofpreviouslynonexistent particleswhenitstrikestheatmosphere.Einstein'stheoriesarediscussedinbook6ofthisseries. Eventoday,whentheenergyconceptisrelativelymatureandstable,anewformofenergyhasbeenproposedbasedonobservations 22 Chapter1ConservationofEnergy

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ofdistantgalaxieswhoselightbeganitsvoyagetousbillionsofyears ago.Astronomershavefoundthattheuniverse'scontinuingexpansion, resultingfromtheBigBang,hasnotbeendeceleratingasrapidlyinthe lastfewbillionyearsaswouldhavebeenexpectedfromgravitational forces.Theysuggestthatanewformofenergymaybeatwork. DiscussionQuestion A I'mnotmakingthisup.XSEnergyDrinkhasadsthatreadlikethis: AlltheEnergy...WithouttheSugar!Only8Calories! Commenton this. 1.4KineticEnergy Thetechnicaltermfortheenergyassociatedwithmotioniskinetic energy,fromtheGreekwordformotion.Therootisthesameas therootofthewordcinema"foramotionpicture,andinFrench thetermforkineticenergyisenergiecinetique."Tondhow muchkineticenergyispossessedbyagivenmovingobject,wemust convertallitskineticenergyintoheatenergy,whichwehavechosen asthestandardreferencetypeofenergy.Wecoulddothis,for example,byringprojectilesintoatankofwaterandmeasuringthe increaseintemperatureofthewaterasafunctionoftheprojectile's massandvelocity.Considerthefollowingdatafromaseriesofthree suchexperiments: m kg v m/s energy J 1.00 1.00 0.50 1.00 2.00 2.00 2.00 1.00 1.00 Comparingtherstexperimentwiththesecond,weseethatdoublingtheobject'svelocitydoesn'tjustdoubleitsenergy,itquadruplesit.Ifwecomparetherstandthirdlines,however,wend thatdoublingthemassonlydoublestheenergy.Thissuggeststhat kineticenergyisproportionaltomassandtothesquareofvelocity, KE / mv 2 ,andfurtherexperimentsofthistypewouldindeed establishsuchageneralrule.Theproportionalityfactorequals0.5 becauseofthedesignofthemetricsystem,sothekineticenergyof amovingobjectisgivenby KE = 1 2 mv 2 Themetricsystemisbasedonthemeter,kilogram,andsecond, withotherunitsbeingderivedfromthose.Comparingtheunitson theleftandrightsidesoftheequationshowsthatthejoulecanbe reexpressedintermsofthebasicunitsaskg m 2 = s 2 Studentsareoftenmystiedbytheoccurrenceofthefactorof 1/2,butitislessobscurethanitlooks.Themetricsystemwas designedsothatsomeoftheequationsrelatingtoenergywould comeoutlookingsimple,attheexpenseofsomeothers,whichhad Section1.4KineticEnergy 23

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tohaveinconvenientconversionfactorsinfront.Ifwewereusing theoldBritishEngineeringSystemofunitsinthiscourse,thenwe'd havetheBritishThermalUnitBTUasourunitofenergy.In thatsystem,theequationyou'dlearnforkineticenergywouldhave aninconvenientproportionalityconstant, KE = )]TJ/F15 10.9091 Tf 5 -8.836 Td [(1.29 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(3 mv 2 with KE measuredinunitsofBTUs, v measuredinfeetpersecond, andsoon.Attheexpenseofthisinconvenientequationforkinetic energy,thedesignersoftheBritishEngineeringSystemgotasimple ruleforcalculatingtheenergyrequiredtoheatwater:oneBTU perdegreeFahrenheitpergallon.Theinventorofkineticenergy, ThomasYoung,actuallydeneditas KE = mv 2 ,whichmeant thatallhisotherequationshadtobedierentfromoursbyafactor oftwo.Allthesesystemsofunitsworkjustneaslongastheyare notcombinedwithoneanotherinaninconsistentway. Energyreleasedbyacometimpactexample7 CometShoemaker-Levy,whichstrucktheplanetJupiterin1994, hadamassofroughly4 10 13 kg,andwasmovingataspeed of60km/s.Comparethekineticenergyreleasedintheimpactto thetotalenergyintheworld'snucleararsenals,whichis2 10 19 J.AssumeforthesakeofsimplicitythatJupiterwasatrest. SinceweassumeJupiterwasatrest,wecanimaginethatthe cometstoppedcompletelyonimpact,and100%ofitskineticenergywasconvertedtoheatandsound.Werstconvertthespeed tomksunits, v =6 10 4 m/s,andthenplugintotheequation tondthatthecomet'skineticenergywasroughly7 10 22 J,or about3000timestheenergyintheworld'snucleararsenals. Isthereanywaytoderivetheequation KE = = 2 mv 2 mathematicallyfromrstprinciples?No,itispurelyempirical.The factorof1/2infrontisdenitelynotderivable,sinceitisdierent indierentsystemsofunits.Theproportionalityto v 2 isnoteven quitecorrect;experimentshaveshowndeviationsfromthe v 2 ruleat highspeeds,aneectthatisrelatedtoEinstein'stheoryofrelativity.Onlytheproportionalityto m isinevitable.Thewholeenergy conceptisbasedontheideathatweaddupenergycontributions fromalltheobjectswithinasystem.Basedonthisphilosophy,it islogicallynecessarythata2-kgobjectmovingat1m/shavethe samekineticenergyastwo1-kgobjectsmovingside-by-sideatthe samespeed. Energyandrelativemotion AlthoughImentionedEinstein'stheoryofrelativityabove,it's morerelevantrightnowtoconsiderhowconservationofenergyrelatestothesimplerGalileanidea,whichwe'vealreadystudied,that motionisrelative.Galileo'sAristotelianenemiesanditisnoexaggerationtocallthemenemies!wouldprobablyhaveobjectedto conservationofenergy.Afterall,theGalileanideathatanobject inmotionwillcontinueinmotionindenitelyintheabsenceofa 24 Chapter1ConservationofEnergy

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DiscussionquestionB forceisnotsodierentfromtheideathatanobject'skineticenergy staysthesameunlessthereisamechanismlikefrictionalheating forconvertingthatenergyintosomeotherform. Moresubtly,however,it'snotimmediatelyobviousthatwhat we'velearnedsofaraboutenergyisstrictlymathematicallyconsistentwiththeprinciplethatmotionisrelative.Supposeweverify thatacertainprocess,saythecollisionoftwopoolballs,conserves energyasmeasuredinacertainframeofreference:thesumofthe balls'kineticenergiesbeforethecollisionisequaltotheirsumafter thecollision.Inrealitywe'dneedtoaddinotherformsofenergy, likeheatandsound,thatareliberatedbythecollision,butlet'skeep itsimple.Butwhatifweweretomeasureeverythinginaframeof referencethatwasinadierentstateofmotion?Aparticularpool ballmighthavelesskineticenergyinthisnewframe;forexample,if thenewframeofreferencewasmovingrightalongwithit,itskinetic energyinthatframewouldbezero.Ontheotherhand,someother ballsmighthaveagreaterkineticenergyinthenewframe.It'snot immediatelyobviousthatthetotalenergybeforethecollisionwill stillequalthetotalenergyafterthecollision.Afterall,theequation forkineticenergyisfairlycomplicated,sinceitinvolvesthesquare ofthevelocity,soitwouldbesurprisingifeverythingstillworked outinthenewframeofreference.It does stillworkout.Homework problem13inthischaptergivesasimplenumericalexample,and thegeneralproofistakenupinch.4,problem15withthesolution giveninthebackofthebook. DiscussionQuestions A Supposethat,likeYoungorEinstein,youweretryingoutdifferent equationsforkineticenergytoseeiftheyagreedwiththeexperimental data.Basedonthemeaningofpositiveandnegativesignsofvelocity, whywouldyoususpectthataproportionalityto mv wouldbelesslikely than mv 2 ? B ThegureshowsapendulumthatisreleasedatAandcaughtbya pegasitpassesthroughthevertical,B.Towhatheightwillthebobrise ontheright? 1.5Power Acarmayhaveplentyofenergyinitsgastank,butstillmaynot beabletoincreaseitskineticenergyrapidly.APorschedoesn't necessarilyhavemoreenergyinitsgastankthanaHyundai,itis justabletotransferitmorequickly.Therateoftransferringenergy fromoneformtoanotheriscalled power .Thedenitioncanbe writtenasanequation, P = E t wheretheuseofthedeltanotationinthesymbol E hastheusual interpretation:thenalamountofenergyinacertainformminus Section1.5Power 25

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theinitialamountthatwaspresentinthatform.Powerhasunits ofJ/s,whichareabbreviatedaswatts,Wrhymeswithlots". Iftherateofenergytransferisnotconstant,thepoweratany instantcanbedenedastheslopeofthetangentlineonagraphof E versus t .Likewise E canbeextractedfromtheareaunderthe P -versust curve. Convertingkilowatt-hourstojoulesexample8 TheelectriccompanybillsyouforenergyinunitsofkilowatthourskilowattsmultipliedbyhoursratherthaninSIunitsof joules.Howmanyjoulesisakilowatt-hour? 1kilowatt-hour=kWhour=J/ss=3.6MJ. Humanwattageexample9 Atypicalpersonconsumes2000kcaloffoodinaday,andconvertsnearlyallofthatdirectlytoheat.Comparetheperson'sheat outputtotherateofenergyconsumptionofa100-wattlightbulb. Lookinguptheconversionfactorfromcaloriestojoules,wend E =2000kcal 1000cal 1kcal 4.18J 1cal =8 10 6 J forourdailyenergyconsumption.Convertingthetimeinterval likewiseintomks, t =1day 24hours 1day 60min 1hour 60s 1min =9 10 4 s. Dividing,wendthatourpowerdissipatedasheatis90J/s=90 W,aboutthesameasalightbulb. Itiseasytoconfusetheconceptsofforce,energy,andpower, especiallysincetheyaresynonymsinordinaryspeech.Thetableon thefollowingpagemayhelptoclearthisup: 26 Chapter1ConservationofEnergy

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force energy power conceptual denition Aforceisaninteraction betweentwoobjectsthat causesapushorapull. Aforcecanbedenedas anythingthatiscapable ofchanginganobject's stateofmotion. Heatinganobject,makingitmovefaster,orincreasingitsdistancefrom anotherobjectthatisattractingitareallexamplesofthingsthatwould requirefuelorphysicaleffort.Allthesethingscan bequantiedusingasinglescaleofmeasurement, andwedescribethemall asformsofenergy. Poweristherateat whichenergyistransformedfromoneform toanotherortransferred fromoneobjecttoanother. operational denition Aspringscalecanbeused tomeasureforce. Ifwedeneaunitofenergyastheamountrequiredtoheatacertain amountofwaterbya 1 C,thenwecanmeasureanyotherquantity ofenergybytransferring itintoheatinwaterand measuringthetemperatureincrease. Measurethechangeinthe amountofsomeformof energypossessedbyan object,anddividebythe amountoftimerequired forthechangetooccur. scalaror vector? vector|hasadirection inspacewhichisthedirectioninwhichitpullsor pushes scalar|hasnodirection inspace scalar|hasnodirection inspace unit newtonsN joulesJ wattsW=joules/s Canitrun out?Doesit costmoney? No.Idon'thaveto payamonthlybillfor themeganewtonsofforce requiredtoholdupmy house. Yes.Wepaymoneyfor gasoline,electricalenergy, batteries,etc.,because theycontainenergy. Morepowermeansyou arepayingmoneyata higherrate.A100-W lightbulbcostsacertain numberofcentsperhour. Canitbea propertyof anobject? No.Aforceisarelationshipbetweentwo interactingobjects. Ahome-runbaseball doesn'thave"force. Yes.Whatahome-run baseballhasiskineticenergy,notforce. Notreally.A100-W lightbulbdoesn'thave" 100W.100J/sistherate atwhichitconvertselectricalenergyintolight. Section1.5Power 27

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Summary SelectedVocabulary energy......Anumericalscaleusedtomeasuretheheat, motion,orotherpropertiesthatwouldrequire fuelorphysicaleorttoputintoanobject;a scalarquantitywithunitsofjoulesJ. power.......Therateoftransferringenergy;ascalarquantitywithunitsofwattsW. kineticenergy..Theenergyanobjectpossessesbecauseofits motion. heat........Aformofenergythatrelatestotemperature. Heatisdierentfromtemperaturebecausean objectwithtwiceasmuchmassrequirestwice asmuchheattoincreaseitstemperatureby thesameamount.Heatismeasuredinjoules, temperatureindegrees.Instandardterminology,thereisanother,nerdistinctionbetweenheatandthermalenergy,whichisdiscussedbelow.Inthisbook,Iinformallyrefer tobothasheat. temperature...Whatathermometermeasures.Objectsleftin contactwitheachothertendtoreachthesame temperature.Cf.heat.Asdiscussedinmore detailinchapter2,temperatureisessentially ameasureoftheaveragekineticenergyper molecule. Notation E .........energy J..........joules,theSIunitofenergy KE ........kineticenergy P .........power W.........watts,theSIunitofpower;equivalenttoJ/s OtherTerminologyandNotation Q or Q .....theamountofheattransferredintooroutof anobject K or T ......alternativesymbolsforkineticenergy,usedin thescienticliteratureandinmostadvanced textbooks thermalenergy.Carefulwritersmakeadistinctionbetween heatandthermalenergy,butthedistinction isoftenignoredincasualspeech,evenamong physicists.Properly,thermalenergyisused tomeanthetotalamountofenergypossessed byanobject,whileheatindicatestheamount ofthermalenergytransferredinorout.The termheatisusedinthisbooktoincludeboth meanings. 28 Chapter1ConservationofEnergy

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Summary Heatinganobject,makingitmovefaster,orincreasingitsdistancefromanotherobjectthatisattractingitareallexamplesof thingsthatwouldrequirefuelorphysicaleort.makingitmove faster,orincreasingitsdistancefromanotherobjectthatisattractingitareallexamplesofthingsthatwouldrequirefuelorphysical eort.Allthesethingscanbequantiedusingasinglescaleof measurement,andwedescribethemallasformsof energy .The SIunitofenergyistheJoule.Thereasonwhyenergyisauseful andimportantquantityisthatitisalwaysconserved.Thatis,it cannotbecreatedordestroyedbutonlytransferredbetweenobjects orchangedfromoneformtoanother.Conservationofenergyisthe mostimportantandbroadlyapplicableofallthelawsofphysics, morefundamentalandgeneraleventhanNewton'slawsofmotion. Heatinganobjectrequiresacertainamountofenergyperdegree oftemperatureandperunitmass,whichdependsonthesubstance ofwhichtheobjectconsists.Heatandtemperaturearecompletely dierentthings.Heatisaformofenergy,anditsSIunitisthejoule J.Temperatureisnotameasureofenergy.Heatingtwiceasmuch ofsomethingrequirestwiceasmuchheat,butdoubletheamount ofasubstancedoesnothavedoublethetemperature. Theenergythatanobjectpossessesbecauseofitsmotionis calledkineticenergy.Kineticenergyisrelatedtothemassofthe objectandthemagnitudeofitsvelocityvectorbytheequation KE = 1 2 mv 2 Poweristherateatwhichenergyistransformedfromoneform toanotherortransferredfromoneobjecttoanother, P = E t .[onlyforconstantpower] TheSIunitofpoweristhewattW. Summary 29

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Problems Key p Acomputerizedanswercheckisavailableonline. R Aproblemthatrequirescalculus. ? Adicultproblem. 1 Thisproblemisnowproblem14inchapter2,onpage47. 2 Cankineticenergyeverbelessthanzero?Explain.[Based onaproblembySerwayandFaughn.] 3 EstimatethekineticenergyofanOlympicsprinter. 4 Youaredrivingyourcar,andyouhitabrickwallheadon, atfullspeed.Thecarhasamassof1500kg.Thekineticenergy releasedisameasureofhowmuchdestructionwillbedonetothecar andtoyourbody.Calculatetheenergyreleasedifyouaretraveling ata40mi/hr,andagainbifyou'regoing80mi/hr.Whatis counterintuitiveaboutthis,andwhatimplicationdoesthishavefor drivingathighspeeds? p 5 Aclosedsystemcanbeabadthing|foranastronaut sealedinsideaspacesuit,gettingridofbodyheatcanbedicult. Supposea60-kgastronautisperformingvigorousphysicalactivity, expending200Wofpower.Ifnoneoftheheatcanescapefromher spacesuit,howlongwillittakebeforeherbodytemperaturerises by6 C F,anamountsucienttokillher?Assumethatthe amountofheatrequiredtoraiseherbodytemperatureby1 Cis thesameasitwouldbeforanequalmassofwater.Expressyour answerinunitsofminutes. p 6 Allstars,includingoursun,showvariationsintheirlight outputtosomedegree.Somestarsvarytheirbrightnessbyafactor oftwoorevenmore,butoursunhasremainedrelativelysteadyduringthehundredyearsorsothataccuratedatahavebeencollected. Nevertheless,itispossiblethatclimatevariationssuchasiceages arerelatedtolong-termirregularitiesinthesun'slightoutput.If thesunwastoincreaseitslightoutputevenslightly,itcouldmelt enoughAntarcticicetooodalltheworld'scoastalcities.Thetotal sunlightthatfallsonAntarcticaamountstoabout1 10 16 watts. Presently,thisheatinputtothepolesisbalancedbythelossof heatviawinds,oceancurrents,andemissionofinfraredlight,so thatthereisnonetmeltingorfreezingoficeatthepolesfromyear toyear.Supposethatthesunchangesitslightoutputbysomesmall percentage,butthereisnochangeintherateofheatlossbythe polarcaps.Estimatethepercentagebywhichthesun'slightoutput wouldhavetoincreaseinordertomeltenoughicetoraisethelevel oftheoceansby10metersoveraperiodof10years.Thiswouldbe enoughtooodNewYork,London,andmanyothercities.Melting 1kgoficerequires3 10 3 J. 30 Chapter1ConservationofEnergy

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7 Abulletiesthroughtheair,passesthroughapaperback book,andthencontinuestoythroughtheairbeyondthebook. Whenisthereaforce?Whenisthereenergy? Solution,p.167 8 Experimentsshowthatthepowerconsumedbyaboat'sengineisapproximatelyproportionaltothirdpowerofitsspeed.We assumethatitismovingatconstantspeed.aWhenaboatiscrusingatconstantspeed,whattypeofenergytransformationdoyou thinkisbeingperformed?bIfyouupgradetoamotorwithdouble thepower,bywhatfactorisyourboat'scrusingspeedincreased? [BasedonaproblembyArnoldArons.] Solution,p.167 9 ObjectAhasakineticenergyof13.4J.ObjectBhasamass thatisgreaterbyafactorof3.77,butismovingmoreslowlyby afactorof2.34.WhatisobjectB'skineticenergy?[Basedona problembyArnoldArons.] Solution,p.167 10 Themoondoesn'treallyjustorbittheEarth.ByNewton's thirdlaw,themoon'sgravitationalforceontheearthisthesameas theearth'sforceonthemoon,andtheearthmustrespondtothe moon'sforcebyaccelerating.Ifweconsidertheearthinmoonin isolationandignoreoutsideforces,thenNewton'srstlawsaystheir commoncenterofmassdoesn'taccelerate,i.e.,theearthwobbles aroundthecenterofmassoftheearth-moonsystemoncepermonth, andthemoonalsoorbitsaroundthispoint.Themoon'smassis81 timessmallerthantheearth's.Comparethekineticenergiesofthe earthandmoon.Weknowthatthecenterofmassisakindof balancepoint,soitmustbeclosertotheearththantothemoon. Infact,thedistancefromtheearthtothecenterofmassis1/81 ofthedistancefromthemoontothecenterofmass,whichmakes senseintuitively,andcanbeprovedrigorouslyusingtheequation onpage89. 11 My1.25kWmicrowaveoventakes126secondstobring250 gofwaterfromroomtemperaturetoaboil.Whatpercentageof thepowerisbeingwasted?Wheremighttherestoftheenergybe going? Solution,p.167 Problems 31

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12 Themultiashphotographshowsacollisionbetweentwo poolballs.Theballthatwasinitiallyatrestshowsupasadark imageinitsinitialposition,becauseitsimagewasexposedseveral timesbeforeitwasstruckandbeganmoving.Bymakingmeasurementsonthegure,determinewhetherornotenergyappearsto havebeenconservedinthecollision.Whatsystematiceectswould limittheaccuracyofyourtest?[FromanexampleinPSSCPhysics.] Problem12. 13 Thisproblemisanumericalexampleoftheimaginaryexperimentdiscussedattheendofsection1.4regardingtherelationship betweenenergyandrelativemotion.Let'ssaythatthepoolballs bothhavemassesof1.00kg.Supposethatintheframeofreference ofthepooltable,thecueballmovesataspeedof1.00m/stoward theeightball,whichisinitiallyatrest.Thecollisionishead-on,and asyoucanverifyforyourselfthenexttimeyou'replayingpool,the resultofsuchacollisionisthattheincomingballstopsdeadand theballthatwasstrucktakesowiththesamespeedoriginally possessedbytheincomingball.Thisisactuallyabitofanidealization.Tokeepthingssimple,we'reignoringthespinoftheballs, andweassumethatnoenergyisliberatedbythecollisionasheator sound.aCalculatethetotalinitialkineticenergyandthetotal nalkineticenergy,andverifythattheyareequal.bNowcarry outthewholecalculationagainintheframeofreferencethatis movinginthesamedirectionthatthecueballwasinitiallymoving, butataspeedof0.50m/s.Inthisframeofreference,bothballs havenonzeroinitialandnalvelocities,whicharedierentfrom whattheywereinthetable'sframe.[Seealsohomeworkproblem 15inch.4.] 32 Chapter1ConservationofEnergy

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14 OnetheoryaboutthedestructionofthespaceshuttleColumbia in2003isthatoneofitswingshadbeendamagedonliftobya chunkoffoaminsulationthatfelloofoneofitsexternalfueltanks. TheNewYorkTimesreportedonJune5,2003,thatNASAengineershadrecreatedtheimpacttoseeifitwoulddamageamock-up oftheshuttle'swing.Beforelastweek'stest,manyengineersat NASAsaidtheythoughtlightweightfoamcouldnotharmtheseeminglytoughcompositepanels,andprivatelypredictedthatthefoam wouldbounceoharmlessly,likeaNerfball."Infact,the1.7-pound pieceoffoam,movingat531milesperhour,didseriousdamage. Amemberoftheboardinvestigatingthedisastersaidthisdemonstratedthatpeople'sintuitivesenseofphysicsissometimesway o."aComputethekineticenergyofthefoam,andbcompare withtheenergyofa170-poundbouldermovingat5.3milesper hourthespeeditwouldhaveifyoudroppeditfromaboutkneelevel.cTheboulderisahundredtimesmoremassive,butits speedisahundredtimessmaller,sowhat'scounterintuitiveabout yourresults? 15 Thegureaboveisfromaclassic1920physicstextbook byMillikanandGale.Itrepresentsamethodforraisingthewater fromtheponduptothewatertower,atahigherlevel,without usingapump.Waterisallowedintothedrivepipe,andonceitis owingfastenough,itforcesthevalveatthebottomclosed.Explain howthisworksintermsofconservationofmassandenergy.Cf. example1onpage15. Problems 33

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Dotheseformsofenergyhaveanythingincommon? Chapter2 SimplifyingtheEnergyZoo Varietyisthespiceoflife,notofscience.Thegureshowsafew examplesfromthebewilderingarrayofformsofenergythatsurroundsus.Thephysicist'spsycherebelsagainsttheprospectofa longlaundrylistoftypesofenergy,eachofwhichwouldrequire itsownequations,concepts,notation,andterminology.Thepoint atwhichwe'vearrivedinthestudyofenergyisanalogoustothe periodinthe1960'swhenahalfadozennewsubatomicparticles werebeingdiscoveredeveryyearinparticleaccelerators.Itwasan embarrassment.Physicistsbegantospeakoftheparticlezoo," anditseemedthatthesubatomicworldwasdistressinglycomplex. Theparticlezoowassimpliedbytherealizationthatmostofthe 35

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newparticlesbeingwhippedupweresimplyclustersofapreviously unsuspectedsetofmorefundamentalparticleswhichwerewhimsicallydubbedquarks,amade-upwordfromalineofpoetrybyJames Joyce,ThreequarksforMasterMark."Theenergyzoocanalso besimplied,anditisthepurposeofthischaptertodemonstrate thehiddensimilaritiesbetweenformsofenergyasseeminglydierentasheatandmotion. a / Avividdemonstrationthat heatisaformofmotion.Asmall amountofboilingwaterispoured intotheemptycan,whichrapidly llsupwithhotsteam.Thecan isthensealedtightly,andsoon crumples.Thiscanbeexplained asfollows.Thehightemperatureofthesteamisinterpretedas ahighaveragespeedofrandom motionsofitsmolecules.Before thelidwasputonthecan,the rapidlymovingsteammolecules pushedtheirwayoutofthecan, forcingtheslowerairmolecules outoftheway.Asthesteaminsidethecanthinnedout,astablesituationwassoonachieved, inwhichtheforcefromtheless densesteammoleculesmoving athighspeedbalancedagainst theforcefromthemoredensebut slowerairmoleculesoutside.The capwasputon,andafterawhile thesteaminsidethecanreached thesametemperatureastheair outside.Theforcefromthecool, thinsteamnolongermatchedthe forcefromthecool,denseairoutside,andtheimbalanceofforces crushedthecan. 2.1HeatisKineticEnergy Whatisheatreally?Isitaninvisibleuidthatyourbarefeetsoak upfromahotsidewalk?Canoneeverremovealltheheatfroman object?Isthereamaximumtothetemperaturescale? Thetheoryofheatasauidseemedtoexplainwhycolderobjectsabsorbedheatfromhotterones,butonceitbecameclearthat heatwasaformofenergy,itbegantoseemunlikelythatamaterial substancecouldtransformitselfintoandoutofallthoseotherforms ofenergylikemotionorlight.Forinstance,acompostpilegetshot, andwedescribethisasacasewhere,throughtheactionofbacteria, chemicalenergystoredintheplantcuttingsistransformedintoheat energy.Theheatingoccursevenifthereisnonearbywarmerobject thatcouldhavebeenleakingheatuid"intothepile. Analternativeinterpretationofheatwassuggestedbythetheory thatmatterismadeofatoms.Sincegasesarethousandsoftimesless densethansolidsorliquids,theatomsorclustersofatomscalled moleculesinagasmustbefarapart.Inthatcase,whatiskeeping alltheairmoleculesfromsettlingintoathinlmontheoorofthe roominwhichyouarereadingthisbook?Thesimplestexplanation isthattheyaremovingveryrapidly,continuallyricochetingoof 36 Chapter2SimplifyingtheEnergyZoo

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b / Randommotionofatoms inagas,aliquid,andasolid. theoor,walls,andceiling.Thoughbizarre,thecloud-of-bullets imageofagasdidgiveanaturalexplanationforthesurprising abilityofsomethingastenuousasagastoexerthugeforces.Your car'stirescanholditupbecauseyouhavepumpedextramolecules intothem.Theinsideofthetiregetshitbymoleculesmoreoften thantheoutside,forcingittostretchandstien. Theoutwardforcesoftheairinyourcar'stiresincreaseeven furtherwhenyoudriveonthefreewayforawhile,heatingupthe rubberandtheairinside.Thistypeofobservationleadsnaturally totheconclusionthathottermatterdiersfromcolderinthatits atoms'randommotionismorerapid.Inaliquid,themotioncould bevisualizedaspeopleinamillingcrowdshovingpasteachother morequickly.Inasolid,wheretheatomsarepackedtogether,the motionisarandomvibrationofeachatomasitknocksagainstits neighbors. Wethusachieveagreatsimplicationinthetheoryofheat.Heat issimplyaformofkineticenergy,thetotalkineticenergyofrandom motionofalltheatomsinanobject.Withthisnewunderstanding, itbecomespossibletoansweratonestrokethequestionsposedat thebeginningofthesection.Yes,itisatleasttheoreticallypossible toremovealltheheatfromanobject.Thecoldestpossibletemperature,knownasabsolutezero,isthatatwhichalltheatomshave zerovelocity,sothattheirkineticenergies, = 2 mv 2 ,areallzero. No,thereisnomaximumamountofheatthatacertainquantityof mattercanhave,andnomaximumtothetemperaturescale,since arbitrarilylargevaluesof v cancreatearbitrarilylargeamountsof kineticenergyperatom. Thekinetictheoryofheatalsoprovidesasimpleexplanationof thetruenatureoftemperature.Temperatureisameasureofthe amountofenergypermolecule,whereasheatisthetotalamountof energypossessedbyallthemoleculesinanobject. Thereisanentirebranchofphysics,calledthermodynamics, thatdealswithheatandtemperatureandformsthebasisfortechnologiessuchasrefrigeration.Thermodynamicsisdiscussedinmore detailinoptionalchapterA,andIhaveprovidedhereonlyabrief overviewofthethermodynamicconceptsthatrelatedirectlytoenergy,glossingoveratleastonepointthatwouldbedealtwithmore carefullyinathermodynamicscourse:itisreallyonlytruefora gasthatalltheheatisintheformofkineticenergy.Insolidsand liquids,theatomsarecloseenoughtoeachothertoexertintense electricalforcesoneachother,andthereisthereforeanothertype ofenergyinvolved,theenergyassociatedwiththeatoms'distances fromeachother.Strictlyspeaking,heatenergyisdenednotas energyassociatedwithrandommotionofmoleculesbutasanyform ofenergythatcanbeconductedbetweenobjectsincontact,without anyforce. Section2.1HeatisKineticEnergy 37

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c / Theskaterhasconverted allhiskineticenergyintopotential energyonthewayupthesideof thepool.PhotobyJ.D.Rogge, www.sonic.net/ shawn. 2.2PotentialEnergy:EnergyofDistanceor Closeness Wehavealreadyseenmanyexamplesofenergyrelatedtothedistancebetweeninteractingobjects.Whentwoobjectsparticipatein anattractivenoncontactforce,energyisrequiredtobringthemfartherapart.Inbothoftheperpetualmotionmachinesthatstarted othepreviouschapter,oneofthetypesofenergyinvolvedwasthe energyassociatedwiththedistancebetweentheballsandtheearth, whichattracteachothergravitationally.Intheperpetualmotion machinewiththemagnetonthepedestal,therewasalsoenergy associatedwiththedistancebetweenthemagnetandtheironball, whichwereattractingeachother. Theoppositehappenswithrepulsiveforces:twosockswiththe sametypeofstaticelectricchargewillrepeleachother,andcannot bepushedclosertogetherwithoutsupplyingenergy. Ingeneral,theterm potentialenergy, withalgebrasymbol PE, is usedfortheenergyassociatedwiththedistancebetweentwoobjects thatattractorrepeleachotherviaaforcethatdependsonthe distancebetweenthem.Forcesthatarenotdeterminedbydistance donothavepotentialenergyassociatedwiththem.Forinstance, thenormalforceactsonlybetweenobjectsthathavezerodistance betweenthem,anddependsonotherfactorsbesidesthefactthat thedistanceiszero.Thereisnopotentialenergyassociatedwith thenormalforce. Thefollowingaresomecommonplaceexamplesofpotentialenergy: gravitationalpotentialenergy: Theskateboarderinthephoto hasrisenfromthebottomofthepool,convertingkineticenergyintogravitationalpotentialenergy.Afterbeingatrest foraninstant,hewillgobackdown,convertingPEbackinto KE. magneticpotentialenergy: Whenamagneticcompassneedleis allowedtorotate,thepolesofthecompasschangetheirdistancesfromtheearth'snorthandsouthmagneticpoles,convertingmagneticpotentialenergyintokineticenergy.Eventuallythekineticenergyisallchangedintoheatbyfriction, andtheneedlesettlesdowninthepositionthatminimizesits potentialenergy. electricalpotentialenergy: Sockscomingoutofthedryercling togetherbecauseofattractiveelectricalforces.Energyisrequiredinordertoseparatethem. potentialenergyofbendingorstretching: Theforcebetween thetwoendsofaspringdependsonthedistancebetween 38 Chapter2SimplifyingtheEnergyZoo

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d / Astheskaterfree-falls, hisPEisconvertedintoKE.The numberswouldbeequallyvalid asadescriptionofhismotionon thewayup. them,i.e.,onthelengthofthespring.Ifacarispressed downonitsshockabsorbersandthenreleased,thepotential energystoredinthespringistransformedintokineticand gravitationalpotentialenergyasthecarbouncesbackup. Ihavedeliberatelyavoidedintroducingthetermpotentialenergyupuntilthispoint,becauseittendstoproduceunfortunate connotationsinthemindsofstudentswhohavenotyetbeeninoculatedwithacarefuldescriptionoftheconstructionofanumerical energyscale.Specically,thereisatendencytogeneralizetheterm inappropriatelytoapplytoanysituationwherethereisthepotential"forsomethingtohappen:Itookabreakfromdigging,but IhadpotentialenergybecauseIknewI'dbereadytoworkhard againinafewminutes." Anequationforgravitationalpotentialenergy Allthevitalpointsaboutpotentialenergycanbemadebyfocusingontheexampleofgravitationalpotentialenergy.Forsimplicity, wetreatonlyverticalmotion,andmotionclosetothesurfaceofthe earth,wherethegravitationalforceisnearlyconstant.Thegeneralizationtothethreedimensionsandvaryingforcesismoreeasily accomplishedusingtheconceptofwork,whichisthesubjectthe nextchapter. TondanequationforgravitationalPE,weexaminethecase offreefall,inwhichenergyistransformedbetweenkineticenergy andgravitationalPE.Whateverenergyislostinoneformisgained inanequalamountintheotherform,sousingthenotation KE tostandfor KE f )]TJ/F20 10.9091 Tf 10.909 0 Td [(KE i andasimilarnotationforPE,wehave [1] KE = )]TJ/F15 10.9091 Tf 8.485 0 Td [( PE grav Itwillbeconvenienttorefertotheobjectasfalling,sothatPE isbeingchangedintoKE,butthemathappliesequallywelltoan objectslowingdownonitswayup.Weknowanequationforkinetic energy, [2] KE = 1 2 mv 2 soifwecanrelate v toheight, y ,wewillbeabletorelate PE to y whichwouldtelluswhatwewanttoknowaboutpotentialenergy. The y componentofthevelocitycanbeconnectedtotheheightvia theconstantaccelerationequation [3] v 2 f = v 2 i +2 a y andNewton'ssecondlawprovidestheacceleration, [4] a = F=m intermsofthegravitationalforce. Section2.2PotentialEnergy:EnergyofDistanceorCloseness 39

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Thealgebraissimplebecausebothequation[2]andequation[3] havevelocitytothesecondpower.Equation[2]canbesolvedfor v 2 togive v 2 =2 KE=m ,andsubstitutingthisintoequation[3],we nd 2 KE f m =2 KE i m +2 a y Makinguseofequations[1]and[4]givesthesimpleresult PE grav = )]TJ/F20 10.9091 Tf 8.485 0 Td [(F y .[changeingravitationalPE resultingfromachangeinheight y ; F isthegravitationalforceontheobject, i.e.,itsweight;validonlynearthesurface oftheearth,where F isconstant] Droppingarockexample1 Ifyoudropa1-kgrockfromaheightof1m,howmanyjoules ofKEdoesithaveonimpactwiththeground?Assumethatany energytransformedintoheatbyairfrictionisnegligible. Ifwechoosethe y axistopointup,then F y isnegative,and equals )]TJ/F39 10.9091 Tf 8.485 0 Td [(kg g = )]TJ/F39 10.9091 Tf 8.485 0 Td [(9.8N.Adecreasein y isrepresentedbya negativevalueof y y = )]TJ/F39 10.9091 Tf 8.485 0 Td [(1m,sothechangeinpotentialenergyis )]TJ/F39 10.9091 Tf 8.485 0 Td [( )]TJ/F39 10.9091 Tf 8.485 0 Td [(9.8N )]TJ/F39 10.9091 Tf 8.485 0 Td [(1m )]TJ/F39 10.9091 Tf 20.311 0 Td [(10J.Theproofthatnewtonsmultipliedbymetersgiveunitsofjoulesisleftasahomeworkproblem.Conservationofenergysaysthatthelossofthisamountof PEmustbeaccompaniedbyacorrespondingincreaseinKEof 10J. Itmaybedismayingtonotehowmanyminussignshadtobe handledcorrectlyeveninthisrelativelysimpleexample:atotal offour.Ratherthandependingonyourselftoavoidanymistakes withsigns,itisbettertocheckwhetherthenalresultmakesense physically.Ifitdoesn't,justreversethesign. Althoughtheequationforgravitationalpotentialenergywasderivedbyimaginingasituationwhereitwastransformedintokinetic energy,theequationcanbeusedinanycontext,becauseallthe typesofenergyarefreelyconvertibleintoeachother. GravitationalPEconverteddirectlyintoheatexample2 A50-kgreghterslidesdowna5-mpoleatconstantvelocity. Howmuchheatisproduced? Sincesheslidesdownatconstantvelocity,thereisnochange inKE.HeatandgravitationalPEaretheonlyformsofenergythat change.Ignoringplusandminussigns,thegravitationalforceon herbodyequals mg ,andtheamountofenergytransformedis mg m=2500J. 40 Chapter2SimplifyingtheEnergyZoo

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Onphysicalgrounds,weknowthattheremusthavebeenanincreasepositivechangeintheheatenergyinherhandsandin theagpole. Herearesomequestionsandanswersabouttheinterpretationof theequation PE grav = )]TJ/F20 10.9091 Tf 8.485 0 Td [(F y forgravitationalpotentialenergy. Question: Inanutshell,whyisthereaminussignintheequation? Answer: ItisbecauseweincreasethePEbymovingtheobjectin the opposite directioncomparedtothegravitationalforce. Question: Whydoweonlygetanequationforthe change inpotentialenergy?Don'tIreallywantanequationforthepotential energyitself? Answer: No,youreallydon't.Thisrelatestoabasicfactabout potentialenergy,whichisthatitisnotawelldenedquantityin theabsolutesense.Onlychangesinpotentialenergyareunambiguouslydened.IfyouandIbothobservearockfalling,andagree thatitdeposits10Jofenergyinthedirtwhenithits,thenwewill beforcedtoagreethatthe10JofKEmusthavecomefromaloss of10joulesofPE.ButImightclaimthatitstartedwith37JofPE andendedwith27,whileyoumightswearjustastruthfullythatit had109Jinitiallyand99attheend.Itispossibletopicksome specicheightasareferencelevelandsaythatthePEiszerothere, butit'seasierandsaferjusttoworkwithchangesinPEandavoid absolutePEaltogether. Question: Youreferredtopotentialenergyastheenergythat two objectshavebecauseoftheirdistancefromeachother.Ifarock falls,theobjectistherock.Where'stheotherobject? Answer: Newton'sthirdlawguaranteesthattherewillalwaysbe twoobjects.Theotherobjectistheplanetearth. Question: Iftheotherobjectistheearth,arewetalkingaboutthe distancefromtherocktothecenteroftheearthorthedistance fromtherocktothesurfaceoftheearth? Answer: Itdoesn'tmatter.Allthatmattersisthechangeindistance, y ,not y .Measuringfromtheearth'scenteroritssurface arejusttwoequallyvalidchoicesofareferencepointfordening absolutePE. Question: WhichobjectcontainsthePE,therockortheearth? Answer: WemayrefercasuallytothePEoftherock,buttechnicallythePEisarelationshipbetweentheearthandtherock,and weshouldrefertotheearthandtherocktogetheraspossessingthe PE. Question: Howwouldthisbeanydierentforaforceotherthan gravity? Answer: Itwouldn't.Theresultwasderivedundertheassumption ofconstantforce,buttheresultwouldbevalidforanyothersituationwheretwoobjectsinteractedthroughaconstantforce.Gravity Section2.2PotentialEnergy:EnergyofDistanceorCloseness 41

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e / Alltheseenergytransformationsturnoutattheatomic leveltobechangesinpotential energyresultingfromchangesin thedistancesbetweenatoms. isunusual,however,inthatthegravitationalforceonanobjectis sonearlyconstantunderordinaryconditions.Themagneticforce betweenamagnetandarefrigerator,ontheotherhand,changes drasticallywithdistance.Themathisalittlemorecomplexfora varyingforce,buttheconceptsarethesame. Question: Supposeapencilisbalancedonitstipandthenfalls over.Thepencilissimultaneouslychangingitsheightandrotating, sotheheightchangeisdierentfordierentpartsoftheobject. Thebottomofthepencildoesn'tloseanyheightatall.Whatdo youdointhissituation? Answer: Thegeneralphilosophyofenergyisthatanobject'senergyisfoundbyaddinguptheenergyofeverylittlepartofit. Youcouldthusaddupthechangesinpotentialenergyofallthe littlepartsofthepenciltondthetotalchangeinpotentialenergy.Luckilythere'saneasierway!Thederivationoftheequation forgravitationalpotentialenergyusedNewton'ssecondlaw,which dealswiththeaccelerationoftheobject'scenterofmassi.e.,its balancepoint.Ifyoujustdene y astheheightchangeofthe centerofmass,everythingworksout.AhugeFerriswheelcanbe rotatedwithoutputtinginortakingoutanyPE,becauseitscenter ofmassisstayingatthesameheight. self-checkA Aballthrownstraightupwillhavethesamespeedonimpactwiththe groundasaballthrownstraightdownatthesamespeed.Howcanthis beexplainedusingpotentialenergy? Answer,p.166 DiscussionQuestion A Youthrowasteelballupintheair.Howcanyouprovebasedon conservationofenergythatithasthesamespeedwhenitfallsbackinto yourhand?Whatifyouthrowafeatherupisenergynotconservedin thiscase? 2.3AllEnergyisPotentialorKinetic Inthesamewaythatwefoundthatachangeintemperature isreallyonlyachangeinkineticenergyattheatomiclevel,we nowndthateveryotherformofenergyturnsouttobeaform ofpotentialenergy.Boiling,forinstance,meansknockingsomeof theatomsormoleculesoutoftheliquidandintothespaceabove, wheretheyconstituteagas.Thereisanetattractiveforcebetween essentiallyanytwoatomsthatarenexttoeachother,whichiswhy matteralwayspreferstobepackedtightlyinthesolidorliquidstate unlesswesupplyenoughpotentialenergytopullitapartintoagas. Thisexplainswhywaterstopsgettinghotterwhenitreachesthe boilingpoint:thepowerbeingpumpedintothewaterbyyourstove beginsgoingintopotentialenergyratherthankineticenergy. Asshowninguree,everystoredformofenergythatween42 Chapter2SimplifyingtheEnergyZoo

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f / Thisgurelookssimilarto thepreviousones,butthescale isamilliontimessmaller.The littleballsaretheneutronsand protonsthatmakeupthetinynucleusatthecenteroftheuranium atom.Whenthenucleussplits ssions,thepotentialenergy changeispartlyelectricaland partlyachangeinthepotential energyderivedfromtheforce thatholdsatomicnucleitogether knownasthestrongnuclear force. g / Apelletofplutonium-238 glowswithitsownheat.Its nuclearpotentialenergyisbeing convertedintoheat,aformof kineticenergy.Pelletsofthistype areusedaspowersupplieson somespaceprobes. counterineverydaylifeturnsouttobeaformofpotentialenergy attheatomiclevel.Theforcesbetweenatomsareelectricaland magneticinnature,sotheseareactuallyelectricalandmagnetic potentialenergies. Althoughlightisatopicofthesecondhalfofthiscourse,itis usefultohaveapreviewhowittsinhere.Lightisawavecomposed ofoscillatingelectricandmagneticelds,sowecanincludeitunder thecategoryofelectricalandmagneticpotentialenergy. Evenifwewishtoincludenuclearreactionsinthepicture,there stillturnouttobeonlyfourfundamentaltypesofenergy: kineticenergy includingheat gravitationalpotentialenergy electricalandmagneticpotentialenergyincluding light nuclearpotentialenergy DiscussionQuestion A Referringbacktothepicturesatthebeginningofthechapter,how doalltheseformsofenergytintotheshortenedlistofcategoriesgiven above? Section2.3AllEnergyisPotentialorKinetic 43

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Summary SelectedVocabulary potentialenergytheenergyhavingtodowiththedistancebetweentwoobjectsthatinteractviaanoncontactforce Notation PE.........potentialenergy OtherTerminologyandNotation U or V ......symbolsusedforpotentialenergyinthescienticliteratureandinmostadvancedtextbooks Summary Historically,theenergyconceptwasonlyinventedtoincludea fewphenomena,butitwaslatergeneralizedmoreandmoretoapply tonewsituations,forexamplenuclearreactions.Thisgeneralizing processresultedinanundesirablylonglistoftypesofenergy,each ofwhichapparentlybehavedaccordingtoitsownrules. Therststepinsimplifyingthepicturecamewiththerealization thatheatwasaformofrandommotionontheatomiclevel,i.e.,heat wasnothingmorethanthekineticenergyofatoms. Asecondandevengreatersimplicationwasachievedwiththe realizationthatalltheotherapparentlymysteriousformsofenergy actuallyhadtodowithchangingthedistancesbetweenatomsor similarprocessesinnuclei.Thistypeofenergy,whichrelatesto thedistancebetweenobjectsthatinteractviaaforce,istherefore ofgreatimportance.Wecallitpotentialenergy. Mostoftheimportantideasaboutpotentialenergycanbeunderstoodbystudyingtheexampleofgravitationalpotentialenergy. Thechangeinanobject'sgravitationalpotentialenergyisgivenby PE grav = )]TJ/F20 10.9091 Tf 8.485 0 Td [(F grav y ,[if F grav isconstant,i.e.,the themotionisallneartheEarth'ssurface] Themostimportantthingtounderstandaboutpotentialenergy isthatthereisnounambiguouswaytodeneitinanabsolutesense. Theonlythingthateveryonecanagreeonishowmuchthepotential energyhaschangedfromonemomentintimetosomelatermoment intime. 44 Chapter2SimplifyingtheEnergyZoo

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Problems Key p Acomputerizedanswercheckisavailableonline. R Aproblemthatrequirescalculus. ? Adicultproblem. 1 Cangravitationalpotentialenergyeverbenegative?Note thatthequestionrefersto PE ,not PE ,sothatyoumustthink abouthowthechoiceofareferencelevelcomesintoplay.[Basedon aproblembySerwayandFaughn.] 2 Aballrollsuparamp,turnsaround,andcomesbackdown. Whendoesithavethegreatestgravitationalpotentialenergy?The greatestkineticenergy?[BasedonaproblembySerwayandFaughn.] 3 aYoureleaseamagnetonatabletopnearabigpieceof iron,andthemagnetslidesacrossthetabletotheiron.Doesthe magneticpotentialenergyincreaseordecrease?Explain. bSupposeinsteadthatyouhavetworepellingmagnets.Yougive themaninitialpushtowardseachother,sotheydeceleratewhileapproachingeachother.Doesthemagneticpotentialenergyincrease ordecrease?Explain. 4 Let E b betheenergyrequiredtoboilonekgofwater.aFind anequationfortheminimumheightfromwhichabucketofwater mustbedroppediftheenergyreleasedonimpactistovaporizeit. Assumethatalltheheatgoesintothewater,notintothedirtit strikes,andignoretherelativelysmallamountofenergyrequiredto heatthewaterfromroomtemperatureto100 C.[Numericalcheck, notforcredit:Pluggingin E b =2.3MJ/kgshouldgivearesultof 230km.] p bShowthattheunitsofyouranswerinpartacomeoutright basedontheunitsgivenfor E b 5 Agrasshopperwithamassof110mgfallsfromrestfroma heightof310cm.Onthewaydown,itdissipates1.1mJofheatdue toairresistance.Atwhatspeed,inm/s,doesithittheground? Solution,p.168 6 Apersononabicycleistocoastdownarampofheight h and thenpassthroughacircularloopofradius r .Whatisthesmallestvalueof h forwhichthecyclistwillcompletetheloopwithout falling?Ignorethekineticenergyofthespinningwheels. p Problems 45

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Problem7. 7 Askateboarderstartsatnearlyrestatthetopofagiant cylinder,andbeginsrollingdownitsside.Ifhestartedexactlyat restandexactlyatthetop,hewouldnevergetgoing!Showthathis boardlosescontactwiththepipeafterhehasdroppedbyaheight equaltoonethirdtheradiusofthepipe. Solution,p.168 ? 8 aAcircularhoopofmass m andradius r spinslikeawheel whileitscenterremainsatrest.Itsperiodtimerequiredforone revolutionis T .Showthatitskineticenergyequals2 2 mr 2 =T 2 bIfsuchahooprollswithitscentermovingatvelocity v ,its kineticenergyequals = 2 mv 2 ,plustheamountofkineticenergy foundintherstpartofthisproblem.Showthatahooprollsdown aninclinedplanewithhalftheaccelerationthatafrictionlesssliding blockwouldhave. ? 9 Studentsareoftentemptedtothinkofpotentialenergyand kineticenergyasiftheywerealwaysrelatedtoeachother,like yinandyang.Toshowthisisincorrect,giveexamplesofphysical situationsinwhichaPEisconvertedtoanotherformofPE,and bKEisconvertedtoanotherformofKE. Solution,p.168 10 LordKelvin,aphysicist,toldthestoryofhowheencountered JamesJoulewhenJoulewasonhishoneymoon.Ashetraveled, Joulewouldstopwithhiswifeatvariouswaterfalls,andmeasure thedierenceintemperaturebetweenthetopofthewaterfalland thestillwateratthebottom.aItwouldsurprisemostpeople tolearnthatthetemperatureincreased.Whyshouldtherebeany sucheect,andwhywouldJoulecare?Howwouldthisrelatetothe energyconcept,ofwhichhewastheprincipalinventor?bHow muchofagainintemperatureshouldtherebebetweenthetop andbottomofa50-meterwaterfall?cWhatassumptionsdidyou havetomakeinordertocalculateyouranswertopartb?Inreality, wouldthetemperaturechangebemorethanorlessthanwhatyou calculated?[BasedonaproblembyArnoldArons.] p 11 Makeanorder-of-magnitudeestimateofthepowerrepresentedbythelossofgravitationalenergyofthewatergoingover NiagaraFalls.Ifthehydroelectricplantatthebottomofthefalls couldconvert100%ofthistoelectricalpower,roughlyhowmany householdscouldbepowered? Solution,p.168 12 Whenyoubuyahelium-lledballoon,thesellerhastoinate itfromalargemetalcylinderofthecompressedgas.Thehelium insidethecylinderhasenergy,ascanbedemonstratedforexample byreleasingalittleofitintotheair:youhearahissingsound, andthatsoundenergymusthavecomefromsomewhere.Thetotal amountofenergyinthecylinderisverylarge,andifthevalveis inadvertentlydamagedorbrokeno,thecylindercanbehavelike bomborarocket. Supposethecompanythatputsthegasinthecylindersprepares 46 Chapter2SimplifyingtheEnergyZoo

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cylinderAwithhalfthenormalamountofpurehelium,andcylinder Bwiththenormalamount.CylinderBhastwiceasmuchenergy, andyetthetemperaturesofbothcylindersarethesame.Explain,at theatomiclevel,whatformofenergyisinvolved,andwhycylinder Bhastwiceasmuch. 13 Atagiventemperature,theaveragekineticenergyper moleculeisaxedvalue,soforinstanceinair,themoremassive oxygenmoleculesaremovingmoreslowlyontheaveragethanthe nitrogenmolecules.Theratioofthemassesofoxygenandnitrogen moleculesis16to14.Nowsupposeavesselcontainingsomeairis surroundedbyavacuum,andthevesselhasatinyholeinit,which allowstheairtoslowlyleakout.Themoleculesarebouncingaround randomly,soagivenmoleculewillhavetotry"manytimesbefore itgetsluckyenoughtoheadoutthroughthehole.Howmanytimes morerapidlydoesthenitrogenescape? 14 Explainintermsofconservationofenergywhysweating coolsyourbody,eventhoughthesweatisatthesametemperature asyourbody.Describetheformsofenergyinvolvedinthisenergy transformation.Whydon'tyougetthesamecoolingeectifyou wipethesweatowithatowel?Hint:Thesweatisevaporating. Problems 47

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Chapter3 Work:TheTransferof MechanicalEnergy 3.1Work:TheTransferofMechanicalEnergy Theconceptofwork Themasscontainedinaclosedsystemisaconservedquantity, butifthesystemisnotclosed,wealsohavewaysofmeasuringthe amountofmassthatgoesinorout.Thewatercompanydoesthis withameterthatrecordsyourwateruse. Likewise,weoftenhaveasystemthatisnotclosed,andwould liketoknowhowmuchenergycomesinorout.Energy,however, isnotaphysicalsubstancelikewater,soenergytransfercannot bemeasuredwiththesamekindofmeter.Howcanwetell,for instance,howmuchusefulenergyatractorcanputout"onone tankofgas? Thelawofconservationofenergyguaranteesthatallthechem49

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a / Workisatransferofenergy. b / Thetractorraisestheweight overthepulley,increasingits gravitationalpotentialenergy. c / Thetractoraccelerates thetrailer,increasingitskinetic energy. d / Thetractorpullsaplow. Energyisexpendedinfrictional heatingoftheplowandthedirt, andinbreakingdirtclodsand liftingdirtuptothesidesofthe furrow. icalenergyinthegasolinewillreappearinsomeform,butnotnecessarilyinaformthatisusefulfordoingfarmwork.Tractors,like cars,areextremelyinecient,andtypically90%oftheenergythey consumeisconverteddirectlyintoheat,whichiscarriedawayby theexhaustandtheairowingovertheradiator.Wewishtodistinguishtheenergythatcomesoutdirectlyasheatfromtheenergy thatservestoaccelerateatrailerortoplowaeld,sowedene atechnicalmeaningoftheordinarywordwork"toexpressthe distinction: denitionofwork Workistheamountofenergytransferredintooroutofa system,notcountingenergytransferredbyheatconduction. self-checkA Basedonthisdenition,isworkavector,orascalar?Whatareits units? Answer,p.166 Theconductionofheatistobedistinguishedfromheatingby friction.Whenahotpotatoheatsupyourhandsbyconduction,the energytransferoccurswithoutanyforce,butwhenfrictionheats yourcar'sbrakeshoes,thereisaforceinvolved.Thetransferofenergywithandwithoutaforcearemeasuredbycompletelydierent methods,sowewishtoincludeheattransferbyfrictionalheating underthedenitionofwork,butnotheattransferbyconduction. Thedenitionofworkcouldthusberestatedastheamountofenergytransferredbyforces. Calculatingworkasforcemultipliedbydistance Theexamplesinguresb-dshowthattherearemanydierent waysinwhichenergycanbetransferred.Evenso,alltheseexamples havetwothingsincommon: 1.Aforceisinvolved. 2.Thetractortravelssomedistanceasitdoesthework. Inb,theincreaseintheheightoftheweight, y ,isthesameas thedistancethetractortravels,whichwe'llcall d .Forsimplicity, wediscussthecasewherethetractorraisestheweightatconstant speed,sothatthereisnochangeinthekineticenergyoftheweight, andweassumethatthereisnegligiblefrictioninthepulley,sothat theforcethetractorappliestotheropeisthesameastherope's upwardforceontheweight.ByNewton'srstlaw,theseforcesare alsoofthesamemagnitudeastheearth'sgravitationalforceonthe weight.Theincreaseintheweight'spotentialenergyisgivenby F y ,sotheworkdonebythetractorontheweightequals Fd ,the productoftheforceandthedistancemoved: W = Fd 50 Chapter3Work:TheTransferofMechanicalEnergy

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Inexamplec,thetractor'sforceonthetraileracceleratesit,increasingitskineticenergy.Iffrictionalforcesonthetrailerarenegligible, thentheincreaseinthetrailer'skineticenergycanbefoundusing thesamealgebrathatwasusedonpage39tondthepotential energyduetogravity.Justasinexampleb,wehave W = Fd Doesthisequationalwaysgivetherightanswer?Well,sortof. Inexampled,therearetwoquantitiesofworkyoumightwantto calculate:theworkdonebythetractorontheplowandthework donebytheplowonthedirt.Thesetwoquantitiescan'tbothequal Fd .Mostoftheenergytransmittedthroughthecablegoesinto frictionalheatingoftheplowandthedirt.Theworkdonebythe plowonthedirtislessthantheworkdonebythetractoronthe plow,byanamountequaltotheheatabsorbedbytheplow.Itturns outthattheequation W = Fd givestheworkdonebythetractor, nottheworkdonebytheplow.Howareyousupposedtoknowwhen theequationwillworkandwhenitwon't?Thesomewhatcomplex answerispostponeduntilsection3.6.Untilthen,wewillrestrict ourselvestoexamplesinwhich W = Fd givestherightanswer; essentiallythereasontheambiguitiescomeupisthatwhenone surfaceisslippingpastanother, d maybehardtodene,because thetwosurfacesmovedierentdistances. e / Thebaseballpitcherputkineticenergyintotheball,sohe didworkonit.Todothegreatest possibleamountofwork,heappliedthegreatestpossibleforce overthegreatestpossibledistance. Wehavealsobeenusingexamplesinwhichtheforceisinthe samedirectionasthemotion,andtheforceisconstant.Iftheforce wasnotconstant,wewouldhavetorepresentitwithafunction,not asymbolthatstandsforanumber.Tosummarize,wehave: ruleforcalculatingworksimplestversion Theworkdonebyaforcecanbecalculatedas W = Fd iftheforceisconstantandinthesamedirectionasthemotion. Someambiguitiesareencounteredincasessuchaskineticfriction. Section3.1Work:TheTransferofMechanicalEnergy 51

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f / Example1. Mechanicalworkdoneinanearthquakeexample1 In1998,geologistsdiscoveredevidenceforabigprehistoric earthquakeinPasadena,between10,000and15,000yearsago. Theyfoundthatthetwosidesofthefaultmoved6.7mrelative tooneanother,andestimatedthattheforcebetweenthemwas 1.3 10 17 N.Howmuchenergywasreleased? Multiplyingtheforcebythedistancegives9 10 17 J.Forcomparison,theNorthridgeearthquakeof1994,whichkilled57peopleanddid40billiondollarsofdamage,released22timesless energy. Machinescanincreaseforce,butnotwork. Figuregshowsapulleyarrangementfordoublingtheforcesuppliedbythetractorbook1,section5.6.Thetensioninthelefthandropeisequalthroughout,assumingnegligiblefriction,sothere aretwoforcespullingthepulleytotheleft,eachequaltotheoriginalforceexertedbythetractorontherope.Thisdoubledforceis transmittedthroughtheright-handropetothestump. g / Thepulleydoublestheforce thetractorcanexertonthe stump. Itmightseemasthoughthisarrangementwouldalsodoublethe workdonebythetractor,butlookagain.Asthetractormoves forward2meters,1meterofropecomesaroundthepulley,andthe pulleymoves1mtotheleft.Althoughthepulleyexertsdoublethe forceonthestump,thepulleyandstumponlymovehalfasfar,so theworkdoneonthestumpisnogreaterthatitwouldhavebeen withoutthepulley. Thesameistrueforanymechanicalarrangementthatincreases ordecreasesforce,suchasthegearsonaten-speedbike.Youcan't getoutmoreworkthanyouputin,becausethatwouldviolate conservationofenergy.Ifyoushiftgearssothatyourforceonthe pedalsisamplied,theresultisthatyoujusthavetospinthepedals moretimes. Noworkisdonewithoutmotion. Itstrikesmoststudentsasnonsensicalwhentheyaretoldthat iftheystandstillandholdaheavybagofcement,theyaredoing noworkonthebag.Evenifitmakessensemathematicallythat W = Fd giveszerowhen d iszero,itseemstoviolatecommon sense.Youwouldcertainlybecometired!Thesolutionissimple. 52 Chapter3Work:TheTransferofMechanicalEnergy

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h / Wheneverenergyistransferredoutofthespring,thesame amounthastobetransferredinto theball,andviceversa.Asthe springcompresses,theballis doingpositiveworkonthespring givingupitsKEandtransferring energyintothespringasPE, andasitdecompressestheball isdoingnegativeworkextracting energy. Physicistshavetakenoverthecommonwordwork"andgivenita newtechnicalmeaning,whichisthetransferofenergy.Theenergy ofthebagofcementisnotchanging,andthatiswhatthephysicist meansbysayingnoworkisdoneonthebag. Thereisatransformationofenergy,butitistakingplaceentirely withinyourownmuscles,whichareconvertingchemicalenergyinto heat.Physiologically,ahumanmuscleisnotlikeatreelimb,which cansupportaweightindenitelywithouttheexpenditureofenergy. Eachmusclecell'scontractionisgeneratedbyzillionsoflittlemolecularmachines,whichtaketurnssupportingthetension.Whena particularmoleculegoesonoroduty,itmoves,andsinceitmoves whileexertingaforce,itisdoingwork.Thereiswork,butitiswork donebyonemoleculeinamusclecellonanother. Positiveandnegativework WhenobjectAtransfersenergytoobjectB,wesaythatAdoes positiveworkonB.BissaidtodonegativeworkonA.Inother words,amachinelikeatractorisdenedasdoingpositivework. Thisuseoftheplusandminussignsrelatesinalogicalandconsistentwaytotheiruseinindicatingthedirectionsofforceandmotion inonedimension.Ingureh,supposewechooseacoordinatesystemwiththe x axispointingtotheright.Thentheforcethespring exertsontheballisalwaysapositivenumber.Theball'smotion, however,changesdirections.Thesymbol d isreallyjustashorter wayofwritingthefamiliarquantity x ,whosepositiveandnegative signsindicatedirection. Whiletheballismovingtotheleft,weuse d< 0torepresent itsdirectionofmotion,andtheworkdonebythespring, Fd ,comes outnegative.Thisindicatesthatthespringistakingkineticenergy outoftheball,andacceptingitintheformofitsownpotential energy. Astheballisreacceleratedtotheright,ithas d> 0, Fd is positive,andthespringdoespositiveworkontheball.Potential energyistransferredoutofthespringanddepositedintheballas kineticenergy. Insummary: ruleforcalculatingworkincludingcasesofnegative work Theworkdonebyaforcecanbecalculatedas W = Fd iftheforceisconstantandalongthesamelineasthemotion. Thequantity d istobeinterpretedasasynonymfor x ,i.e., positiveandnegativesignsareusedtoindicatethedirection ofmotion.Someambiguitiesareencounteredincasessuchas kineticfriction. Section3.1Work:TheTransferofMechanicalEnergy 53

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i / Becausetheforceisin theoppositedirectioncompared tothemotion,thebrakeshoe doesnegativeworkonthedrum, i.e.,acceptsenergyfromitinthe formofheat. self-checkB Ingureh,whatabouttheworkdonebytheballonthespring? Answer,p.166 Therearemanyexampleswherethetransferofenergyoutofan objectcancelsoutthetransferofenergyin.Whenthetractorpulls theplowwitharope,theropedoesnegativeworkonthetractor andpositiveworkontheplow.Thetotalworkdonebytheropeis zero,whichmakessense,sinceitisnotchangingitsenergy. Itmayseemthatwhenyourarmsdonegativeworkbylowering abagofcement,thecementisnotreallytransferringenergyinto yourbody.Ifyourbodywasstoringpotentialenergylikeacompressedspring,youwouldbeabletoraiseandloweraweightall day,recyclingthesameenergy.Thebagofcementdoestransfer energyintoyourbody,butyourbodyacceptsitasheat,notaspotentialenergy.Thetensioninthemusclesthatcontrolthespeedof themotionalsoresultsintheconversionofchemicalenergytoheat, forthesamephysiologicalreasonsdiscussedpreviouslyinthecase whereyoujustholdthebagstill. Oneoftheadvantagesofelectriccarsovergasoline-poweredcars isthatitisjustaseasytoputenergybackinabatteryasitisto takeenergyout.Whenyousteponthebrakesinagascar,thebrake shoesdonegativeworkontherestofthecar.Thekineticenergyof thecaristransmittedthroughthebrakesandacceptedbythebrake shoesintheformofheat.Theenergycannotberecovered.Electric cars,however,aredesignedtouseregenerativebraking.Thebrakes don'tusefrictionatall.Theyareelectrical,andwhenyoustepon thebrake,thenegativeworkdonebythebrakesmeanstheyaccept theenergyandputitinthebatteryforlateruse.Thisisoneofthe reasonswhyanelectriccarisfarbetterfortheenvironmentthana gascar,eveniftheultimatesourceoftheelectricalenergyhappens tobetheburningofoilintheelectriccompany'splant.Theelectric carrecyclesthesameenergyoverandover,andonlydissipatesheat duetoairfrictionandrollingresistance,notbraking.Theelectric company'spowerplantcanalsobettedwithexpensivepollutionreductionequipmentthatwouldbeprohibitivelyexpensiveorbulky forapassengercar. 54 Chapter3Work:TheTransferofMechanicalEnergy

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k / Aforcecandopositive, negative,orzerowork,dependingonitsdirectionrelativetothe directionofthemotion. DiscussionQuestions A Besidesthepresenceofaforce,whatotherthingsdifferentiatethe processesoffrictionalheatingandheatconduction? B Criticizethefollowingincorrectstatement:Aforcedoesn'tdoany workunlessit'scausingtheobjecttomove. C Tostopyourcar,youmustrsthavetimetoreact,andthenittakes sometimeforthecartoslowdown.Bothofthesetimescontributetothe distanceyouwilltravelbeforeyoucanstop.Thegureshowshowthe averagestoppingdistanceincreaseswithspeed.Becausethestopping distanceincreasesmoreandmorerapidlyasyougofaster,theruleof onecarlengthper10m.p.h.ofspeedisnotconservativeenoughathigh speeds.Intermsofworkandkineticenergy,whatisthereasonforthe morerapidincreaseathighspeeds? DiscussionquestionC. 3.2WorkinThreeDimensions Aforceperpendiculartothemotiondoesnowork. Supposeworkisbeingdonetochangeanobject'skineticenergy. Aforceinthesamedirectionasitsmotionwillspeeditup,anda forceintheoppositedirectionwillslowitdown.Aswehavealready seen,thisisdescribedasdoingpositiveworkordoingnegativework ontheobject.Alltheexamplesdiscussedupuntilnowhavebeen ofmotioninonedimension,butinthreedimensionstheforcecan beatanyangle withrespecttothedirectionofmotion. Whatiftheforceisperpendiculartothedirectionofmotion?We havealreadyseenthataforceperpendiculartothemotionresults incircularmotionatconstantspeed.Thekineticenergydoesnot change,andweconcludethatnoworkisdonewhentheforceis perpendiculartothemotion. Sofarwehavebeenreasoningaboutthecaseofasingleforce actingonanobject,andchangingonlyitskineticenergy.Theresult ismoregenerallytrue,however.Forinstance,imagineahockeypuck slidingacrosstheice.Theicemakesanupwardnormalforce,but doesnottransferenergytoorfromthepuck. Section3.2WorkinThreeDimensions 55

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m / Self-check.Breaking Trail,byWalterE.Bohl. l / Workisonlydonebythe componentoftheforceparallelto themotion. Forcesatotherangles Supposetheforceisatsomeotheranglewithrespecttothe motion,say =45 .Suchaforcecouldbebrokendownintotwo components,onealongthedirectionofthemotionandtheother perpendiculartoit.Theforcevectorequalsthevectorsumofits twocomponents,andtheprincipleofvectoradditionofforcesthus tellsusthattheworkdonebythetotalforcecannotbeanydierent thanthesumoftheworksthatwouldbedonebythetwoforcesby themselves.Sincethecomponentperpendiculartothemotiondoes nowork,theworkdonebytheforcemustbe W = F k j d j ,[workdonebyaconstantforce] wherethevector d issimplyalesscumbersomeversionofthenotation r .Thisresultcanberewrittenviatrigonometryas W = j F jj d j cos .[workdonebyaconstantforce] Eventhoughthisequationhasvectorsinit,itdependsonlyon theirmagnitudes,andthemagnitudeofavectorisascalar.Work isthereforestillascalarquantity,whichonlymakessenseifitis denedasthetransferofenergy.Tengallonsofgasolinehavethe abilitytodoacertainamountofmechanicalwork,andwhenyou pullintoafull-servicegasstationyoudon'thavetosayFill'erup with10gallonsofsouth-goinggas." Studentsoftenwonderwhythisequationinvolvesacosinerather thanasine,oraskifitwouldeverbeasine.Invectoraddition,the treatmentofsinesandcosinesseemedmoreequalanddemocratic, sowhyisthecosinesospecialnow?Theansweristhatifweare goingtodescribe,say,avelocityvector,wemustgiveboththe component parallel tothe x axisandthecomponent perpendicular tothe x axisi.e.,the y component.Incalculatingwork,however, theforcecomponentperpendiculartothemotionisirrelevant|it changesthedirectionofmotionwithoutincreasingordecreasingthe energyoftheobjectonwhichitacts.Inthiscontext,itis only the parallelforcecomponentthatmatters,soonlythecosineoccurs. self-checkC aWorkisthetransferofenergy.Accordingtothisdenition,isthe horseinthepicturedoingworkonthepack?bIfyoucalculatework bythemethoddescribedinthissection,isthehorseinguremdoing workonthepack? Answer,p.166 Pushingabroomexample2 Ifyouexertaforceof21Nonapushbroom,atanangle35 degreesbelowhorizontal,andwalkfor5.0m,howmuchworkdo youdo?Whatisthephysicalsignicanceofthisquantityofwork? Usingthesecondequationabove,theworkdoneequals N.0mcos35 =86J. 56 Chapter3Work:TheTransferofMechanicalEnergy

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Theformofenergybeingtransferredisheatintheoorandthe broom'sbristles.Thiscomesfromthechemicalenergystoredin yourbody.Themajorityofthecaloriesyouburnaredissipated directlyasheatinsideyourbodyratherthandoinganyworkon thebroom.The86Jisonlytheamountofenergytransferred throughthebroom'shandle. Aviolinexample3 Asaviolinistdrawsthebowacrossastring,thebowhairsexert bothanormalforceandakineticfrictionalforceonthestring.The normalforceisperpendiculartothedirectionofmotion,anddoes nowork.However,thefrictionalforceisinthesamedirectionas themotionofthebow,soitdoeswork:energyistransferredto thestring,causingittovibrate. Onewayofplayingaviolinmoreloudlyistouselongerstrokes. Since W = Fd ,thegreaterdistanceresultsinmorework. Asecondwayofgettingaloudersoundistopressthebowmore rmlyagainstthestrings.Thisincreasesthenormalforce,and althoughthenormalforceitselfdoesnowork,anincreaseinthe normalforcehasthesideeffectofincreasingthefrictionalforce, therebyincreasing W = Fd Theviolinistmovesthebowbackandforth,andsoundisproducedonboththeup-bowthestroketowardtheplayer'sleft andthedown-bowtotheright.Onemay,forexample,playa seriesofnotesinalternationbetweenup-bowsanddown-bows. However,ifthenotesareofunequallength,theupanddownmotionstendtobeunequal,andiftheplayerisnotcareful,shecan runoutofbowinthemiddleofanote!Tokeepthisfromhappening,onecanmovethebowmorequicklyontheshorternotes, buttheresultingincreasein d willmaketheshorternoteslouder thantheyshouldbe.Askilledplayercompensatesbyreducing theforce. 3.3VaryingForce Upuntilnowwehavedonenoactualcalculationsofworkincases wheretheforcewasnotconstant.Thequestionofhowtotreat suchcasesismathematicallyanalogoustotheissueofhowtogeneralizetheequationdistance=velocitytimetocaseswherethe velocitywasnotconstant.There,wefoundthatthecorrectgeneralizationwastondtheareaunderthegraphofvelocityversus time.Theequivalentthingcanbedonewithwork: generalruleforcalculatingwork Theworkdonebyaforce F equalstheareaunderthecurve onagraphof F k versus x .Someambiguitiesareencountered incasessuchaskineticfriction. Section3.3VaryingForce 57

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n / Thespringdoesworkon thecart.Unliketheballin section3.1,thecartisattached tothespring. o / Theareaoftheshaded trianglegivestheworkdoneby thespringasthecartmoves fromtheequilibriumpositionto position x Theexamplesinthissectionareonesinwhichtheforceisvarying,butisalwaysalongthesamelineasthemotion,so F isthe sameas F k self-checkD InwhichofthefollowingexampleswoulditbeOKtocalculatework using Fd ,andinwhichoneswouldyouhavetousetheareaunderthe F )]TJ/F102 9.9627 Tf 9.962 0 Td [(x graph? aAshingboatcruiseswithanetdraggingbehindit. bAmagnetleapsontoarefrigeratorfromadistance. cEarth'sgravitydoesworkonanoutward-boundspaceprobe. Answer,p.166 Animportantandstraightforwardexampleisthecalculationof theworkdonebyaspringthatobeysHooke'slaw, F )]TJ/F20 10.9091 Tf 20 0 Td [(k x )]TJ/F20 10.9091 Tf 10.909 0 Td [(x o Theminussignisbecausethisistheforcebeingexertedbythe spring,nottheforcethatwouldhavetoactonthespringtokeep itatthisposition.Thatis,ifthepositionofthecartinguren istotherightofequilibrium,thespringpullsbacktotheleft,and vice-versa. Wecalculatetheworkdonewhenthespringisinitiallyatequilibriumandthendeceleratesthecarasthecarmovestotheright. Theworkdonebythespringonthecartequalstheminusareaof theshadedtriangle,becausethetrianglehangsbelowthe x axis. Theareaofatriangleishalfitsbasemultipliedbyitsheight,so W = )]TJ/F15 10.9091 Tf 9.681 7.38 Td [(1 2 k x )]TJ/F20 10.9091 Tf 10.909 0 Td [(x o 2 Thisistheamountofkineticenergylostbythecartasthespring deceleratesit. Itwasstraightforwardtocalculatetheworkdonebythespringin thiscasebecausethegraphof F versus x wasastraightline,giving atriangulararea.Butifthecurvehadnotbeensogeometrically simple,itmightnothavebeenpossibletondasimpleequationfor theworkdone,oranequationmighthavebeenderivableonlyusing calculus.Optionalsection3.4givesanimportantexampleofsuch anapplicationofcalculus. Energyproductioninthesunexample4 Thesunproducesenergythroughnuclearreactionsinwhichnucleicollideandsticktogether.Theguredepictsonesuchreaction,inwhichasingleprotonhydrogennucleuscollideswith acarbonnucleus,consistingofsixprotonsandsixneutrons. Neutronsandprotonsattractotherneutronsandprotonsviathe strongnuclearforce,soastheprotonapproachesthecarbonnucleusitisaccelerated.Inthelanguageofenergy,wesaythat 58 Chapter3Work:TheTransferofMechanicalEnergy

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p / Example4. itlosesnuclearpotentialenergyandgainskineticenergy.Together,thesevenprotonsandsixneutronsmakeanitrogennucleus.Withinthenewlyput-togethernucleus,theneutronsand protonsarecontinuallycolliding,andthenewproton'sextrakineticenergyisrapidlysharedoutamongalltheneutronsand protons.Soonafterward,thenucleuscalmsdownbyreleasing someenergyintheformofagammaray,whichhelpstoheatthe sun. Thegraphshowstheforcebetweenthecarbonnucleusandthe protonastheprotonisonitswayin,withthedistanceinunitsof femtometersfm=10 )]TJ/F39 7.9701 Tf 6.586 0 Td [(15 m.Amusingly,theforceturnsouttobe afewnewtons:onthesameorderofmagnitudeastheforceswe encounterordinarilyonthehumanscale.Keepinmind,however, thataforcethisbigexertedonasinglesubatomicparticlesuchas aprotonwillproduceatrulyfantasticaccelerationontheorder of10 27 m = s 2 !. Whydoestheforcehaveapeakaround x =3fm,andbecome Section3.3VaryingForce 59

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smalleroncetheprotonhasactuallymergedwiththenucleus? At x =3fm,theprotonisattheedgeofthecrowdofprotonsand neutrons.Itfeelsmanyattractiveforcesfromtheleft,andnone fromtheright.Theforcesadduptoalargevalue.Howeverif itlaterndsitselfatthecenterofthenucleus, x =0,thereare forcespullingitfromalldirections,andtheseforcevectorscancel out. Wecannowcalculatetheenergyreleasedinthisreactionbyusingtheareaunderthegraphtodeterminetheamountofmechanicalworkdonebythecarbonnucleusontheproton.Forsimplicity,weassumethattheprotoncameinaimedatthecenterof thenucleus,andweignorethefactthatithastoshovesomeneutronsandprotonsoutofthewayinordertogetthere.Thearea underthecurveisabout17squares,andtheworkrepresented byeachsquareis N )]TJ/F39 7.9701 Tf 6.586 0 Td [(15 m=10 )]TJ/F39 7.9701 Tf 6.586 0 Td [(15 J, sothetotalenergyreleasedisabout )]TJ/F39 7.9701 Tf 6.587 0 Td [(15 J = squaresquares=1.7 10 )]TJ/F39 7.9701 Tf 6.587 0 Td [(14 J. Thismaynotseemlikemuch,butrememberthatthisisonlya reactionbetweenthenucleioftwooutofthezillionsofatomsin thesun.Forcomparison,atypical chemical reactionbetween twoatomsmighttransformontheorderof10 )]TJ/F39 7.9701 Tf 6.586 0 Td [(19 Jofelectrical potentialenergyintoheat100,000timeslessenergy! Asanalnote,youmaywonderwhyreactionssuchastheseonly occurinthesun.Thereasonisthatthereisarepulsiveelectrical forcebetweennuclei.Whentwonucleiareclosetogether,the electricalforcesaretypicallyaboutamilliontimesweakerthanthe nuclearforces,butthenuclearforcesfalloffmuchmorequickly withdistancethantheelectricalforces,sotheelectricalforceis thedominantoneatlongerranges.Thesunisaveryhotgas,so therandommotionofitsatomsisextremelyrapid,andacollision betweentwoatomsissometimesviolentenoughtoovercomethis initialelectricalrepulsion. 3.4 R ApplicationsofCalculus Thestudentwhohasstudiedintegralcalculuswillrecognizethat thegraphicalrulegivenintheprevioussectioncanbereexpressed asanintegral, W = Z x 2 x 1 F d x Wecanthenimmediatelyndbythefundamentaltheoremofcalculusthatforceisthederivativeofworkwithrespecttoposition, F = d W d x 60 Chapter3Work:TheTransferofMechanicalEnergy

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Forexample,acraneraisingaone-tonblockonthemoonwould betransferringpotentialenergyintotheblockatonlyonesixththe ratethatwouldberequiredonEarth,andthiscorrespondstoone sixththeforce. Althoughtheworkdonebythespringcouldbecalculatedwithoutcalculususingtheareaofatriangle,therearemanycaseswhere themethodsofcalculusareneededinordertondananswerin closedform.Themostimportantexampleistheworkdoneby gravitywhenthechangeinheightisnotsmallenoughtoassumea constantforce.Newton'slawofgravityis F = GMm r 2 whichcanbeintegratedtogive W = Z r 2 r 1 GMm r 2 d r = GMm 1 r 2 )]TJ/F15 10.9091 Tf 14.204 7.38 Td [(1 r 1 Section3.4 R ApplicationsofCalculus 61

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3.5WorkandPotentialEnergy Thetechniquesforcalculatingworkcanalsobeappliedtothecalculationofpotentialenergy.Ifacertainforcedependsonlyon thedistancebetweenthetwoparticipatingobjects,thentheenergy releasedbychangingthedistancebetweenthemisdenedasthepotentialenergy,andtheamountofpotentialenergylostequalsminus theworkdonebytheforce, PE = )]TJ/F20 10.9091 Tf 8.484 0 Td [(W Theminussignoccursbecausepositiveworkindicatesthatthepotentialenergyisbeingexpendedandconvertedtosomeotherform. Itissometimesconvenienttopicksomearbitrarypositionasa referenceposition,andderiveanequationforonceandforallthat givesthepotentialenergyrelativetothisposition PE x = )]TJ/F20 10.9091 Tf 8.485 0 Td [(W ref x .[potentialenergyatapoint x ] Tondtheenergytransferredintooroutofpotentialenergy,one thensubtractstwodierentvaluesofthisequation. Theseequationsmightalmostmakeitlookasthoughworkand energywerethesamething,buttheyarenot.First,potentialenergy measurestheenergythatasystem has storedinit,whilework measureshowmuchenergyis transferred inorout.Second,the techniquesforcalculatingworkcanbeusedtondtheamountof energytransferredinmanysituationswherethereisnopotential energyinvolved,aswhenwecalculatetheamountofkineticenergy transformedintoheatbyacar'sbrakeshoes. Atoygunexample5 Atoygunusesaspringwithaspringconstantof10N/mto shootaping-pongballofmass5g.Thespringiscompressedto 10cmshorterthanitsequilibriumlengthwhenthegunisloaded. Atwhatspeedistheballreleased? Theequilibriumpointisthenaturalchoiceforareferencepoint. Usingtheequationfoundpreviouslyforthework,wehave PE x = 1 2 k x )]TJ/F102 10.9091 Tf 10.909 0 Td [(x o 2 Thespringlosescontactwiththeballattheequilibriumpoint,so thenalpotentialenergyis PE f =0. Theinitialpotentialenergyis PE i = 1 2 N = m.10m 2 =0.05J. 62 Chapter3Work:TheTransferofMechanicalEnergy

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Thelossinpotentialenergyof0.05Jmeansanincreaseinkinetic energyofthesameamount.Thevelocityoftheballisfoundby solvingtheequation KE = = 2 mv 2 for v v = r 2 KE m = s .05J 0.005kg =4m = s. Gravitationalpotentialenergyexample6 Wehavealreadyfoundtheequation PE = )]TJ/F102 10.9091 Tf 8.485 0 Td [(F y forthegravitationalpotentialenergywhenthechangeinheightisnotenough tocauseasignicantchangeinthegravitationalforce F .What ifthechangeinheightisenoughsothatthisassumptionisno longervalid?Usetheequation W = GMm = r 2 )]TJ/F39 10.9091 Tf 11.401 0 Td [(1 = r 1 derived insection3.4tondthepotentialenergy,using r = 1 asareferencepoint. Thepotentialenergyequalsminustheworkthatwouldhaveto bedonetobringtheobjectfrom r 1 = 1 to r = r 2 ,whichis PE = )]TJ/F102 10.9091 Tf 9.681 7.38 Td [(GMm r Thisissimplerthantheequationforthework,whichisanexampleofwhyitisadvantageoustorecordanequationforpotential energyrelativetosomereferencepoint,ratherthananequation forwork. Althoughtheequationsderivedintheprevioustwoexamples mayseemarcaneandnotparticularlyusefulexceptfortoydesignersandrocketscientists,theirusefulnessisactuallygreaterthan itappears.Theequationforthepotentialenergyofaspringcan beadaptedtoanyothercaseinwhichanobjectiscompressed, stretched,twisted,orbent.Whileyouarenotlikelytousethe equationforgravitationalpotentialenergyforanythingpractical,it isdirectlyanalogoustoanequationthatisextremelyusefulinchemistry,whichistheequationforthepotentialenergyofanelectron atadistance r fromthenucleusofitsatom.Asdiscussedinmore detaillaterinthecourse,theelectricalforcebetweentheelectron andthenucleusisproportionalto1 =r 2 ,justlikethegravitational forcebetweentwomasses.Sincetheequationfortheforceisofthe sameform,soistheequationforthepotentialenergy. Section3.5WorkandPotentialEnergy 63

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q / 64ThetwinVoyagerspace probeswereperhapsthegreatestscienticsuccessesofthe spaceprogram.Overaperiod ofdecades,theyewbyallthe planetsoftheoutersolarsystem, probablyaccomplishingmoreof scienticinterestthantheentire spaceshuttleprogramatatiny fractionofthecost.BothVoyager probescompletedtheirnalplanetaryybyswithspeedsgreater thantheescapevelocityatthat distancefromthesun,andso headedonoutofthesolarsystem onhyperbolicorbits,nevertoreturn.Radiocontacthasbeenlost, andtheyarenowlikelytotravel interstellarspaceforbillionsof yearswithoutcollidingwithanythingorbeingdetectedbyanyintelligentspecies. DiscussionQuestions A Whatdoesthegraphof PE = = 2 k x )]TJ/F102 9.9627 Tf 9.962 0 Td [(x o 2 looklikeasafunction of x ?Discussthephysicalsignicanceofitsfeatures. B Whatdoesthegraphof PE = )]TJ/F102 9.9627 Tf 7.749 0 Td [(GMm = r looklikeasafunctionof r ? Discussthephysicalsignicanceofitsfeatures.Howwouldtheequation andgraphchangeifsomeotherreferencepointwaschosenratherthan r = 1 ? C Startingatadistance r fromaplanetofmass M ,howfastmustan objectbemovinginordertohaveahyperbolicorbit,i.e.,onethatnever comesbacktotheplanet?Thisvelocityiscalledtheescapevelocity.Interpretingtheresult,doesitmatterinwhatdirectionthevelocityis?Does itmatterwhatmasstheobjecthas?Doestheobjectescapebecauseitis movingtoofastforgravitytoactonit? D Doesaspringhaveanescapevelocity? E Calculus-basedquestion:Iftheformofenergybeingtransferredis potentialenergy,thentheequations F =d W = d x and W = R F d x become F = )]TJ/F39 9.9626 Tf 7.749 0 Td [(d PE = d x and PE = )]TJ/F26 9.9626 Tf 9.41 8.025 Td [(R F d x .Howwouldyouthenapplythefollowingcalculusconcepts:zeroderivativeatminimaandmaxima,andthe secondderivativetestforconcavityupordown. 3.6 ? WhenDoesWorkEqualForceTimes Distance? Intheexampleofthetractorpullingtheplowdiscussedonpage 51,theworkdidnotequal Fd .Thepurposeofthissectionisto explainmorefullyhowthequantity Fd canandcannotbeused. Tosimplifythings,Iwrite Fd throughoutthissection,butmore generallyeverythingsaidherewouldbetruefortheareaunderthe 64 Chapter3Work:TheTransferofMechanicalEnergy

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graphof F k versus d Thefollowingtwotheoremsallowmostoftheambiguitytobe clearedup. thework-kinetic-energytheorem Thechangeinkineticenergyassociatedwiththemotionofan object'scenterofmassisrelatedtothetotalforceactingon itandtothedistancetraveledbyitscenterofmassaccording totheequation KE cm = F total d cm ThiscanbeprovedbasedonNewton'ssecondlawandtheequation KE = = 2 mv 2 .Notethatdespitethetraditionalname,it doesnotnecessarilytelltheamountofworkdone,sincetheforces actingontheobjectcouldbechangingothertypesofenergybesides theKEassociatedwithitscenterofmassmotion. Thesecondtheoremdoesrelatedirectlytowork: Whenacontactforceactsbetweentwoobjectsandthetwo surfacesdonotslippasteachother,theworkdoneequals Fd where d isthedistancetraveledbythepointofcontact. Thisonehasnogenerallyacceptedname,sowerefertoitsimply asthesecondtheorem. Agreatnumberofphysicalsituationscanbeanalyzedwiththese twotheorems,andoftenitisadvantageoustoapplybothofthem tothesamesituation. Aniceskaterpushingofffromawallexample7 Thework-kineticenergytheoremtellsushowtocalculatethe skater'skineticenergyifweknowtheamountofforceandthe distancehercenterofmasstravelswhilesheispushingoff. Thesecondtheoremtellsusthatthewalldoesnoworkonthe skater.Thismakessense,sincethewalldoesnothaveany sourceofenergy. Absorbinganimpactwithoutrecoiling?example8 Isitpossibletoabsorbanimpactwithoutrecoiling?Forinstance,wouldabrickwallgiveatallifhitbyaping-pongball? Therewillalwaysbearecoil.Intheexampleproposed,thewall willsurelyhavesomeenergytransferredtoitintheformofheat andvibration.Thesecondtheoremtellsusthatwecanonlyhave nonzeroworkifthedistancetraveledbythepointofcontactis nonzero. Section3.6 ? WhenDoesWorkEqualForceTimesDistance? 65

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Draggingarefrigeratoratconstantvelocityexample9 Newton'srstlawtellsusthatthetotalforceontherefrigerator mustbezero:yourforceiscancelingtheoor'skineticfrictional force.Thework-kineticenergytheoremisthereforetruebutuseless.Ittellsusthatthereiszerototalforceontherefrigerator, andthattherefrigerator'skineticenergydoesn'tchange. Thesecondtheoremtellsusthattheworkyoudoequalsyour hand'sforceontherefrigeratormultipliedbythedistancetraveled. Sinceweknowtheoorhasnosourceofenergy,theonlywayfor theoorandrefrigeratortogainenergyisfromtheworkyoudo. Wecanthuscalculatethetotalheatdissipatedbyfrictioninthe refrigeratorandtheoor. Notethatthereisnowaytondhowmuchoftheheatisdissipatedintheoorandhowmuchintherefrigerator. Acceleratingacartexample10 Ifyoupushonacartandaccelerateit,therearetwoforcesacting onthecart:yourhand'sforce,andthestaticfrictionalforceofthe groundpushingonthewheelsintheoppositedirection. Applyingthesecondtheoremtoyourforcetellsushowtocalculatetheworkyoudo. Applyingthesecondtheoremtotheoor'sforcetellsusthatthe oordoesnoworkonthecart.Thereisnomotionatthepoint ofcontact,becausetheatomsintheoorarenotmoving.The atomsinthesurfaceofthewheelarealsomomentarilyatrest whentheytouchtheoor.Thismakessense,sincetheoor doesnothaveanysourceofenergy. Thework-kineticenergytheoremreferstothetotalforce,andbecausetheoor'sbackwardforcecancelspartofyourforce,the totalforceislessthanyourforce.Thistellsusthatonlypartof yourworkgoesintothekineticenergyassociatedwiththeforward motionofthecart'scenterofmass.Therestgoesintorotationof thewheels. 3.7 ? TheDotProduct Upuntilnow,wehavenotfoundanyphysicallyusefulwaytodene themultiplicationoftwovectors.Itwouldbepossible,forinstance, tomultiplytwovectorscomponentbycomponenttoformathird vector,buttherearenophysicalsituationswheresuchamultiplicationwouldbeuseful. Theequation W = j F jj d j cos isanexampleofasortofmultiplicationofvectorsthatisuseful.Theresultisascalar,nota vector,andthisisthereforeoftenreferredtoasthe scalarproduct ofthevectors F and d .Thereisastandardshorthandnotationfor 66 Chapter3Work:TheTransferofMechanicalEnergy

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thisoperation, A B = j A jj B j cos ,[denitionofthenotation A B ; istheanglebetweenvectors A and B ] andbecauseofthisnotation,amorecommontermforthisoperation isthe dotproduct .Indotproductnotation,theequationforwork issimply W = F d Thedotproducthasthefollowinggeometricinterpretation: A B = j A j componentof B parallelto A = j B j componentof A parallelto B Thedotproducthassomeofthepropertiespossessedbyordinary multiplicationofnumbers, A B = B A A B + C = A B + A C c A B = c A B butitlacksoneother:theabilitytoundomultiplicationbydividing. Ifyouknowthecomponentsoftwovectors,youcaneasilycalculatetheirdotproductasfollows: A B = A x B x + A y B y + A z B z Thiscanbeprovedbyrstanalyzingthespecialcasewhereeach vectorhasonlyan x component,andthesimilarcasesfor y and z Wecanthenusetherule A B + C = A B + A C tomakea generalizationbywritingeachvectorasthesumofits x y ,and z components.Seehomeworkproblem17. Section3.7 ? TheDotProduct 67

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Summary SelectedVocabulary work........theamountofenergytransferredintoorout ofasystem,excludingenergytransferredby heatconduction Notation W .........work Summary Workisameasureofthetransferofmechanicalenergy,i.e.,the transferofenergybyaforceratherthanbyheatconduction.When theforceisconstant,workcanusuallybecalculatedas W = F k j d j ,[onlyiftheforceisconstant] where d issimplyalesscumbersomenotationfor r ,thevector fromtheinitialpositiontothenalposition.Thus, Aforceinthesamedirectionasthemotiondoespositivework, i.e.,transfersenergyintotheobjectonwhichitacts. Aforceintheoppositedirectioncomparedtothemotiondoes negativework,i.e.,transfersenergyoutoftheobjectonwhich itacts. Whenthereisnomotion,nomechanicalworkisdone.The humanbodyburnscalorieswhenitexertsaforcewithout moving,butthisisaninternalenergytransferofenergywithin thebody,andthusdoesnotfallwithinthescienticdenition ofwork. Aforceperpendiculartothemotiondoesnowork. Whentheforceisnotconstant,theaboveequationshouldbegeneralizedastheareaunderthegraphof F k versus d Machinessuchaspulleys,levers,andgearsmayincreaseordecreaseaforce,buttheycanneverincreaseordecreasetheamount ofworkdone.Thatwouldviolateconservationofenergyunlessthe machinehadsomesourceofstoredenergyorsomewaytoaccept andstoreupenergy. Therearesomesituationsinwhichtheequation W = F k j d j is ambiguousornottrue,andtheseissuesarediscussedrigorouslyin section3.6.However,problemscanusuallybeavoidedbyanalyzing thetypesofenergybeingtransferredbeforeplungingintothemath. Inanycasethereisnosubstituteforaphysicalunderstandingof theprocessesinvolved. Thetechniquesdevelopedforcalculatingworkcanalsobeappliedtothecalculationofpotentialenergy.Wexsomeposition 68 Chapter3Work:TheTransferofMechanicalEnergy

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asareferenceposition,andcalculatethepotentialenergyforsome otherposition, x ,as PE x = )]TJ/F20 10.9091 Tf 8.484 0 Td [(W ref x Thefollowingtwoequationsforpotentialenergyhavebroader signicancethanmightbesuspectedbasedonthelimitedsituations inwhichtheywerederived: PE = 1 2 k x )]TJ/F20 10.9091 Tf 10.909 0 Td [(x o 2 [potentialenergyofaspringhavingspringconstant k ,whenstretchedorcompressedfromtheequilibrium position x o ;analogousequationsapplyforthetwisting, bending,compression,orstretchingofanyobject.] PE = )]TJ/F20 10.9091 Tf 9.68 7.38 Td [(GMm r [gravitationalpotentialenergyofobjectsofmasses M and m ,separatedbyadistance r ;ananalogousequation appliestotheelectricalpotentialenergyofanelectron inanatom.] Summary 69

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Problems Key p Acomputerizedanswercheckisavailableonline. R Aproblemthatrequirescalculus. ? Adicultproblem. 1 Twospeedboatsareidentical,butonehasmorepeopleaboard thantheother.Althoughthetotalmassesofthetwoboatsare unequal,supposethattheyhappentohavethesamekineticenergy. Inaboat,asinacar,it'simportanttobeabletostopintimeto avoidhittingthings.aIfthefrictionalforcefromthewateristhe sameinbothcases,howwilltheboats'stoppingdistancescompare? Explain.bComparethetimesrequiredfortheboatstostop. 2 Ineachofthefollowingsituations,istheworkbeingdone positive,negative,orzero?aabullpawstheground;bashing boatpullsanetthroughthewaterbehindit;cthewaterresists themotionofthenetthroughit;dyoustandbehindapickup truckandlowerabaleofhayfromthetruck'sbedtotheground. Explain.[BasedonaproblembySerwayandFaughn.] 3 Intheearth'satmosphere,themoleculesareconstantlymoving around.Becausetemperatureisameasureofkineticenergyper molecule,theaveragekineticenergyofeachtypeofmoleculeisthe same,e.g.,theaverageKEoftheO 2 moleculesisthesameasthe averageKEoftheN 2 molecules.aIfthemassofanO 2 molecule iseighttimesgreaterthanthatofaHeatom,whatistheratioof theiraveragespeeds?Whichwayistheratio,i.e.,whichistypically movingfaster?bUseyourresultfrompartatoexplainwhyany heliumoccurringnaturallyintheatmospherehaslongsinceescaped intoouterspace,nevertoreturn.Heliumisobtainedcommercially byextractingitfromrocks.Youmaywanttodoproblem21rst, forinsight. 4 Weipingliftsarockwithaweightof1.0Nthroughaheightof 1.0m,andthenlowersitbackdowntothestartingpoint.Bubba pushesatable1.0macrosstheooratconstantspeed,requiring aforceof1.0N,andthenpushesitbacktowhereitstarted.a ComparethetotalworkdonebyWeipingandBubba.bCheck thatyouranswerstopartamakesense,usingthedenitionofwork: workisthetransferofenergy.Inyouranswer,you'llneedtodiscuss whatspecictypeofenergyisinvolvedineachcase. 5 Inoneofhismoreamboyantmoments,GalileowroteWho doesnotknowthatahorsefallingfromaheightofthreeorfour cubitswillbreakhisbones,whileadogfallingfromthesameheight oracatfromaheightofeightortencubitswillsuernoinjury? Equallyharmlesswouldbethefallofagrasshopperfromatoweror thefallofanantfromthedistanceofthemoon."Findthespeed ofanantthatfallstoearthfromthedistanceofthemoonatthe 70 Chapter3Work:TheTransferofMechanicalEnergy

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Problem8:Acylinderfrom the1965Rambler'sengine.The pistonisshowninitspushedout position.Thetwobulgesatthe topareforthevalvesthatletfresh air-gasmixturein.Basedona gurefromMotorService'sAutomotiveEncyclopedia,Toboldt andPurvis. momentwhenitisabouttoentertheatmosphere.Assumeitis releasedfromapointthatisnotactuallynearthemoon,sothe moon'sgravityisnegligible. p 6 [Problem6hasbeendeleted.] 7 aThecrewofan18thcenturywarshipisraisingtheanchor. Theanchorhasamassof5000kg.Thewateris30mdeep.The chaintowhichtheanchorisattachedhasamassperunitlengthof 150kg/m.Beforetheystartraisingtheanchor,whatisthetotal weightoftheanchorplustheportionofthechainhangingoutofthe ship?Assumethatthebuoyancyoftheanchorandisnegligible. bAftertheyhaveraisedtheanchorby1m,whatistheweight theyareraising? cDene y =0whentheanchorisrestingonthebottom,and y =+30mwhenithasbeenraiseduptotheship.Drawagraph oftheforcethecrewhastoexerttoraisetheanchorandchain,as afunctionof y .Assumethattheyareraisingitslowly,sowater resistanceisnegligible.Itwillnotbeaconstant!Nowndthe areaunderthegraph,anddeterminetheworkdonebythecrewin raisingtheanchor,injoules. dConvertyouranswerfromcintounitsofkcal. p 8 Inthepowerstrokeofacar'sgasolineengine,thefuel-airmixtureisignitedbythesparkplug,explodes,andpushesthepiston out.Theexplodingmixture'sforceonthepistonheadisgreatest atthebeginningoftheexplosion,anddecreasesasthemixtureexpands.Itcanbeapproximatedby F = a=x ,where x isthedistance fromthecylindertothepistonhead,and a isaconstantwithunits ofN.m.Actually a=x 1.4 wouldbemoreaccurate,buttheproblem worksoutmorenicelywith a=x !Thepistonbeginsitsstrokeat x = x 1 ,andendsat x = x 2 .The1965Ramblerhadsixcylinders, eachwith a =220N m, x 1 =1.2cm,and x 2 =10.2cm. aDrawaneat,accurategraphof F vs x ,ongraphpaper. bFromtheareaunderthecurve,derivetheamountofworkdone inonestrokebyonecylinder. p cAssumetheengineisrunningat4800r.p.m.,sothatduring oneminute,eachofthesixcylindersperforms2400powerstrokes. Powerstrokesonlyhappeneveryotherrevolution.Findtheengine'spower,inunitsofhorsepowerhp=746W. p dThecompressionratioofanengineisdenedas x 2 =x 1 .Explain inwordswhythecar'spowerwouldbeexactlythesameif x 1 and x 2 were,say,halvedortripled,maintainingthesamecompression ratioof8.5.Explainwhythiswould not quitebetruewiththemore realisticforceequation F = a=x 1.4 9 Themagnitudeoftheforcebetweentwomagnetsseparated byadistance r canbeapproximatedas kr )]TJ/F18 7.9701 Tf 6.587 0 Td [(3 forlargevaluesof r Theconstant k dependsonthestrengthsofthemagnetsandthe relativeorientationsoftheirnorthandsouthpoles.Twomagnets Problems 71

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arereleasedonaslipperysurfaceataninitialdistance r i ,andbegin slidingtowardseachother.Whatwillbethetotalkineticenergy ofthetwomagnetswhentheyreachanaldistance r f ?Ignore friction. R 10 Acarstartsfromrestat t =0,andstartsspeedingupwith constantacceleration.aFindthecar'skineticenergyintermsof itsmass, m ,acceleration, a ,andthetime, t .bYouranswerin thepreviouspartalsoequalstheamountofwork, W ,donefrom t =0untiltime t .Takethederivativeofthepreviousexpression tondthepowerexpendedbythecarattime t .cSupposetwo carswiththesamemassbothstartfromrestatthesametime,but onehastwiceasmuchaccelerationastheother.Atanymoment, howmanytimesmorepowerisbeingdissipatedbythemorequickly acceleratingcar?Theanswerisnot2. R 11 Aspaceprobeofmass m isdroppedintoapreviouslyunexploredsphericalcloudofgasanddust,andacceleratestoward thecenterofthecloudundertheinuenceofthecloud'sgravity. Measurementsofitsvelocityallowitspotentialenergy, U ,tobe determinedasafunctionofthedistance r fromthecloud'scenter. Themassinthecloudisdistributedinasphericallysymmetricway, soitsdensity, r ,dependsonlyon r andnotontheangularcoordinates.Showthatbynding U r ,onecaninfer r asfollows: r = 1 4 Gmr 2 d d r r 2 d U d r R ? 12 Arailgunisadevicelikeatrainonatrack,withthetrain propelledbyapowerfulelectricalpulse.Veryhighspeedshavebeen demonstratedintestmodels,andrailgunshavebeenproposedas analternativetorocketsforsendingintoouterspaceanyobject thatwouldbestrongenoughtosurvivetheextremeaccelerations. Supposethattherailguncapsuleislaunchedstraightup,andthat theforceofairfrictionactingonitisgivenby F = be )]TJ/F21 7.9701 Tf 6.587 0 Td [(cx ,where x isthealtitude, b and c areconstants,and e isthebaseofnatural logarithms.Theexponentialdecayoccursbecausetheatmosphere getsthinnerwithincreasingaltitude.Inreality,theforcewould probablydropoevenfasterthananexponential,becausethecapsulewouldbeslowingdownsomewhat.Findtheamountofkinetic energylostbythecapsuleduetoairfrictionbetweenwhenitis launchedandwhenitiscompletelybeyondtheatmosphere.Gravityisnegligible,sincetheairfrictionforceismuchgreaterthanthe gravitationalforce. R 13 Acertainbinarystarsystemconsistsoftwostarswithmasses m 1 and m 2 ,separatedbyadistance b .Acomet,originallynearlyat restindeepspace,dropsintothesystemandatacertainpointin timearrivesatthemidpointbetweenthetwostars.Forthatmoment 72 Chapter3Work:TheTransferofMechanicalEnergy

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intime,nditsvelocity, v ,symbolicallyintermsof b m 1 m 2 ,and fundamentalconstants.[Numericalcheck:For m 1 =1.5 10 30 kg, m 2 =3.0 10 30 kg,and b =2.0 10 11 myoushouldnd v =7.7 10 4 m/s.] 14 Anairplaneiesinthepositivedirectionalongthe x axis, throughcrosswindsthatexertaforce F = a + bx ^ x + c + dx ^ y Findtheworkdonebythewindontheplane,andbytheplaneon thewind,intravelingfromtheorigintoposition x R 15 In1935,Yukawaproposedanearlytheoryoftheforcethat heldtheneutronsandprotonstogetherinthenucleus.Hisequationforthepotentialenergyoftwosuchparticles,atacenter-tocenterdistance r ,was PE r = gr )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 e )]TJ/F21 7.9701 Tf 6.587 0 Td [(r=a ,where g parametrizesthe strengthoftheinteraction, e isthebaseofnaturallogarithms,and a isabout10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(15 m.Findtheforcebetweentwonucleonsthatwould beconsistentwiththisequationforthepotentialenergy. R 16 Provethatthedotproductdenedinsection3.7isrotationallyinvariantinthesenseofbook1,section7.5. 17 Fillinthedetailsoftheproofof A B = A x B x + A y B y + A z B z onpage67. 18 Doesitmakesensetosaythatworkisconserved? Solution,p.169 19 aSupposeworkisdoneinone-dimensionalmotion.What happenstotheworkifyoureversethedirectionofthepositive coordinateaxis?Baseyouranswerdirectlyonthedenitionofwork. bNowanswerthequestionbasedonthe W = Fd rule. 20 >Amicrowaveovenworksbytwistingmoleculesonewayand thentheother,counterclockwiseandthenclockwiseabouttheirown centers,millionsoftimesasecond.Ifyouputanicecubeorastick ofbutterinamicrowave,you'llobservethattheovendoesn'theat thesolidveryquickly,althougheventuallymeltingbeginsinone smallspot.Onceameltedspotforms,itgrowsrapidly,whilethe restofthesolidremainssolid.Inotherwords,itappearsbasedon thisexperimentthatamicrowaveovenheatsaliquidmuchmore rapidlythanasolid.Explainwhythisshouldhappen,basedonthe atomic-leveldescriptionofheat,solids,andliquids.See,e.g.,gure bonpage37. Pleasedon'trepeatthefollowingcommonmistakesinyourexplanation: Inasolid,theatomsarepackedmoretightlyandhaveless spacebetweenthem. Nottrue.Iceoatsbecauseit's less densethanwater. Inaliquid,theatomsaremovingmuchfaster. No,thedierenceinaveragespeedbetweeniceat )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 Candwaterat1 C isonly0.4%. Problems 73

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Problem22. 21 Startingatadistance r fromaplanetofmass M ,howfast mustanobjectbemovinginordertohaveahyperbolicorbit,i.e., onethatnevercomesbacktotheplanet?Thisvelocityiscalled theescapevelocity.Interpretingtheresult,doesitmatterinwhat directionthevelocityis?Doesitmatterwhatmasstheobjecthas? Doestheobjectescapebecauseitismovingtoofastforgravityto actonit? p 22 Thegure,redrawnfrom Gray'sAnatomy ,showsthetensionofwhichamuscleiscapable.Thevariable x isdenedasthe contractionofthemusclefromitsmaximumlength L ,sothatat x =0themusclehaslength L ,andat x = L themusclewouldtheoreticallyhavezerolength.Inreality,themusclecanonlycontract to x = cL ,where c islessthan1.Whenthemuscleisextendedto itsmaximumlength,at x =0,itiscapableofthegreatesttension, T o .Asthemusclecontracts,however,itbecomesweaker.Graysuggestsapproximatingthisfunctionasalineardecrease,whichwould theoreticallyextrapolatetozeroat x = L .aFindthemaximum workthemusclecandoinonecontraction,intermsof c L ,and T o p bShowthatyouranswertopartahastherightunits. cShowthatyouranswertopartahastherightbehaviorwhen c =0andwhen c =1. dGrayalsostatesthattheabsolutemaximumtension T o has beenfoundtobeapproximatelyproportionaltothemuscle'scrosssectionalarea A whichispresumablymeasuredat x =0,with proportionalityconstant k .Approximatingthemuscleasacylinder,showthatyouranswerfrompartacanbereexpressedinterms ofthevolume, V ,eliminating L and A p eEvaluateyourresultnumericallyforabicepsmusclewithavolumeof200cm 3 ,with c =0.8and k =100N = cm 2 asestimatedby Gray. p 74 Chapter3Work:TheTransferofMechanicalEnergy

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Poolballsexchangemomentum. Chapter4 Conservationof Momentum Inmanysubeldsofphysicsthesedays,itispossibletoreadan entireissueofajournalwithouteverencounteringanequationinvolvingforceorareferencetoNewton'slawsofmotion.Inthelast hundredandftyyears,anentirelydierentframeworkhasbeen developedforphysics,basedonconservationlaws. Thenewapproachisnotjustpreferredbecauseitisinfashion. Itappliesinsideanatomornearablackhole,whereNewton'slaws donot.Evenineverydaysituationsthenewapproachcanbesuperior.Wehavealreadyseenhowperpetualmotionmachinescouldbe designedthatweretoocomplextobeeasilydebunkedbyNewton's laws.Thebeautyofconservationlawsisthattheytellussomething mustremainthesame,regardlessofthecomplexityoftheprocess. Sofarwehavediscussedonlytwoconservationlaws,thelawsof conservationofmassandenergy.Isthereanyreasontobelievethat furtherconservationlawsareneededinordertoreplaceNewton's lawsasacompletedescriptionofnature?Yes.Conservationofmass andenergydonotrelateinanywaytothethreedimensionsofspace, becausebotharescalars.Conservationofenergy,forinstance,does notpreventtheplanetearthfromabruptlymakinga90-degreeturn andheadingstraightintothesun,becausekineticenergydoesnot dependondirection.Inthischapter,wedevelopanewconserved quantity,calledmomentum,whichisavector. 75

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4.1Momentum Aconservedquantityofmotion Yourrstencounterwithconservationofmomentummayhave comeasasmallchildunjustlyconnedtoashoppingcart.Youspot somethinginterestingtoplaywith,likethedisplaycaseofimported winedownattheendoftheaisle,anddecidetopushthecartover there.ButbeingimprisonedbyDadinthecartwasnottheonly injusticethatday.Therewasafargreaterconspiracytothwart youryoungid,onethatoriginatedinthelawsofnature.Pushing forwarddidnudgethecartforward,butitpushedyoubackward. Ifthewheelsofthecartwerewelllubricated,itwouldn'tmatter howyoujerked,yanked,orkickedofromthebackofthecart. Youcouldnotcauseanyoverallforwardmotionoftheentiresystem consistingofthecartwithyouinside. IntheNewtonianframework,wedescribethisasarisingfrom Newton'sthirdlaw.Thecartmadeaforceonyouthatwasequal andoppositetoyourforceonit.Intheframeworkofconservation laws,wecannotattributeyourfrustrationtoconservationofenergy. Itwouldhavebeenperfectlypossibleforyoutotransformsomeof theinternalchemicalenergystoredinyourbodytokineticenergy ofthecartandyourbody. Thefollowingcharacteristicsofthesituationsuggestthatthere maybeanewconservationlawinvolved: Aclosedsystemisinvolved. Allconservationlawsdealwith closedsystems.Youandthecartareaclosedsystem,sincethe well-oiledwheelspreventtheoorfrommakinganyforwardforce onyou. Somethingremainsunchanged. Theoverallvelocityofthe systemstartedoutbeingzero,andyoucannotchangeit.This vaguereferencetooverallvelocity"canbemademoreprecise: itisthevelocityofthesystem'scenterofmassthatcannotbe changed. Somethingcanbetransferredbackandforthwithout changingthetotalamount. Ifwedeneforwardaspositive andbackwardasnegative,thenonepartofthesystemcangain positivemotionifanotherpartacquiresnegativemotion.Ifwe don'twanttoworryaboutpositiveandnegativesigns,wecan imaginethatthewholecartwasinitiallyglidingforwardonits well-oiledwheels.Bykickingofromthebackofthecart,you couldincreaseyourownvelocity,butthisinevitablycausesthe carttoslowdown. 76 Chapter4ConservationofMomentum

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Itthusappearsthatthereissomenumericalmeasureofanobject's quantityofmotionthatisconservedwhenyouaddupalltheobjects withinasystem. Momentum Althoughvelocityhasbeenreferredto,itisnotthetotalvelocity ofaclosedsystemthatremainsconstant.Ifitwas,thenringa gunwouldcausetheguntorecoilatthesamevelocityasthebullet! Thegundoesrecoil,butatamuchlowervelocitythanthebullet. Newton'sthirdlawtellsus F gunonbullet = )]TJ/F20 10.9091 Tf 8.485 0 Td [(F bulletongun andassumingaconstantforceforsimplicity,Newton'ssecondlaw allowsustochangethisto m bullet v bullet t = )]TJ/F20 10.9091 Tf 8.485 0 Td [(m gun v gun t Thusifthegunhas100timesmoremassthanthebullet,itwill recoilatavelocitythatis100timessmallerandintheopposite direction,representedbytheoppositesign.Thequantity mv is thereforeapparentlyausefulmeasureofmotion,andwegiveita name, momentum ,andasymbol, p .AsfarasIknow,theletter p"wasjustchosenatrandom,sincem"wasalreadybeingusedfor mass.Thesituationsdiscussedsofarhavebeenone-dimensional, butinthree-dimensionalsituationsitistreatedasavector. denitionofmomentumformaterialobjects Themomentumofamaterialobject,i.e.,apieceofmatter,isdened as p = m v theproductoftheobject'smassanditsvelocityvector. Theunitsofmomentumarekg m = s,andthereisunfortunatelyno abbreviationforthisclumsycombinationofunits. Thereasoningleadinguptothedenitionofmomentumwasall basedonthesearchforaconservationlaw,andtheonlyreasonwhy webothertodenesuchaquantityisthatexperimentsshowitis conserved: thelawofconservationofmomentum Inanyclosedsystem,thevectorsumofallthemomentaremains constant, p 1 i + p 2 i + ::: = p 1 f + p 2 f + ::: where i labelstheinitialand f thenalmomenta.Aclosedsystem isoneonwhichnoexternalforcesact. Section4.1Momentum 77

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Thischapterrstaddressestheone-dimensionalcase,inwhichthe directionofthemomentumcanbetakenintoaccountbyusingplus andminussigns.Wethenpasstothreedimensions,necessitating theuseofvectoraddition. Asubtlepointaboutconservationlawsisthattheyallreferto closedsystems,"butclosed"meansdierentthingsindierent cases.Whendiscussingconservationofmass,closed"meansasystemthatdoesn'thavemattermovinginoroutofit.Withenergy, wemeanthatthereisnoworkorheattransferoccurringacross theboundaryofthesystem.Formomentumconservation,closed" meanstherearenoexternal forces reachingintothesystem. Acannonexample1 Acannonofmass1000kgresa10-kgshellatavelocityof 200m/s.Atwhatspeeddoesthecannonrecoil? Thelawofconservationofmomentumtellsusthat p cannon i + p shell i = p cannon f + p shell f Choosingacoordinatesysteminwhichthecannonpointsinthe positivedirection,thegiveninformationis p cannon i =0 p shell i =0 p shell f =2000kg m = s. Wemusthave p cannon f = )]TJ/F39 10.9091 Tf 8.485 0 Td [(2000kg m = s,sotherecoilvelocityof thecannonis )]TJ/F39 10.9091 Tf 8.485 0 Td [(2m/s. Iondriveforpropellingspacecraftexample2 Theexperimentalsolar-powerediondriveoftheDeepSpace1 spaceprobeexpelsitsxenongasexhaustataspeedof30,000 m/s,tentimesfasterthantheexhaustvelocityforatypicalchemicalfuelrocketengine.Roughlyhowmanytimesgreateristhemaximumspeedthisspacecraftcanreach,comparedwithachemicalfueledprobewiththesamemassoffuelreactionmassavailableforpushingoutthebackasexhaust? Momentumequalsmassmultipliedbyvelocity.Bothspacecraft areassumedtohavethesameamountofreactionmass,andthe iondrive'sexhausthasavelocitytentimesgreater,sothemomentumofitsexhaustistentimesgreater.Beforetheengine startsring,neithertheprobenortheexhausthasanymomentum,sothetotalmomentumofthesystemiszero.Byconservationofmomentum,thetotalmomentummustalsobezeroafter 78 Chapter4ConservationofMomentum

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a / TheiondriveengineoftheNASADeepSpace1probe,shown underconstructionleftandbeingtestedinavacuumchamberright priortoitsOctober1998launch.Intendedmainlyasatestvehiclefornew technologies,thecraftneverthelesscarriedoutasuccessfulscientic programthatincludedaybyofacomet. alltheexhausthasbeenexpelled.Ifwedenethepositivedirectionasthedirectionthespacecraftisgoing,thenthenegative momentumoftheexhaustiscanceledbythepositivemomentumofthespacecraft.Theiondriveallowsanalspeedthatis tentimesgreater.Thissimpliedanalysisignoresthefactthat thereactionmassexpelledlaterintheburnisnotmovingbackwardasfast,becauseoftheforwardspeedofthealready-moving spacecraft. Generalizationofthemomentumconcept Aswithalltheconservationlaws,thelawofconservationof momentumhasevolvedovertime.Inthe1800'sitwasfoundthat abeamoflightstrikinganobjectwouldgiveitsomemomentum, eventhoughlighthasnomass,andwouldthereforehavenomomenSection4.1Momentum 79

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b / Steamandothergases boilingoffofthenucleusofHalley'scomet.Thisclose-upphoto wastakenbytheEuropeanGiotto spaceprobe,whichpassedwithin 596kmofthenucleusonMarch 13,1986. c / Halley'scomet,inamuch lessmagniedviewfroma ground-basedtelescope. tumaccordingtotheabovedenition.Ratherthandiscardingthe principleofconservationofmomentum,thephysicistsofthetime decidedtoseeifthedenitionofmomentumcouldbeextendedto includemomentumcarriedbylight.Theprocessisanalogousto theprocessoutlinedonpage21foridentifyingnewformsofenergy. Therststepwasthediscoverythatlightcouldimpartmomentum tomatter,andthesecondstepwastoshowthatthemomentum possessedbylightcouldberelatedinadenitewaytoobservable propertiesofthelight.Theyfoundthatconservationofmomentumcouldbesuccessfullygeneralizedbyattributingtoabeamof lightamomentumvectorinthedirectionofthelight'smotionand havingamagnitudeproportionaltotheamountofenergythelight possessed.Themomentumoflightisnegligibleunderordinarycircumstances,e.g.,aashlightleftonforanhourwouldonlyabsorb about10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(5 kg m = sofmomentumasitrecoiled. Thetailofacometexample3 Momentumisnotalwaysequalto mv .Likemanycomets,Halley's comethasaveryelongatedellipticalorbit.Aboutoncepercentury,itsorbitbringsitclosetothesun.Thecomet'shead,or nucleus,iscomposedofdirtyice,sotheenergydepositedbythe intensesunlightboilsoffsteamandothergases,b.Thesunlight doesnotjustcarryenergy,howeveritalsocarriesmomentum.Oncethegasboilsoff,themomentumofthesunlightimpactingonitpushesitawayfromthesun,formingatail,c.By analogywithmatter,forwhichmomentumequals mv ,youwould expectthatmasslesslightwouldhavezeromomentum,butthe equation p = mv isnotthecorrectoneforlight,andlightdoes havemomentum.Somecometsalsohaveasecondtail,which ispropelledbyelectricalforcesratherthanbythemomentumof sunlight. Thereasonforbringingthisupisnotsothatyoucanplug numbersintoaformulasintheseexoticsituations.Thepointis thattheconservationlawshaveprovensosturdyexactlybecause theycaneasilybeamendedtotnewcircumstances.Newton's lawsarenolongeratthecenterofthestageofphysicsbecausethey didnothavethesameadaptability.Moregenerally,themoralof thisstoryistheprovisionalnatureofscientictruth. Itshouldalsobenotedthatconservationofmomentumisnot aconsequenceofNewton'slaws,asisoftenassertedintextbooks. Newton'slawsdonotapplytolight,andthereforecouldnotpossiblybeusedtoproveanythingaboutaconceptasgeneralasthe conservationofmomentuminitsmodernform. ModernChangesintheMomentumConcept Einsteinplayedaroleintwomajorchangesinthemomentumconcept inthe1900's. 80 Chapter4ConservationofMomentum

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FirstEinsteinshowedthattheequation p = mv wouldnotworkfor asystemcontainingobjectsmovingatveryhighspeedsrelativetoone another.Hecameupwithanewequation,towhich mv isonlythe low-velocityapproximation. Thesecondchange,andafarstrangerone,wastherealization thatattheatomiclevel,motionisinescapablyrandom.Theelectron inahydrogenatomdoesn'treallyorbitthenucleus,itformsavague cloudaroundit.Itmightseemthatthiswouldprovenonconservation ofmomentum,butinfacttherandomwanderingsoftheprotonareexactlycoordinatedwiththoseoftheelectronsothatthetotalmomentumstaysexactlyconstant.Inanatomoflead,thereare82electrons plusthenucleus,allchangingtheirmomentarandomlyfrommomentto moment,butallcoordinatingmysteriouslywitheachothertokeepthe vectorsumconstant.Inthe1930s,Einsteinpointedoutthatthetheoriesoftheatomthenbeingdevelopedwouldrequirethiskindofspooky coordination,andusedthisasanargumentthattherewassomething physicallyunreasonableinthenewideas.Experiments,however,have shownthatthespookyeffectsdohappen,andEinstein'sobjectionsare rememberedtodayonlyasahistoricalcuriousity. Momentumcomparedtokineticenergy Momentumandkineticenergyarebothmeasuresofthequantityofmotion,andasideshowintheNewton-Leibnitzcontroversy overwhoinventedcalculuswasanargumentoverwhether mv i.e., momentumor mv 2 i.e.,kineticenergywithoutthe1/2infront wasthetrue"measureofmotion.Themodernstudentcancertainlybeexcusedforwonderingwhyweneedbothquantities,when theircomplementarynaturewasnotevidenttothegreatestminds ofthe1700's.Thefollowingtablehighlightstheirdierences. kineticenergy... momentum... isascalar. isavector isnotchangedbyaforceperpendiculartothemotion,whichchanges onlythedirectionofthevelocity vector. ischangedbyanyforce,sincea changeineitherthemagnitudeor thedirectionofthevelocityvector willresultinachangeinthemomentumvector. isalwayspositive,andcannotcancel out. cancelswithmomentumintheoppositedirection. canbetradedforotherformsofenergythatdonotinvolvemotion.KE isnotaconservedquantitybyitself. isalwaysconservedinaclosedsystem. isquadrupledifthevelocityisdoubled. isdoubledifthevelocityisdoubled. Aspinningtopexample4 Aspinningtophaszerototalmomentum,becauseforeverymovingpoint,thereisanotherpointontheoppositesidethatcancels itsmomentum.Itdoes,however,havekineticenergy. Section4.1Momentum 81

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Momentumandkineticenergyinringarieexample5 Therieandbullethavezeromomentumandzerokineticenergy tostartwith.Whenthetriggerispulled,thebulletgainssomemomentumintheforwarddirection,butthisiscanceledbytherie's backwardmomentum,sothetotalmomentumisstillzero.The kineticenergiesofthegunandbulletarebothpositivescalars, however,anddonotcancel.Thetotalkineticenergyisallowedto increase,becausekineticenergyisbeingtradedforotherforms ofenergy.Initiallythereischemicalenergyinthegunpowder. Thischemicalenergyisconvertedintoheat,sound,andkinetic energy.Thegun'sbackward'kineticenergydoesnotrefrigerate theshooter'sshoulder! Thewobblyearthexample6 Asthemooncompleteshalfacirclearoundtheearth,itsmotion reversesdirection.Thisdoesnotinvolveanychangeinkinetic energy,andtheearth'sgravitationalforcedoesnotdoanywork onthemoon.Thereversedvelocityvectordoes,however,imply areversedmomentumvector,soconservationofmomentumin theclosedearth-moonsystemtellsusthattheearthmustalso changeitsmomentum.Infact,theearthwobblesinalittleorbitaboutapointbelowitssurfaceonthelineconnectingitand themoon.Thetwobodies'momentumvectorsalwayspointin oppositedirectionsandcanceleachotherout. Theearthandmoongetadivorceexample7 Whycan'tthemoonsuddenlydecidetoyoffonewayandthe earththeotherway?Itisnotforbiddenbyconservationofmomentum,becausethemoon'snewlyacquiredmomentuminone directioncouldbecanceledoutbythechangeinthemomentum oftheearth,supposingtheearthheadedtheoppositedirection attheappropriate,slowerspeed.Thecatastropheisforbiddenby conservationofenergy,becauseboththeirenergieswouldhave toincreasegreatly. Momentumandkineticenergyofaglacierexample8 Acubic-kilometerglacierwouldhaveamassofabout10 12 kg.If itmovesataspeedof10 )]TJ/F39 7.9701 Tf 6.587 0 Td [(5 m/s,thenitsmomentumis10 7 kg m = s.Thisisthekindofheroic-scaleresultweexpect,perhaps theequivalentofthespaceshuttletakingoff,orallthecarsinLA drivinginthesamedirectionatfreewayspeed.Itskineticenergy, however,isonly50J,theequivalentofthecaloriescontained inapoppyseedortheenergyinadropofgasolinetoosmall tobeseenwithoutamicroscope.Thesurprisinglysmallkinetic energyisbecausekineticenergyisproportionaltothesquareof thevelocity,andthesquareofasmallnumberisanevensmaller number. 82 Chapter4ConservationofMomentum

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d / ThisHubbleSpaceTelescopephotoshowsasmall galaxyyellowblobinthelower rightthathascollidedwitha largergalaxyspiralnearthe center,producingawaveofstar formationbluetrackduetothe shockwavespassingthrough thegalaxies'cloudsofgas.This isconsideredacollisioninthe physicssense,eventhoughitis statisticallycertainthatnostarin eithergalaxyeverstruckastarin theother.Thisisbecausethe starsareverysmallcomparedto thedistancesbetweenthem. DiscussionQuestions A Ifalltheairmoleculesintheroomsettleddowninathinlmonthe oor,wouldthatviolateconservationofmomentumaswellasconservationofenergy? B Arefrigeratorhascoilsinbackthatgethot,andheatismolecular motion.Thesemovingmoleculeshavebothenergyandmomentum.Why doesn'ttherefrigeratorneedtobetiedtothewalltokeepitfromrecoiling fromthemomentumitlosesouttheback? 4.2CollisionsinOneDimension Physicistsemploythetermcollision"inabroadersensethan ordinaryusage,applyingittoanysituationwhereobjectsinteract foracertainperiodoftime.Abathittingabaseball,aradioactively emittedparticledamagingDNA,andagunandabulletgoingtheir separatewaysareallexamplesofcollisionsinthissense.Physical contactisnotevenrequired.Acometswingingpastthesunona hyperbolicorbitisconsideredtoundergoacollision,eventhoughit nevertouchesthesun.Allthatmattersisthatthecometandthe sunexertedgravitationalforcesoneachother. Thereasonforbroadeningthetermcollision"inthiswayis thatallofthesesituationscanbeattackedmathematicallyusing thesameconservationlawsinsimilarways.Intherstexample, conservationofmomentumisallthatisrequired. Gettingrear-endedexample9 Ms.Changisrear-endedatastoplightbyMr.Nelson,andsues tomakehimpayhermedicalbills.Hetestiesthathewasonly going35milesperhourwhenhehitMs.Chang.Shethinkshe wasgoingmuchfasterthanthat.Thecarsskiddedtogetherafter theimpact,andmeasurementsofthelengthoftheskidmarks andthecoefcientoffrictionshowthattheirjointvelocityimmediatelyaftertheimpactwas19milesperhour.Mr.Nelson'sNissan weighs3100pounds,andMs.Chang'sCadillacweighs5200 pounds.IsMr.Nelsontellingthetruth? Sincethecarsskiddedtogether,wecanwritedowntheequation forconservationofmomentumusingonlytwovelocities, v forMr. Nelson'svelocitybeforethecrash,and v 0 fortheirjointvelocity afterward: m N v = m N v 0 + m C v 0 Solvingfortheunknown, v ,wend v = 1+ m C m N v 0 Althoughwearegiventheweightsinpounds,aunitofforce,the ratioofthemassesisthesameastheratiooftheweights,and wend v =51milesperhour.Heislying. Section4.2CollisionsinOneDimension 83

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GoryDetailsoftheProofin Example10 Theequation A + B = C + D says thatthechangeinoneball'svelocityisequalandoppositetothe changeintheother's.Weinventa symbol x = C )]TJ/F102 9.9627 Tf 10.196 0 Td [(A forthechange inball1'svelocity.Thesecond equationcanthenberewrittenas A 2 + B 2 = A + x 2 + B )]TJ/F102 9.9627 Tf 8.353 0 Td [(x 2 .Squaringoutthequantitiesinparenthesesandthensimplifying,weget 0= Ax )]TJ/F102 9.9627 Tf 10.488 0 Td [(Bx + x 2 .Theequation hasthetrivialsolution x =0,i.e., neitherball'svelocityischanged, butthisisphysicallyimpossiblebecausetheballscannottravelthrough eachotherlikeghosts.Assuming x 6 =0,wecandivideby x and solvefor x = B )]TJ/F102 9.9627 Tf 10.38 0 Td [(A .Thismeans thatball1hasgainedanamount ofvelocityexactlysufcienttomatch ball2'sinitialvelocity,andviceversa.Theballsmusthaveswapped velocities. Theaboveexamplewassimplebecausebothcarshadthesame velocityafterward.Inmanyone-dimensionalcollisions,however,the twoobjectsdonotstick.Ifwewishtopredicttheresultofsucha collision,conservationofmomentumdoesnotsuce,becauseboth velocitiesafterthecollisionareunknown,sowehaveoneequation intwounknowns. Conservationofenergycanprovideasecondequation,butits applicationisnotasstraightforward,becausekineticenergyisonly theparticularformofenergythathastodowithmotion.Inmany collisions,partofthekineticenergythatwaspresentbeforethe collisionisusedtocreateheatorsound,ortobreaktheobjects orpermanentlybendthem.Cars,infact,arecarefullydesignedto crumpleinacollision.Crumplingthecarusesupenergy,andthat's goodbecausethegoalistogetridofallthatkineticenergyina relativelysafeandcontrolledway.Attheoppositeextreme,asuperballissuper"becauseitemergesfromacollisionwithalmostall itsoriginalkineticenergy,havingonlystoreditbrieyaspotential energywhileitwasbeingsquashedbytheimpact. Collisionsofthesuperballtype,inwhichalmostnokineticenergyisconvertedtootherformsofenergy,canthusbeanalyzed morethoroughly,becausetheyhave KE f = KE i ,asopposedto thelessusefulinequality KE f
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and B butswappedaround.Aformalproofofthisfactisgiven inthesidebar.Inthespecialcasewhereball2isinitiallyatrest, thistellsusthatball1isstoppeddeadbythecollision,andball 2headsoffatthevelocityoriginallypossessedbyball1.This behaviorwillbefamiliartoplayersofpool. Often,asintheexampleabove,thedetailsofthealgebraare theleastinterestingpartoftheproblem,andconsiderablephysical insightcanbegainedsimplybycountingthenumberofunknowns andcomparingtothenumberofequations.Supposeabeginnerat poolnoticesacasewherehercueballhitsaninitiallystationary ballandstopsdead.Wow,whatagoodtrick,"shethinks.I betIcouldneverdothatagaininamillionyears."Butshetries again,andndsthatshecan'thelpdoingitevenifshedoesn't wantto.Luckilyshehasjustlearnedaboutcollisionsinherphysics course.Onceshehaswrittendowntheequationsforconservation ofenergyandnolossofkineticenergy,shereallydoesn'thaveto completethealgebra.Sheknowsthatshehastwoequationsin twounknowns,sotheremustbeawell-denedsolution.Onceshe hasseentheresultofonesuchcollision,sheknowsthatthesame thingmusthappeneverytime.Thesamethingwouldhappenwith collidingmarblesorcroquetballs.Itdoesn'tmatterifthemassesor velocitiesaredierent,becausethatjustmultipliesbothequations bysomeconstantfactor. Thediscoveryoftheneutron ThiswasthetypeofreasoningemployedbyJamesChadwickin his1932discoveryoftheneutron.Atthetime,theatomwasimaginedtobemadeoutoftwotypesoffundamentalparticles,protons andelectrons.Theprotonswerefarmoremassive,andclustered togetherintheatom'score,ornucleus.Attractiveelectricalforces causedtheelectronstoorbitthenucleusincircles,inmuchthe samewaythatgravitationalforceskepttheplanetsfromcruising outofthesolarsystem.Experimentsshowedthattheheliumnucleus,forinstance,exertedexactlytwiceasmuchelectricalforceon anelectronasanucleusofhydrogen,thesmallestatom,andthiswas explainedbysayingthatheliumhadtwoprotonstohydrogen'sone. Thetroublewasthataccordingtothismodel,heliumwouldhave twoelectronsandtwoprotons,givingitpreciselytwicethemassof ahydrogenatomwithoneofeach.Infact,heliumhasaboutfour timesthemassofhydrogen. Chadwicksuspectedthattheheliumnucleuspossessedtwoadditionalparticlesofanewtype,whichdidnotparticipateinelectrical forcesatall,i.e.,wereelectricallyneutral.Iftheseparticleshadvery nearlythesamemassasprotons,thenthefour-to-onemassratioof heliumandhydrogencouldbeexplained.In1930,anewtypeof radiationwasdiscoveredthatseemedtotthisdescription.Itwas electricallyneutral,andseemedtobecomingfromthenucleioflight Section4.2CollisionsinOneDimension 85

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elementsthathadbeenexposedtoothertypesofradiation.Atthis time,however,reportsofnewtypesofparticleswereadimeadozen, andmostofthemturnedouttobeeitherclustersmadeofpreviouslyknownparticlesorelsepreviouslyknownparticleswithhigher energies.Manyphysicistsbelievedthatthenew"particlethathad attractedChadwick'sinterestwasreallyapreviouslyknownparticle calledagammaray,whichwaselectricallyneutral.Sincegamma rayshavenomass,Chadwickdecidedtotrytodeterminethenew particle'smassandseeifitwasnonzeroandapproximatelyequal tothemassofaproton. Unfortunatelyasubatomicparticleisnotsomethingyoucan justputonascaleandweigh.Chadwickcameupwithaningenious solution.Themassesofthenucleiofthevariouschemicalelements werealreadyknown,andtechniqueshadalreadybeendevelopedfor measuringthespeedofarapidlymovingnucleus.Hethereforeset outtobombardsamplesofselectedelementswiththemysterious newparticles.Whenadirect,head-oncollisionoccurredbetween amysteryparticleandthenucleusofoneofthetargetatoms,the nucleuswouldbeknockedoutoftheatom,andhewouldmeasure itsvelocity. Suppose,forinstance,thatwebombardasampleofhydrogen atomswiththemysteryparticles.Sincetheparticipantsinthe collisionarefundamentalparticles,thereisnowayforkineticenergy tobeconvertedintoheatoranyotherformofenergy,andChadwick thushadtwoequationsinthreeunknowns: equation#1:conservationofmomentum equation#2:nolossofkineticenergy unknown#1:massofthemysteryparticle unknown#2:initialvelocityofthemysteryparticle unknown#3:nalvelocityofthemysteryparticle Thenumberofunknownsisgreaterthanthenumberofequations,sothereisnouniquesolution.Butbycreatingcollisionswith nucleiofanotherelement,nitrogen,hegainedtwomoreequations attheexpenseofonlyonemoreunknown: equation#3:conservationofmomentuminthenewcollision equation#4:nolossofkineticenergyinthenewcollision unknown#4:nalvelocityofthemysteryparticleinthenew collision Hewasthusabletosolveforalltheunknowns,includingthe massofthemysteryparticle,whichwasindeedwithin1%ofthe massofaproton.Henamedthenewparticletheneutron,sinceit iselectricallyneutral. 86 Chapter4ConservationofMomentum

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e / Chadwick'ssubatomicpooltable.Adiskofthenaturallyoccurringmetalpoloniumprovidesasourceofradiationcapableofkicking neutronsoutoftheberylliumnuclei.Thetypeofradiationemittedby thepoloniumiseasilyabsorbedbyafewmmofair,sotheairhastobe pumpedoutoftheleft-handchamber.Theneutrons,Chadwick'smystery particles,penetratematterfarmorereadily,andyoutthroughthewall andintothechamberontheright,whichislledwithnitrogenorhydrogen gas.Whenaneutroncollideswithanitrogenorhydrogennucleus,it kicksitoutofitsatomathighspeed,andthisrecoilingnucleusthenrips apartthousandsofotheratomsofthegas.Theresultisanelectrical pulsethatcanbedetectedinthewireontheright.Physicistshadalready calibratedthistypeofapparatussothattheycouldtranslatethestrength oftheelectricalpulseintothevelocityoftherecoilingnucleus.The wholeapparatusshowninthegurewouldtinthepalmofyourhand,in dramaticcontrasttotoday'sgiantparticleaccelerators. DiscussionQuestion A Goodpoolplayerslearntomakethecueballspin,whichcancause itnottostopdeadinahead-oncollisionwithastationaryball.Ifthisdoes notviolatethelawsofphysics,whathiddenassumptionwasthereinthe exampleabove? Section4.2CollisionsinOneDimension 87

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g / Twohockeypuckscollide. Theirmutualcenterofmass tracesthestraightpathshownby thedashedline. 4.3 ? RelationshipofMomentumtotheCenter ofMass f / Inthismultiple-ashphotograph,weseethewrenchfrom aboveasitiesthroughtheair, rotatingasitgoes.Itscenter ofmass,markedwiththeblack cross,travelsalongastraightline, unliketheotherpointsonthe wrench,whichexecuteloops. Wehavealreadydiscussedtheideaofthecenterofmassinthe rstbookofthisseries,butusingtheconceptofmomentumwecan nowndamathematicalmethodfordeningthecenterofmass, explainwhythemotionofanobject'scenterofmassusuallyexhibits simplermotionthananyotherpoint,andgainaverysimpleand powerfulwayofunderstandingcollisions. Therststepistorealizethatthecenterofmassconceptcan beappliedtosystemscontainingmorethanoneobject.Evensomethinglikeawrench,whichwethinkofasoneobject,isreallymade ofmanyatoms.Thecenterofmassisparticularlyeasytovisualize inthecaseshownontheleft,wheretwoidenticalhockeypuckscollide.Itisclearongroundsofsymmetrythattheircenterofmass mustbeatthemidpointbetweenthem.Afterall,wepreviouslydenedthecenterofmassasthebalancepoint,andifthetwohockey puckswerejoinedwithaverylightweightrodwhoseownmasswas negligible,theywouldobviouslybalanceatthemidpoint.Itdoesn't matterthatthehockeypucksaretwoseparateobjects.Itisstill truethatthemotionoftheircenterofmassisexceptionallysimple, justlikethatofthewrench'scenterofmass. The x coordinateofthehockeypucks'centerofmassisthus givenby x cm = x 1 + x 2 = 2,i.e.,thearithmeticaverageoftheir x coordinates.Whyisitsmotionsosimple?Ithastodowith conservationofmomentum.Sincethehockeypucksarenotbeing actedonbyanynetexternalforce,theyconstituteaclosedsystem, andtheirtotalmomentumisconserved.Theirtotalmomentumis mv 1 + mv 2 = m v 1 + v 2 = m x 1 t + x 2 t = m t x 1 + x 2 = m 2 x cm t = m total v cm Inotherwords,thetotalmomentumofthesystemisthesameas ifallitsmasswasconcentratedatthecenterofmasspoint.Since 88 Chapter4ConservationofMomentum

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thetotalmomentumisconserved,the x componentofthecenterof mass'svelocityvectorcannotchange.Thesameisalsotrueforthe othercomponents,sothecenterofmassmustmovealongastraight lineatconstantspeed. Theaboverelationshipbetweenthetotalmomentumandthe motionofthecenterofmassappliestoanysystem,evenifitisnot closed. totalmomentumrelatedtocenterofmassmotion Thetotalmomentumofanysystemisrelatedtoitstotalmass andthevelocityofitscenterofmassbytheequation p total = m total v cm Whataboutasystemcontainingobjectswithunequalmasses, orcontainingmorethantwoobjects?Thereasoningabovecanbe generalizedtoaweightedaverage x cm = m 1 x 1 + m 2 x 2 + ::: m 1 + m 2 + ::: withsimilarequationsforthe y and z coordinates. Momentumindifferentframesofreference Absolutemotionissupposedtobeundetectable,i.e.,thelaws ofphysicsaresupposedtobeequallyvalidinallinertialframes ofreference.Ifwerstcalculatesomemomentainoneframeof referenceandndthatmomentumisconserved,andthenrework thewholeprobleminsomeotherframeofreferencethatismoving withrespecttotherst,thenumericalvaluesofthemomentawill allbedierent.Evenso,momentumwillstillbeconserved.Allthat mattersisthatweworkasingleprobleminoneconsistentframeof reference. Onewayofprovingthisistoapplytheequation p total = m total v cm .IfthevelocityofframeBrelativetoframeAis v BA thentheonlyeectofchangingframesofreferenceistochange v cm fromitsoriginalvalueto v cm + v BA .Thisaddsaconstant ontothemomentumvector,whichhasnoeectonconservationof momentum. Section4.3 ? RelationshipofMomentumtotheCenterofMass 89

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i / Theslingshoteffectviewed inthesun'sframeofreference. Jupiterismovingtotheleft,and thecollisionishead-on. j / Theslingshotviewedin theframeofthecenterofmassof theJupiter-spacecraftsystem. h / Movingyourheadsothat youarealwayslookingdown fromrightabovethecenterof mass,youobservethecollision ofthetwohockeypucksinthe centerofmassframe. Thecenterofmassframeofreference Aparticularlyusefulframeofreferenceinmanycasesisthe framethatmovesalongwiththecenterofmass,calledthecenter ofmassc.m.frame.Inthisframe,thetotalmomentumiszero. Thefollowingexamplesshowhowthecenterofmassframecanbe apowerfultoolforsimplifyingourunderstandingofcollisions. Acollisionofpoolballsviewedinthec.m.frameexample11 Ifyoumoveyourheadsothatyoureyeisalwaysabovethepoint halfwayinbetweenthetwopoolballs,youareviewingthingsin thecenterofmassframe.Inthisframe,theballscometowardthe centerofmassatequalspeeds.Bysymmetry,theymustthereforerecoilatequalspeedsalongthelinesonwhichtheyentered. Sincetheballshaveessentiallyswappedpathsinthecenterof massframe,thesamemustalsobetrueinanyotherframe.This isthesameresultthatrequiredlaboriousalgebratoprovepreviouslywithouttheconceptofthecenterofmassframe. Theslingshoteffectexample12 Itisacounterintuitivefactthataspacecraftcanpickupspeed byswingingaroundaplanet,ifarrivesintheoppositedirection comparedtotheplanet'smotion.Althoughthereisnophysical contact,wetreattheencounterasaone-dimensionalcollision, andanalyzeitinthecenterofmassframe.Figureishowssuch acollision,withaspaceprobewhippingaroundJupiter.Inthe sun'sframeofreference,Jupiterismoving. Whataboutthecenterofmassframe?SinceJupiterissomuch moremassivethanthespacecraft,thecenterofmassisessentiallyxedatJupiter'scenter,andJupiterhaszerovelocityinthe centerofmassframe,asshowningurej.Thec.m.frameis movingtotheleftcomparedtothesun-xedframeusedini,so thespacecraft'sinitialvelocityisgreaterinthisframe. Thingsaresimplerinthecenterofmassframe,becauseitismore symmetric.Inthecomplicatedsun-xedframe,theincomingleg oftheencounterisrapid,becausethetwobodiesarerushingtowardeachother,whiletheirseparationontheoutboundlegis moregradual,becauseJupiteristryingtocatchup.Inthec.m. frame,Jupiterissittingstill,andthereisperfectsymmetrybetweentheincomingandoutgoinglegs,sobysymmetrywehave v 1 f = )]TJ/F102 10.9091 Tf 8.485 0 Td [(v 1 i .Goingbacktothesun-xedframe,thespacecraft's nalvelocityisincreasedbytheframes'motionrelativetoeach other.Inthesun-xedframe,thespacecraft'svelocityhasincreasedgreatly. Theresultcanalsobeunderstoodintermsofworkandenergy. InJupiter'sframe,Jupiterisnotdoinganyworkonthespacecraft asitroundsthebackoftheplanet,becausethemotionisperpendiculartotheforce.Butinthesun'sframe,thespacecraft's velocityvectoratthesamemomenthasalargecomponenttothe 90 Chapter4ConservationofMomentum

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k / Powerandforcearethe ratesatwhichenergyand momentumaretransferred. left,soJupiterisdoingworkonit. DiscussionQuestions A Makeupanumericalexampleoftwounequalmassesmovinginone dimensionatconstantvelocity,andverifytheequation p total = m total v cm overatimeintervalofonesecond. B Amoremassivetennisracquetorbaseballbatmakesthebally offfaster.Explainwhythisistrue,usingthecenterofmassframe.For simplicity,assumethattheracquetorbatissimplysittingstillbeforethe collision,andthatthehitter'shandsdonotmakeanyforcelargeenough tohaveasignicanteffectovertheshortdurationoftheimpact. 4.4MomentumTransfer Therateofchangeofmomentum Aswithconservationofenergy,weneedawaytomeasureand calculatethetransferofmomentumintooroutofasystemwhenthe systemisnotclosed.Inthecaseofenergy,theanswerwasrather complicated,andentirelydierenttechniqueshadtobeusedfor measuringthetransferofmechanicalenergyworkandthetransfer ofheatbyconduction.Formomentum,thesituationisfarsimpler. Inthesimplestcase,thesystemconsistsofasingleobjectacted onbyaconstantexternalforce.Sinceitisonlytheobject'svelocity thatcanchange,notitsmass,themomentumtransferredis p = m v whichwiththehelpof a = F =m andtheconstant-accelerationequation a = v = t becomes p = m a t = F t Thustherateoftransferofmomentum,i.e.,thenumberofkg m = s absorbedpersecond,issimplytheexternalforce, F = p t [relationshipbetweentheforceonanobjectandthe rateofchangeofitsmomentum;validonlyiftheforce isconstant] ThisisjustarestatementofNewton'ssecondlaw,andinfactNewtonoriginallystateditthisway.Asshowningurek,therelationshipbetweenforceandmomentumisdirectlyanalogoustothat betweenpowerandenergy. Thesituationisnotmateriallyalteredforasystemcomposed ofmanyobjects.Theremaybeforcesbetweentheobjects,butthe internalforcescannotchangethesystem'smomentum.Iftheydid, Section4.4MomentumTransfer 91

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thenremovingtheexternalforceswouldresultinaclosedsystem thatcouldchangeitsownmomentum,likethemythicalmanwho couldpullhimselfupbyhisownbootstraps.Thatwouldviolate conservationofmomentum.Theequationabovebecomes F total = p total t [relationshipbetweenthetotalexternalforceonasystemandtherateofchangeofitstotalmomentum;valid onlyiftheforceisconstant] Walkingintoalamppostexample13 Startingfromrest,youbeginwalking,bringingyourmomentum upto100kg m = s.Youwalkstraightintoalamppost.Whyisthe momentumchangeof )]TJ/F39 10.9091 Tf 8.485 0 Td [(100kg m = scausedbythelamppostso muchmorepainfulthanthechangeof+100kg m = swhenyou startedwalking? Thesituationisone-dimensional,sowecandispensewiththe vectornotation.Itprobablytakesyouabout1stospeedupinitially,sotheground'sforceonyouis F = p = t 100N.Your impactwiththelamppost,however,isoverintheblinkofaneye, say1/10sorless.Dividingbythismuchsmaller t givesamuch largerforce,perhapsthousandsofnewtons.Thenegativesign simplyindicatesthattheforceisintheoppositedirection. Thisisalsotheprincipleofairbagsincars.Thetimerequiredfor theairbagtodecelerateyourheadisfairlylong,thetimerequired foryourfacetotravel20or30cm.Withoutanairbag,yourface wouldhitthedashboard,andthetimeintervalwouldbethemuch shortertimetakenbyyourskulltomoveacoupleofcentimeters whileyourfacecompressed.Notethateitherway,thesameamount ofmechanicalworkhastobedoneonyourhead:enoughtoeliminate allitskineticenergy. Iondriveforspacecraftexample14 TheiondriveoftheDeepSpace1spacecraft,picturedonpage 79anddiscussedinexample2,producesathrustof90mN millinewtons.Itcarriesabout80kgofreactionmass,whichit ejectsataspeedof30,000m/s.Forhowlongcantheengine continuesupplyingthisamountofthrustbeforerunningoutofreactionmasstoshoveouttheback? Solvingtheequation F = p = t fortheunknown t ,andtreatingforceandmomentumasscalarssincetheproblemisone92 Chapter4ConservationofMomentum

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m / The F )]TJ/F102 9.9627 Tf 14.684 0 Td [(t graphfora tennisracquethittingaballmight looklikethis.Theamountof momentumtransferredequals theareaunderthecurve. l / Example15. dimensional,wend t = p F = m exhaust v exhaust F = kg,000m = s 0.090N =2.7 10 7 s =300days Atopplingboxexample15 Ifyouplaceaboxonafrictionlesssurface,itwillfalloverwitha verycomplicatedmotionthatishardtopredictindetail.Weknow, however,thatitscenterofmassmovesinthesamedirectionas itsmomentumvectorpoints.Therearetwoforces,anormalforce andagravitationalforce,bothofwhicharevertical.Thegravitationalforceisactuallymanygravitationalforcesactingonall theatomsinthebox.Thetotalforcemustbevertical,sothe momentumvectormustbepurelyverticaltoo,andthecenterof masstravelsvertically.Thisistrueeveniftheboxbouncesand tumbles.[BasedonanexamplebyKleppnerandKolenkow.] Theareaundertheforce-timegraph Fewrealcollisionsinvolveaconstantforce.Forexample,when atennisballhitsaracquet,thestringsstretchandtheballattens dramatically.TheyarebothactinglikespringsthatobeyHooke's law,whichsaysthattheforceisproportionaltotheamountof stretchingorattening.Theforceisthereforesmallatrst,ramps uptoamaximumwhentheballisabouttoreversedirections,and rampsbackdownagainastheballisonitswaybackout.The equation F = p= t ,derivedundertheassumptionofconstant acceleration,doesnotapplyhere,andtheforcedoesnotevenhave asinglewell-denednumericalvaluethatcouldbepluggedintothe equation. Aswithsimilar-lookingequationssuchas v = p= t ,theequation F = p= t iscorrectlygeneralizedbysayingthattheforceis theslopeofthe p )]TJ/F20 10.9091 Tf 10.909 0 Td [(t graph. Conversely,ifwewishtond p fromagraphsuchastheonein gurem,oneapproachwouldbetodividetheforcebythemassof theball,rescalingthe F axistocreateagraphofaccelerationversus time.Theareaundertheacceleration-versus-timegraphgivesthe changeinvelocity,whichcanthenbemultipliedbythemassto ndthechangeinmomentum.Anunnecessarycomplicationwas introduced,however,becausewebeganbydividingbythemass andendedbymultiplyingbyit.Itwouldhavemadejustasmuch sensetondtheareaundertheoriginal F )]TJ/F20 10.9091 Tf 11.055 0 Td [(t graph,whichwould havegivenusthemomentumchangedirectly. Section4.4MomentumTransfer 93

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n / Example16. DiscussionQuestion A Manycollisions,likethecollisionofabatwithabaseball,appearto beinstantaneous.Mostpeoplealsowouldnotimaginethebatandballas bendingorbeingcompressedduringthecollision.Considerthefollowing possibilities: 1.Thecollisionisinstantaneous. 2.Thecollisiontakesaniteamountoftime,duringwhichtheballand batretaintheirshapesandremainincontact. 3.Thecollisiontakesaniteamountoftime,duringwhichtheballand batarebendingorbeingcompressed. Howcantwooftheseberuledoutbasedonenergyormomentumconsiderations? 4.5MomentuminThreeDimensions Inthissectionwediscusshowtheconceptsappliedpreviouslyto one-dimensionalsituationscanbeusedaswellinthreedimensions. Oftenvectoradditionisallthatisneededtosolveaproblem: Anexplosionexample16 AstronomersobservetheplanetMarsastheMartiansghta nuclearwar.TheMartianbombsaresopowerfulthattheyripthe planetintothreeseparatepiecesofliquiedrock,allhavingthe samemass.Ifonefragmentiesoffwithvelocitycomponents v 1 x =0 v 1 y =1.0 10 4 km = hr, andthesecondwith v 2 x =1.0 10 4 km = hr v 2 y =0, allinthecenterofmassframewhatisthemagnitudeofthethird one'svelocity? Inthecenterofmassframe,theplanetinitiallyhadzeromomentum.Aftertheexplosion,thevectorsumofthemomentamuststill bezero.Vectoradditioncanbedonebyaddingcomponents,so mv 1 x + mv 2 x + mv 3 x =0,and mv 1 y + mv 2 y + mv 3 y =0, wherewehaveusedthesamesymbol m foralltheterms,becausethefragmentsallhavethesamemass.Themassescan beeliminatedbydividingeachequationby m ,andwend v 3 x = )]TJ/F39 10.9091 Tf 8.485 0 Td [(1.0 10 4 km = hr v 3 y = )]TJ/F39 10.9091 Tf 8.485 0 Td [(1.0 10 4 km = hr 94 Chapter4ConservationofMomentum

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o / Example17. whichgivesamagnitudeof j v 3 j = q v 2 3 x + v 2 3 y =1.4 10 4 km = hr Thecenterofmass Inthreedimensions,wehavethevectorequations F total = p total t and p total = m total v cm Thefollowingisanexampleoftheiruse. Thebolaexample17 Thebola,similartotheNorthAmericanlasso,isusedbySouth Americangauchostocatchsmallanimalsbytanglinguptheir legsinthethreeleatherthongs.Themotionofthewhirlingbola throughtheairisextremelycomplicated,andwouldbeachallengetoanalyzemathematically.Themotionofitscenterof mass,however,ismuchsimpler.Theonlyforcesonitaregravitational,so F total = m total g Usingtheequation F total = p total = t ,wend p total = t = m total g andsincethemassisconstant,theequation p total = m total v cm allowsustochangethisto m total v cm = t = m total g Themasscancels,and v cm = t issimplytheaccelerationofthe centerofmass,so a cm = g Inotherwords,themotionofthesystemisthesameasifallits masswasconcentratedatandmovingwiththecenterofmass. Thebolahasaconstantdownwardaccelerationequalto g ,and iesalongthesameparabolaasanyotherprojectilethrownwith thesameinitialcenterofmassvelocity.Throwingabolawiththe correctrotationispresumablyadifcultskill,butmakingithitits targetisnoharderthanitiswithaballorasinglerock. [BasedonanexamplebyKleppner&Kolenkow.] Section4.5MomentuminThreeDimensions 95

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Countingequationsandunknowns Countingequationsandunknownsisjustasusefulasinone dimension,buteveryobject'smomentumvectorhasthreecomponents,soanunknownmomentumvectorcountsasthreeunknowns. Conservationofmomentumisasinglevectorequation,butitsays thatallthreecomponentsofthetotalmomentumvectorstayconstant,sowecountitasthreeequations.Ofcourseifthemotion happenstobeconnedtotwodimensions,thenweneedonlycount vectorsashavingtwocomponents. Atwo-carcrashwithstickingexample18 Supposetwocarscollide,sticktogether,andskidofftogether.If weknowthecars'initialmomentumvectors,wecancountequationsandunknownsasfollows: unknown#1: x componentofcars'nal,totalmomentum unknown#2: y componentofcars'nal,totalmomentum equation#1:conservationofthetotal p x equation#2:conservationofthetotal p y Sincethenumberofequationsequalsthenumberofunknowns, theremustbeoneuniquesolutionfortheirtotalmomentumvector afterthecrash.Inotherwords,thespeedanddirectionatwhich theircommoncenterofmassmovesofftogetherisunaffectedby factorssuchaswhetherthecarscollidecenter-to-centerorcatch eachotheralittleoff-center. Shootingpoolexample19 Twopoolballscollide,andasbeforeweassumethereisnodecreaseinthetotalkineticenergy,i.e.,noenergyconvertedfrom KEintootherforms.Asinthepreviousexample,weassumewe aregiventheinitialvelocitiesandwanttondthenalvelocities. Theequationsandunknownsare: unknown#1: x componentofball#1'snalmomentum unknown#2: y componentofball#1'snalmomentum unknown#3: x componentofball#2'snalmomentum unknown#4: y componentofball#2'snalmomentum equation#1:conservationofthetotal p x equation#2:conservationofthetotal p y equation#3:nodecreaseintotalKE Notethatwedonotcounttheballs'nalkineticenergiesasunknowns,becauseknowingthemomentumvector,onecanalways ndthevelocityandthusthekineticenergy.Thenumberofequationsislessthanthenumberofunknowns,sonouniqueresultis guaranteed.Thisiswhatmakespoolaninterestinggame.By 96 Chapter4ConservationofMomentum

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p / Example20. aimingthecueballtoonesideofthetargetballyoucanhave somecontrolovertheballs'speedsanddirectionsofmotionafter thecollision. Itisnotpossible,however,tochooseanycombinationofnal speedsanddirections.Forinstance,acertainshotmaygivethe correctdirectionofmotionforthetargetball,makingitgointoa pocket,butmayalsohavetheundesiredside-effectofmakingthe cueballgoinapocket. Calculationswiththemomentumvector Thefollowingexampleillustrateshowaforceisrequiredto changethedirectionofthemomentumvector,justasonewould berequiredtochangeitsmagnitude. Aturbineexample20 Inahydroelectricplant,waterowingoveradamdrivesaturbine,whichrunsageneratortomakeelectricpower.Thegure showsasimpliedphysicalmodelofthewaterhittingtheturbine, inwhichitisassumedthatthestreamofwatercomesinata 45 anglewithrespecttotheturbineblade,andbouncesoffata 90 angleatnearlythesamespeed.Thewaterowsatarate R inunitsofkg/s,andthespeedofthewateris v .Whatarethe magnitudeanddirectionofthewater'sforceontheturbine? Inatimeinterval t,themassofwaterthatstrikesthebladeis R t,andthemagnitudeofitsinitialmomentumis mv = vR t Thewater'snalmomentumvectorisofthesamemagnitude,but intheperpendiculardirection.ByNewton'sthirdlaw,thewater's forceonthebladeisequalandoppositetotheblade'sforceon thewater.Sincetheforceisconstant,wecanusetheequation F bladeonwater = p water t Choosingthe x axistobetotherightandthe y axistobeup,this canbebrokendownintocomponentsas F bladeonwater, x = p water, x t = )]TJ/F102 10.9091 Tf 8.485 0 Td [(vR t )]TJ/F39 10.9091 Tf 10.909 0 Td [(0 t = )]TJ/F102 10.9091 Tf 8.485 0 Td [(vR and F bladeonwater, y = p water, y t = 0 )]TJ/F39 10.9091 Tf 10.909 0 Td [( )]TJ/F102 10.9091 Tf 8.485 0 Td [(vR t t = vR Section4.5MomentuminThreeDimensions 97

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Thewater'sforceonthebladethushascomponents F wateronblade, x = vR F wateronblade, y = )]TJ/F102 10.9091 Tf 8.485 0 Td [(vR Insituationslikethis,itisalwaysagoodideatocheckthatthe resultmakessensephysically.The x componentofthewater's forceonthebladeispositive,whichiscorrectsinceweknowthe bladewillbepushedtotheright.The y componentisnegative, whichalsomakessensebecausethewatermustpushtheblade down.Themagnitudeofthewater'sforceonthebladeis j F wateronblade j = p 2 vR anditsdirectionisata45-degreeangledownandtotheright. DiscussionQuestions A Theguresshowajetofwaterstrikingtwodifferentobjects.How doesthetotaldownwardforcecompareinthetwocases?Howcouldthis factbeusedtocreateabetterwaterwheel?Suchawaterwheelisknown asaPeltonwheel. DiscussionquestionA. 4.6 R ApplicationsofCalculus BynowyouwillhavelearnedtorecognizedthecircumlocutionsI useinthesectionswithoutcalculusinordertointroducecalculuslikeconceptswithoutusingthenotation,terminology,ortechniques ofcalculus.Itwillthereforecomeasnosurprisetoyouthattherate ofchangeofmomentumcanberepresentedwithaderivative, F total = d p total d t Andofcoursethebusinessabouttheareaunderthe F )]TJ/F20 10.9091 Tf 10.922 0 Td [(t curveis reallyanintegral, p total = R F total d t ,whichcanbemadeintoan integralofavectorinthemoregeneralthree-dimensionalcase: p total = Z F total d t Inthecaseofamaterialobjectthatisneitherlosingnorpickingup mass,thesearejusttriviallyrearrangedversionsoffamiliarequations,e.g., F = m d v= d t rewrittenas F =d mv = d t .Thefollowing isalesstrivialexample,where F = ma alonewouldnothavebeen veryeasytoworkwith. 98 Chapter4ConservationofMomentum

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Rainfallingintoamovingcartexample21 If1kg/sofrainfallsverticallyintoa10-kgcartthatisrolling withoutfrictionataninitialspeedof1.0m/s,whatistheeffecton thespeedofthecartwhentherainrststartsfalling? Therainandthecartmakehorizontalforcesoneachother,but thereisnoexternalhorizontalforceontherain-plus-cartsystem, sothehorizontalmotionobeys F = d mv d t =0 Weusetheproductruletond 0= d m d t v + m d v d t Wearetryingtondhow v changes,sowesolvefor dv = dt d v d t = )]TJ/F102 10.9091 Tf 11.041 7.38 Td [(v m d m d t = )]TJ/F26 10.9091 Tf 10.303 15.382 Td [( 1m = s 10kg 1kg = s = )]TJ/F39 10.9091 Tf 8.485 0 Td [(0.1m = s 2 Thisisonlyatthemomentwhentherainstartstofall. Finallywenotethattherearecaseswhere F = ma isnotjust lessconvenientthan F =d p= d t butinfact F = ma iswrongand F =d p= d t isright.Agoodexampleistheformationofacomet's tailbysunlight.Wecannotuse F = ma todescribethisprocess, sincewearedealingwithacollisionoflightwithmatter,whereas Newton'slawsonlyapplytomatter.Theequation F =d p= d t ,on theotherhand,allowsustondtheforceexperiencedbyanatomof gasinthecomet'stailifweknowtherateatwhichthemomentum vectorsoflightraysarebeingturnedaroundbyreectionfromthe atom. Section4.6 R ApplicationsofCalculus 99

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Summary SelectedVocabulary momentum...ameasureofmotion,equalto mv formaterial objects collision.....aninteractionbetweenmovingobjectsthat lastsforacertaintime centerofmass..thebalancepointoraveragepositionofthe massinasystem Notation p ..........themomentumvector cm.........centerofmass,asin x cm a cm ,etc. OtherTerminologyandNotation impulse, I J ..theamountofmomentumtransferred, p elasticcollision.oneinwhichnoKEisconvertedintoother formsofenergy inelasticcollisiononeinwhichsomeKEisconvertedtoother formsofenergy Summary Iftwoobjectsinteractviaaforce,Newton'sthirdlawguaranteesthatanychangeinone'svelocityvectorwillbeaccompanied byachangeintheother'swhichisintheoppositedirection.Intuitively,thismeansthatifthetwoobjectsarenotactedonbyany externalforce,theycannotcooperatetochangetheiroverallstateof motion.Thiscanbemadequantitativebysayingthatthequantity m 1 v 1 + m 2 v 2 mustremainconstantaslongastheonlyforcesare theinternalonesbetweenthetwoobjects.Thisisaconservation law,calledtheconservationofmomentum,andliketheconservationofenergy,ithasevolvedovertimetoincludemoreandmore phenomenaunknownatthetimetheconceptwasinvented.The momentumofamaterialobjectis p = m v butthisismorelikeastandardforcomparisonofmomentarather thanadenition.Forinstance,lighthasmomentum,buthasno mass,andtheaboveequationisnottherightequationforlight.The lawofconservationofmomentumsaysthatthetotalmomentumof anyclosedsystem,i.e.,thevectorsumofthemomentumvectorsof allthethingsinthesystem,isaconstant. Animportantapplicationofthemomentumconceptistocollisions,i.e.,interactionsbetweenmovingobjectsthatlastforacertain amountoftimewhiletheobjectsareincontactorneareachother. Conservationofmomentumtellsusthatcertainoutcomesofacollisionareimpossible,andinsomecasesmayevenbesucientto predictthemotionafterthecollision.Inothercases,conservation ofmomentumdoesnotprovideenoughequationstondalltheunknowns.Insomecollisions,suchasthecollisionofasuperballwith 100 Chapter4ConservationofMomentum

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theoor,verylittlekineticenergyisconvertedintootherformsof energy,andthisprovidesonemoreequation,whichmaysuceto predicttheoutcome. Thetotalmomentumofasystemcanberelatedtoitstotalmass andthevelocityofitscenterofmassbytheequation p total = m total v cm Thecenterofmass,introducedonanintuitivebasisinbook1as thebalancepoint"ofanobject,canbegeneralizedtoanysystem containinganynumberofobjects,andisdenedmathematically astheweightedaverageofthepositionsofallthepartsofallthe objects, x cm = m 1 x 1 + m 2 x 2 + ::: m 1 + m 2 + ::: withsimilarequationsforthe y and z coordinates. Theframeofreferencemovingwiththecenterofmassofaclosed systemisalwaysavalidinertialframe,andmanyproblemscanbe greatlysimpliedbyworkingthemintheinertialframe.Forexample,anycollisionbetweentwoobjectsappearsinthec.m.frameas ahead-onone-dimensionalcollision. Whenasystemisnotclosed,therateatwhichmomentumis transferredinoroutissimplythetotalforcebeingexertedexternally onthesystem.Iftheforceisconstant, F total = p total t Whentheforceisnotconstant,theforceequalstheslopeofthe tangentlineonagraphof p versus t ,andthechangeinmomentum equalstheareaunderthe F )]TJ/F20 10.9091 Tf 10.909 0 Td [(t graph. Summary 101

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Problems Key p Acomputerizedanswercheckisavailableonline. R Aproblemthatrequirescalculus. ? Adicultproblem. 1 Deriveaformulaexpressingthekineticenergyofanobjectin termsofitsmomentumandmass. p 2 Twopeopleinarowboatwishtomovearoundwithoutcausing theboattomove.Whatshouldbetrueabouttheirtotalmomentum?Explain. 3 Alearjettravelingdueeastat300mi/hrcollideswitha jumbojetwhichwasheadingsouthwestat150mi/hr.Thejumbo jet'smassis5.0timesgreaterthanthatofthelearjet.Whenthey collide,thelearjetsticksintothefuselageofthejumbojet,andthey falltoearthtogether.Theirenginesstopfunctioningimmediately afterthecollision.Onamap,whatwillbethedirectionfromthe locationofthecollisiontotheplacewherethewreckagehitsthe ground?Giveanangle. p 4 Abulletleavesthebarrelofagunwithakineticenergyof90 J.Thegunbarrelis50cmlong.Thegunhasamassof4kg,the bullet10g. aFindthebullet'snalvelocity. p bFindthebullet'snalmomentum. p cFindthemomentumoftherecoilinggun. dFindthekineticenergyoftherecoilinggun,andexplainwhy therecoilinggundoesnotkilltheshooter. p Problem5 5 Thegraphshowstheforce,inmeganewtons,exertedbya rocketengineontherocketasafunctionoftime.Iftherocket's massif4000kg,atwhatspeedistherocketmovingwhentheengine 102 Chapter4ConservationofMomentum

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Problem8 stopsring?Assumeitgoesstraightup,andneglecttheforceof gravity,whichismuchlessthanameganewton. p 6 Cosmicraysareparticlesfromouterspace,mostlyprotonsand atomicnuclei,thatarecontinuallybombardingtheearth.Mostof them,althoughtheyaremovingextremelyfast,havenodiscernible eecteveniftheyhityourbody,becausetheirmassesaresosmall. Theirenergiesvary,however,andaverysmallminorityofthem haveextremelylargeenergies.Insomecasestheenergyisasmuch asseveralJoules,whichiscomparabletotheKEofawellthrown rock!Ifyouareinaplaneatahighaltitudeandaresoincredibly unluckyastobehitbyoneoftheserareultra-high-energycosmic rays,whatwouldyounotice,themomentumimpartedtoyourbody, theenergydissipatedinyourbodyasheat,orboth?Baseyourconclusionsonnumericalestimates,notjustrandomspeculation.At thesehighspeeds,oneshouldreallytakeintoaccountthedeviationsfromNewtonianphysicsdescribedbyEinstein'sspecialtheory ofrelativity.Don'tworryaboutthat,though. 7 Showthatforabodymadeupofmany equal masses,the equationforthecenterofmassbecomesasimpleaverageofallthe positionsofthemasses. 8 Thegureshowsaviewfromaboveofacollisionaboutto happenbetweentwoairhockeypucksslidingwithoutfriction.They havethesamespeed, v i ,beforethecollision,butthebigpuckis2.3 timesmoremassivethanthesmallone.Theirsideshavestickystu onthem,sowhentheycollide,theywillsticktogether.Atwhat anglewilltheyemergefromthecollision?Inadditiontogivinga numericalanswer,pleaseindicatebydrawingonthegurehowyour angleisdened. Solution,p.169 9 Aexibleropeofmass m andlength L slideswithoutfriction overtheedgeofatable.Let x bethelengthoftheropethatis hangingovertheedgeatagivenmomentintime. aShowthat x satisestheequationofmotiond 2 x= d t 2 = gx=L [Hint:Use F =d p= d t ,whichallowsyoutohandlethetwopartsof theropeseparatelyeventhoughmassismovingoutofonepartand intotheother.] bGiveaphysicalexplanationforthefactthatalargervalueof x ontheright-handsideoftheequationleadstoagreatervalueof theaccelerationontheleftside. cWhenwetakethesecondderivativeofthefunction x t weare supposedtogetessentiallythesamefunctionbackagain,except foraconstantoutinfront.Thefunction e x hasthepropertythat itisunchangedbydierentiation,soitisreasonabletolookfor solutionstothisproblemthatareoftheform x = be ct ,where b and c areconstants.Showthatthisdoesindeedprovideasolutionfor twospecicvaluesof c andforanyvalueof b dShowthatthesumofanytwosolutionstotheequationofmotion Problems 103

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isalsoasolution. eFindthesolutionforthecasewheretheropestartsatrestat t =0withsomenonzerovalueof x R ? 10 Averymassiveobjectwithvelocity v collideshead-onwith anobjectatrestwhosemassisverysmall.Nokineticenergyis convertedintootherforms.Provethatthelow-massobjectrecoils withvelocity2 v .[Hint:Usethecenter-of-massframeofreference.] 11 Whenthecontentsofarefrigeratorcooldown,thechanged molecularspeedsimplychangesinbothmomentumandenergy. Why,then,doesafridgetransfer power throughitsradiatorcoils, butnot force ? Solution,p.169 12 A10-kgbowlingballmovingat2.0m/shitsa1.0-kgbowling pin,whichisinitiallyatrest.Theotherpinsareallgonealready, andthecollisionishead-on,sothatthemotionisone-dimensional. Assumethatnegligibleamountsofheatandsoundareproduced. Findthevelocityofthepinimmediatelyafterthecollision. 13 Arocketejectsexhaustwithanexhaustvelocity u .The rateatwhichtheexhaustmassisusedmassperunittimeis b Weassumethattherocketacceleratesinastraightlinestarting fromrest,andthatnoexternalforcesactonit.Lettherocket's initialmassfuelplusthebodyandpayloadbe m i ,and m f beits nalmass,afterallthefuelisusedup.aFindtherocket'snal velocity, v ,intermsof u m i ,and m f .bAtypicalexhaustvelocity forchemicalrocketenginesis4000m/s.Estimatetheinitialmass ofarocketthatcouldaccelerateaone-tonpayloadto10%ofthe speedoflight,andshowthatthisdesignwon'twork.Forthesake oftheestimate,ignorethemassofthefueltanks. R ? 14 Areworkshootsupintotheair,andjustbeforeitexplodes ithasacertainmomentumandkineticenergy.Whatcanyousay aboutthemomentaandkineticenergiesofthepiecesimmediately aftertheexplosion?[BasedonaproblemfromPSSCPhysics.] Solution,p.169 15 Supposeasystemconsistingofpointlikeparticleshasatotal kineticenergy K cm measuredinthecenter-of-massframeofreference.Sincetheyarepointlike,theycannothaveanyenergydueto internalmotion. aProvethatinadierentframeofreference,movingwithvelocity u relativetothecenter-of-massframe,thetotalkineticenergy equals K cm + M j u j 2 = 2,where M isthetotalmass.[Hint:Youcan saveyourselfalotofwritingifyouexpressthetotalkineticenergy usingthedotproduct.] Solution,p.169 bUsethistoprovethatifenergyisconservedinoneframeof reference,thenitisconservedineveryframeofreference.Thetotal energyequalsthetotalkineticenergyplusthesumofthepotential 104 Chapter4ConservationofMomentum

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energiesduetotheparticles'interactionswitheachother,which weassumedependsonlyonthedistancebetweenparticles.[Fora simplernumericalexample,seeproblem13inch.1.] ? 16 Thebigdierencebetweentheequationsformomentumand kineticenergyisthatoneisproportionalto v andoneto v 2 .Both, however,areproportionalto m .Supposesomeonetellsyouthat there'sathirdquantity,funkosity,denedas f = m 2 v ,andthat funkosityisconserved.Howdoyouknowyourlegisbeingpulled? Solution,p.169 17 Amass m movingatvelocity v collideswithastationary targethavingthesamemass m .Findthemaximumamountof energythatcanbereleasedasheatandsound. Problems 105

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AtornadotouchesdowninSpringHill,Kansas,May20,1957. Chapter5 ConservationofAngular Momentum Sure,andmaybethesunwon'tcomeuptomorrow."Ofcourse, thesunonlyappearstogoupanddownbecausetheearthspins, sotheclicheshouldreallyrefertotheunlikelihoodoftheearth's stoppingitsrotationabruptlyduringthenight.Whycan'titstop? Itwouldn'tviolateconservationofmomentum,becausetheearth's rotationdoesn'taddanythingtoitsmomentum.WhileCalifornia spinsinonedirection,someequallymassivepartofIndiagoesthe oppositeway,cancelingitsmomentum.AhalttoEarth'srotation wouldentailadropinkineticenergy,butthatenergycouldsimply beconvertedintosomeotherform,suchasheat. Otherexamplesalongtheselinesarenothardtond.Ahydrogenatomspinsatthesamerateforbillionsofyears.Ahigh-diver whoisrotatingwhenhecomesotheboarddoesnotneedtomake 107

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anyphysicaleorttocontinuerotating,andindeedwouldbeunable tostoprotatingbeforehehitthewater. Theseobservationshavethehallmarksofaconservationlaw: Aclosedsystemisinvolved. Nothingismakinganeortto twisttheearth,thehydrogenatom,orthehigh-diver.Theyare isolatedfromrotation-changinginuences,i.e.,theyareclosed systems. Somethingremainsunchanged. Thereappearstobeanumericalquantityformeasuringrotationalmotionsuchthatthetotal amountofthatquantityremainsconstantinaclosedsystem. Somethingcanbetransferredbackandforthwithout changingthetotalamount. Ingurea,thejumperwantsto gethisfeetoutinfrontofhimsohecankeepfromdoingaface plant"whenhelands.Bringinghisfeetforwardwouldinvolvea certainquantityofcounterclockwiserotation,buthedidn'tstart outwithanyrotationwhenhelefttheground.Supposeweconsidercounterclockwiseaspositiveandclockwiseasnegative.The onlywayhislegscanacquiresomepositiverotationisifsomeother partofhisbodypicksupanequalamountofnegativerotation. Thisiswhyheswingshisarmsupbehindhim,clockwise. a / Anearlyphotographofanold-fashionedlong-jump. Whatnumericalmeasureofrotationalmotionisconserved?Car enginesandold-fashionedLPrecordshavespeedsofrotationmeasuredinrotationsperminuter.p.m.,butthenumberofrotationsperminuteorpersecondisnotaconservedquantity.A twirlinggureskater,forinstance,canpullherarmsintoincrease herr.p.m.'s.Therstsectionofthischapterdealswiththenumericaldenitionofthequantityofrotationthatresultsinavalid conservationlaw. 108 Chapter5ConservationofAngularMomentum

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b / Anoverheadviewofa pieceofputtybeingthrownat adoor.Eventhoughtheputty isneitherspinningnortraveling alongacurve,wemustdeneit ashavingsomekindofrotation becauseitisabletomakethe doorrotate. c / Asseenbysomeonestanding attheaxis,theputtychanges itsangularposition.Wethereforedeneitashavingangular momentum. 5.1ConservationofAngularMomentum Whenmostpeoplethinkofrotation,theythinkofasolidobject likeawheelrotatinginacirclearoundaxedpoint.Examplesof thistypeofrotation,calledrigidrotationorrigid-bodyrotation,includeaspinningtop,aseatedchild'sswingingleg,andahelicopter's spinningpropeller.Rotation,however,isamuchmoregeneralphenomenon,andincludesnoncircularexamplessuchasacometin anellipticalorbitaroundthesun,oracyclone,inwhichthecore completesacirclemorequicklythantheouterparts. Ifthereisanumericalmeasureofrotationalmotionthatisa conservedquantity,thenitmustincludenonrigidcaseslikethese, sincenonrigidrotationcanbetradedbackandforthwithrigidrotation.Forinstance,thereisatrickforndingoutifaneggis raworhardboiled.Ifyouspinahardboiledeggandthenstopit brieywithyournger,itstopsdead.Butifyoudothesamewith arawegg,itspringsbackintorotationbecausethesoftinteriorwas stillswirlingaroundwithinthemomentarilymotionlessshell.The patternofowoftheliquidpartispresumablyverycomplexand nonuniformduetotheasymmetricshapeoftheeggandthedierentconsistenciesoftheyolkandthewhite,butthereisapparently somewaytodescribetheliquid'stotalamountofrotationwitha singlenumber,ofwhichsomepercentageisgivenbacktotheshell whenyoureleaseit. Thebeststrategyistodeviseawayofdeningtheamountof rotationofasinglesmallpartofasystem.Theamountofrotation ofasystemsuchasacyclonewillthenbedenedasthetotalofall thecontributionsfromitsmanysmallparts. Thequestforaconservedquantityofrotationevenrequiresus tobroadentherotationconcepttoincludecaseswherethemotion doesn'trepeatorevencurvearound.Ifyouthrowapieceofputty atadoor,thedoorwillrecoilandstartrotating.Theputtywas travelingstraight,notinacircle,butifthereistobeageneral conservationlawthatcancoverthissituation,itappearsthatwe mustdescribetheputtyashavinghadsomerotation,"whichit thengaveuptothedoor.Thebestwayofthinkingaboutitisto attributerotationtoanymovingobjectorpartofanobjectthat changesitsangleinrelationtotheaxisofrotation.Intheputtyand-doorexample,thehingeofthedooristhenaturalpointtothink ofasanaxis,andtheputtychangesitsangleasseenbysomeone standingatthehinge.Forthisreason,theconservedquantityweare investigatingiscalled angular momentum.Thesymbolforangular momentumcan'tbe a or m ,sincethoseareusedforacceleration andmass,sothesymbol L isarbitrarilychoseninstead. Imaginea1-kgblobofputty,thrownatthedoorataspeedof 1m/s,whichhitsthedooratadistanceof1 m fromthehinge. Wedenethisblobtohave1unitofangularmomentum.When Section5.1ConservationofAngularMomentum 109

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d / Aputtyblobthrowndirectlyattheaxishasnoangular motion,andthereforenoangular momentum.Itwillnotcausethe doortorotate. e / Onlythecomponentof thevelocityvectorperpendicular tothedashedlineshouldbe countedintothedenitionof angularmomentum. ithitsthedoor,thedoorwillrecoilandstartrotating.Wecan usethespeedatwhichthedoorrecoilsasameasureoftheangular momentumtheblobbroughtin. 1 Experimentsshow,notsurprisingly,thata2-kgblobthrownin thesamewaymakesthedoorrotatetwiceasfast,sotheangular momentumoftheputtyblobmustbeproportionaltomass, L / m Similarly,experimentsshowthatdoublingthevelocityofthe blobwillhaveadoublingeectontheresult,soitsangularmomentummustbeproportionaltoitsvelocityaswell, L / mv Youhaveundoubtedlyhadtheexperienceofapproachingaclosed doorwithoneofthosebar-shapedhandlesonitandpushingonthe wrongside,thesideclosetothehinges.Youfeellikeanidiot,becauseyouhavesolittleleveragethatyoucanhardlybudgethedoor. Thesamewouldbetruewiththeputtyblob.Experimentswould showthattheamountofrotationtheblobcangivetothedooris proportionaltothedistance, r ,fromtheaxisofrotation,soangular momentummustalsobeproportionalto r L / mvr Wearealmostdone,butthereisonemissingingredient.We knowongroundsofsymmetrythataputtyballthrowndirectly inwardtowardthehingewillhavenoangularmomentumtogive tothedoor.Afterall,therewouldnotevenbeanywaytodecidewhethertheball'srotationwasclockwiseorcounterclockwise inthissituation.Itisthereforeonlythecomponentoftheblob's velocityvectorperpendiculartothedoorthatshouldbecountedin itsangularmomentum, L = mv ? r Moregenerally, v ? shouldbethoughtofasthecomponentofthe object'svelocityvectorthatisperpendiculartothelinejoiningthe objecttotheaxisofrotation. Wendthatthisequationagreeswiththedenitionoftheoriginalputtyblobashavingoneunitofangularmomentum,andwecan nowseethattheunitsofangularmomentumarekg m = s m,i.e., kg m 2 = s.Thisgivesusawayofcalculatingtheangularmomentum ofanymaterialobjectoranysystemconsistingofmaterialobjects: 1 Weassumethatthedoorismuchmoremassivethantheblob.Underthis assumption,thespeedatwhichthedoorrecoilsismuchlessthantheoriginal speedoftheblob,sotheblobhaslostessentiallyallitsangularmomentum,and givenittothedoor. 110 Chapter5ConservationofAngularMomentum

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f / Agureskaterpullsinher armssothatshecanexecutea spinmorerapidly. angularmomentumofamaterialobject Theangularmomentumofamovingparticleis L = mv ? r where m isitsmass, v ? isthecomponentofitsvelocityvector perpendiculartothelinejoiningittotheaxisofrotation,and r is itsdistancefromtheaxis.Positiveandnegativesignsareusedto describeoppositedirectionsofrotation. Theangularmomentumofanite-sizedobjectorasystem ofmanyobjectsisfoundbydividingitupintomanysmallparts, applyingtheequationtoeachpart,andaddingtondthetotal amountofangularmomentum. Notethat r isnotnecessarilytheradiusofacircle.Asimplied bythequaliers,matterisn'ttheonlythingthatcanhaveangular momentum.Lightcanalsohaveangularmomentum,andtheabove equationwouldnotapplytolight. Conservationofangularmomentumhasbeenveriedoverand overagainbyexperiment,andisnowbelievedtobeoneofthethree mostfundamentalprinciplesofphysics,alongwithconservationof energyandmomentum. Agureskaterpullsherarmsinexample1 Whenagureskateristwirling,thereisverylittlefrictionbetween herandtheice,sosheisessentiallyaclosedsystem,andher angularmomentumisconserved.Ifshepullsherarmsin,sheis decreasing r foralltheatomsinherarms.Itwouldviolateconservationofangularmomentumifshethencontinuedrotatingat thesamespeed,i.e.,takingthesameamountoftimeforeach revolution,becauseherarms'contributionstoherangularmomentumwouldhavedecreased,andnootherpartofherwould haveincreaseditsangularmomentum.Thisisimpossiblebecauseitwouldviolateconservationofangularmomentum.Ifher totalangularmomentumistoremainconstant,thedecreasein r forherarmsmustbecompensatedforbyanoverallincreasein herrateofrotation.Thatis,bypullingherarmsin,shesubstantiallyreducesthetimeforeachrotation. Section5.1ConservationofAngularMomentum 111

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h / Example3.Aviewofthe earth-moonsystemfromabove thenorthpole.Alldistances havebeenhighlydistortedfor legibility.Theearth'srotationis counterclockwisefromthispoint ofviewarrow.Themoon'sgravitycreatesabulgeontheside nearit,becauseitsgravitational pullisstrongerthere,andan anti-bulgeonthefarside,since itsgravitythereisweaker.For simplicity,let'sfocusonthetidal bulgeclosertothemoon.Its frictionalforceistryingtoslow downtheearth'srotation,soits forceontheearth'ssolidcrustis towardthebottomofthegure. ByNewton'sthirdlaw,thecrust mustthusmakeaforceonthe bulgewhichistowardthetopof thegure.Thiscausesthebulge tobepulledforwardataslight angle,andthebulge'sgravity thereforepullsthemoonforward, acceleratingitsorbitalmotion abouttheearthandingingit outward. g / Example2. Changingtheaxisexample2 Anobject'sangularmomentumcanbedifferentdependingonthe axisaboutwhichitrotates.Figuregshowsshowstwodoubleexposurephotographsaviolaplayertippingthebowinorderto crossfromonestringtoanother.Muchmoreangularmomentum isrequiredwhenplayingnearthebow'shandle,calledthefrog, asinthepanelontheright;notonlyaremostoftheatomsinthe bowatgreaterdistances, r ,fromtheaxisofrotation,buttheones inthetipalsohavemoremomentum, p .Itisdifcultfortheplayer toquicklytransferalargeangularmomentumintothebow,and thentransferitbackoutjustasquickly.Inthelanguageofsection 5.4,largetorquesarerequired.Thisisoneofthereasonsthat stringplayerstendtostaynearthemiddleofthebowasmuchas possible. Earth'sslowingrotationandtherecedingmoonexample3 Asnotedinchapter1,theearth'srotationisactuallyslowingdown verygradually,withthekineticenergybeingdissipatedasheatby frictionbetweenthelandandthetidalbulgesraisedintheseas bytheearth'sgravity.Doesthismeanthatangularmomentumis notreallyperfectlyconserved?No,itjustmeansthattheearth isnotquiteaclosedsystembyitself.Ifweconsidertheearth andmoonasasystem,thentheangularmomentumlostbythe earthmustbegainedbythemoonsomehow.Infactveryprecise measurementsofthedistancebetweentheearthandthemoon havebeencarriedoutbybouncinglaserbeamsoffofamirror lefttherebyastronauts,andthesemeasurementsshowthatthe moonisrecedingfromtheearthatarateof4centimetersper year!Themoon'sgreatervalueof r meansthatithasagreater 112 Chapter5ConservationofAngularMomentum

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angularmomentum,andtheincreaseturnsouttobeexactlythe amountlostbytheearth.Inthedaysofthedinosaurs,thedays weresignicantlyshorter,andthemoonwascloserandappeared biggerinthesky. Butwhatforceiscausingthemoontospeedup,drawingitout intoalargerorbit?Itisthegravitationalforcesoftheearth'stidal bulges.Theeffectisdescribedqualitativelyinthecaptionofthe gure.Theresultwouldobviouslybeextremelydifculttocalculatedirectly,andthisisoneofthosesituationswhereaconservationlawallowsustomakeprecisequantitativestatementsabout theoutcomeofaprocesswhenthecalculationoftheprocess itselfwouldbeprohibitivelycomplex. Restrictiontorotationinaplane Isangularmomentumavector,orascalar?Itdoeshavea directioninspace,butit'sadirectionofrotation,notastraight-line directionlikethedirectionsofvectorssuchasvelocityorforce.It turnsoutthatthereisawayofdeningangularmomentumasa vector,butinthisbooktheexampleswillbeconnedtoasingle planeofrotation,i.e.,eectivelytwo-dimensionalsituations.Inthis specialcase,wecanchoosetovisualizetheplaneofrotationfrom onesideortheother,andtodeneclockwiseandcounterclockwise rotationashavingoppositesignsofangularmomentum. DiscussionQuestion A Conservationofplainoldmomentum, p ,canbethoughtofasthe greatlyexpandedandmodieddescendantofGalileo'soriginalprinciple ofinertia,thatnoforceisrequiredtokeepanobjectinmotion.Theprincipleofinertiaiscounterintuitive,andtherearemanysituationsinwhichit appearssuperciallythataforce is neededtomaintainmotion,asmaintainedbyAristotle.Thinkofasituationinwhichconservationofangular momentum, L ,alsoseemstobeviolated,makingitseemincorrectlythat somethingexternalmustactonaclosedsystemtokeepitsangularmomentumfromrunningdown. 5.2AngularMomentuminPlanetaryMotion Wenowdiscusstheapplicationofconservationofangularmomentumtoplanetarymotion,bothbecauseofitsintrinsicimportance andbecauseitisagoodwaytodevelopavisualintuitionforangular momentum. Kepler'slawofequalareasstatesthattheareasweptoutby aplanetinacertainlengthoftimeisalwaysthesame.Angular momentumhadnotbeeninventedinKepler'stime,andhedidnot evenknowthemostbasicphysicalfactsabouttheforcesatwork.He thoughtofthislawasanentirelyempiricalandunexpectedlysimple wayofsummarizinghisdata,arulethatsucceededindescribing andpredictinghowtheplanetsspedupandsloweddownintheir Section5.2AngularMomentuminPlanetaryMotion 113

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i / Theplanet'sangularmomentumisrelatedtotherateat whichitsweepsoutarea. ellipticalpaths.Itisnowfairlysimple,however,toshowthatthe equalarealawamountstoastatementthattheplanet'sangular momentumstaysconstant. Thereisnosimplegeometricalrulefortheareaofapiewedge cutoutofanellipse,butifweconsideraveryshorttimeinterval, asshowningurei,theshadedshapesweptoutbytheplanetis verynearlyatriangle.Wedoknowhowtocomputetheareaofa triangle.Itisonehalftheproductofthebaseandtheheight: area= 1 2 bh Wewishtorelatethistoangularmomentum,whichcontains thevariables r and v ? .Ifweconsiderthesuntobetheaxisof rotation,thenthevariable r isidenticaltothebaseofthetriangle, r = b .Referringtothemagniedportionofthegure, v ? canbe relatedto h ,becausethetworighttrianglesaresimilar: h distancetraveled = v ? j v j Theareacanthusberewrittenas area= 1 2 r v ? distancetraveled j v j Thedistancetraveledequals j v j t ,sothissimpliesto area= 1 2 rv ? t Wehavefoundthefollowingrelationshipbetweenangularmomentumandtherateatwhichareaissweptout: L =2 m area t Thefactorof2infrontissimplyamatterofconvention,sinceany conservedquantitywouldbeanequallyvalidconservedquantityif youmultiplieditbyaconstant.Thefactorof m wasnotrelevant toKepler,whodidnotknowtheplanets'masses,andwhowasonly describingthemotionofoneplanetatatime. WethusndthatKepler'sequal-arealawisequivalenttoastatementthattheplanet'sangularmomentumremainsconstant.But wait,whyshoulditremainconstant?|theplanetisnotaclosed system,sinceitisbeingactedonbythesun'sgravitationalforce. Therearetwovalidanswers.Therstisthatitisactuallythetotalangularmomentumofthesunplustheplanetthatisconserved. Thesun,however,ismillionsoftimesmoremassivethanthetypical planet,soitacceleratesverylittleinresponsetotheplanet'sgravitationalforce.Itisthusagoodapproximationtosaythatthesun 114 Chapter5ConservationofAngularMomentum

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DiscussionquestionA. doesn'tmoveatall,sothatnoangularmomentumistransferred betweenitandtheplanet. Thesecondansweristhattochangetheplanet'sangularmomentumrequiresnotjustaforcebutaforceappliedinacertain way.Insection5.4wediscussthetransferofangularmomentumby aforce,butthebasicideahereisthataforcedirectlyintowardthe axisdoesnotchangetheangularmomentum. DiscussionQuestions A Supposeanobjectissimplytravelinginastraightlineatconstant speed.Ifwepicksomepointnotonthelineandcallittheaxisofrotation, isareasweptoutbytheobjectataconstantrate?Woulditmatterifwe choseadifferentaxis? B Thegureisastrobephotoofapendulumbob,takenfromunderneaththependulumlookingstraightup.Theblackstringcan'tbeseen inthephotograph.Thebobwasgivenaslightsidewayspushwhenit wasreleased,soitdidnotswinginaplane.Thebrightspotmarksthe center,i.e.,thepositionthebobwouldhaveifithungstraightdownatus. Doesthebob'sangularmomentumappeartoremainconstantifweconsiderthecentertobetheaxisofrotation?Whatifwechooseadifferent axis? DiscussionquestionB. 5.3TwoTheoremsAboutAngularMomentum Withplainoldmomentum, p ,wehadthefreedomtoworkinany inertialframeofreferenceweliked.Thesameobjectcouldhave dierentvaluesofmomentumintwodierentframes,iftheframes werenotatrestwithrespecttoeachother.Conservationofmomentum,however,wouldbetrueineitherframe.Aslongaswe employedasingleframeconsistentlythroughoutacalculation,everythingwouldwork. Thesameistrueforangularmomentum,andinadditionthere isanambiguitythatarisesfromthedenitionofanaxisofrotation. Forawheel,thenaturalchoiceofanaxisofrotationisobviously theaxle,butwhataboutaneggrotatingonitsside?Theegg Section5.3TwoTheoremsAboutAngularMomentum 115

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j / Example4. hasanasymmetricshape,andthusnoclearlydenedgeometric center.Asimilarissuearisesforacyclone,whichdoesnoteven haveasharplydenedshape,orforacomplicatedmachinewith manygears.Thefollowingtheorem,therstoftwopresentedin thissectionwithoutproof,explainshowtodealwiththisissue. AlthoughIhaveputdescriptivetitlesaboveboththeorems,they havenogenerallyacceptednames. thechoiceofaxistheorem Itisentirelyarbitrarywhatpointonedenesastheaxisfor purposesofcalculatingangularmomentum.Ifaclosedsystem'sangularmomentumisconservedwhencalculatedwith onechoiceofaxis,thenitwillalsobeconservedforanyother choice.Likewise,anyinertialframeofreferencemaybeused. Collidingasteroidsdescribedwithdifferentaxesexample4 ObserversonplanetsAand B bothseethetwoasteroidscolliding.Theasteroidsareofequalmassandtheirimpactspeedsare thesame.Astronomersoneachplanetdecidetodenetheirown planetastheaxisofrotation.PlanetAistwiceasfarfromthecollisionasplanet B .Theasteroidscollideandstick.Forsimplicity, assumeplanetsAand B arebothatrest. WithplanetAastheaxis,thetwoasteroidshavethesameamount ofangularmomentum,butonehaspositiveangularmomentum andtheotherhasnegative.Beforethecollision,thetotalangular momentumisthereforezero.Afterthecollision,thetwoasteroids willhavestoppedmoving,andagainthetotalangularmomentumiszero.Thetotalangularmomentumbothbeforeandafter thecollisioniszero,soangularmomentumisconservedifyou chooseplanetAastheaxis. TheonlydifferencewithplanetBasaxisisthat r issmallerbya factoroftwo,soalltheangularmomentaarehalved.Eventhough theangularmomentaaredifferentthantheonescalculatedby planetA,angularmomentumisstillconserved. Theearthspinsonitsownaxisonceaday,butsimultaneously travelsinitscircularone-yearorbitaroundthesun,soanygiven partofittracesoutacomplicatedloopypath.Itwouldseemdicult tocalculatetheearth'sangularmomentum,butitturnsoutthat thereisanintuitivelyappealingshortcut:wecansimplyaddupthe angularmomentumduetoitsspinplusthatarisingfromitscenter ofmass'scircularmotionaroundthesun.Thisisaspecialcaseof thefollowinggeneraltheorem: thespintheorem Anobject'sangularmomentumwithrespecttosomeoutside axisAcanbefoundbyaddinguptwoparts: Therstpartistheobject'sangularmomentumfound byusingitsowncenterofmassastheaxis,i.e.,theangular 116 Chapter5ConservationofAngularMomentum

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k / Everyonehasastrong tendencytothinkofthediveras rotatingabouthisowncenterof mass.However,heisyingin anarc,andhealsohasangular momentumbecauseofthis motion. l / Thisrigidobjecthasangularmomentumbothbecauseitis spinningaboutitscenterofmass andbecauseitismovingthrough space. momentumtheobjecthasbecauseitisspinning. Theotherpartequalstheangularmomentumthatthe objectwouldhavewithrespecttotheaxisAifithadallits massconcentratedatandmovingwithitscenterofmass. Asystemwithitscenterofmassatrestexample5 Inthespecialcaseofanobjectwhosecenterofmassisatrest, thespintheoremimpliesthattheobject'sangularmomentumis thesameregardlessofwhataxiswechoose.Thisisaneven strongerstatementthanthechoiceofaxistheorem,whichonly guaranteesthatangularmomentumisconservedforanygiven choiceofaxis,withoutspecifyingthatitisthesameforallsuch choices. Angularmomentumofarigidobjectexample6 Amotorcyclewheelhasalmostallitsmassconcentratedat theoutside.Ifthewheelhasmass m andradius r ,andthetime requiredforonerevolutionis T ,whatisthespinpartofitsangular momentum? Thisisanexampleofthecommonlyencounteredspecialcase ofrigidmotion,asopposedtotherotationofasystemlikeahurricaneinwhichthedifferentpartstakedifferentamountsoftime togoaround.Wedon'treallyhavetogothroughalaborious processofaddingupcontributionsfromallthemanypartsofa wheel,becausetheyareallataboutthesamedistancefromthe axis,andareallmovingaroundtheaxisataboutthesamespeed. Thevelocityisallperpendiculartothespokes, v ? = v =circumference = T =2 r = T andtheangularmomentumofthewheelaboutitscenteris L = mv ? r = m r = T r =2 mr 2 = T Notethatalthoughthefactorsof2 inthisexpressionispeculiar toawheelwithitsmassconcentratedontherim,theproportionalityto m=T wouldhavebeenthesameforanyotherrigidlyrotating object.Althoughanobjectwithanoncircularshapedoesnothave aradius,itisalsotrueingeneralthatangularmomentumisproportionaltothesquareoftheobject'ssizeforxedvaluesof m and T .Forinstancedoublinganobject'ssizedoublesboththe v ? and r factorsinthecontributionofeachofitspartstothetotalangular momentum,resultinginanoverallfactoroffourincrease. Thegureshowssomeexamplesofangularmomentaofvarious shapesrotatingabouttheircentersofmass.Theequationsfortheir Section5.3TwoTheoremsAboutAngularMomentum 117

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angularmomentawerederivedusingcalculus,asdescribedinmy calculus-basedbookSimpleNature.Donotmemorizetheseequations! Thehammerthrowexample7 Inthemen'sOlympichammerthrow,asteelballofradius6.1cm isswungontheendofawireoflength1.22m.Whatfractionof theball'sangularmomentumcomesfromitsrotation,asopposed toitsmotionthroughspace? It'salwaysimportanttosolveproblemssymbolicallyrst,and pluginnumbersonlyattheend,solettheradiusoftheballbe b andthelengthofthewire ` .Ifthetimetheballtakestogoonce aroundthecircleis T ,thenthisisalsothetimeittakestorevolve oncearounditsownaxis.Itsspeedis v =2 `= T ,soitsangular momentumduetoitsmotionthroughspaceis mv ` =2 m ` 2 = T Itsangularmomentumduetoitsrotationarounditsowncenteris = 5 mb 2 = T .Theratioofthesetwoangularmomentais = 5 b =` 2 =1.0 10 )]TJ/F39 7.9701 Tf 6.587 0 Td [(3 .Theangularmomentumduetotheball's spinisextremelysmall. 118 Chapter5ConservationofAngularMomentum

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m / Example8. Topplingarodexample8 Arodoflength b andmass m standsupright.Wewanttostrike therodatthebottom,causingittofallandlandat.Findthe momentum, p ,thatshouldbedelivered,intermsof m b ,and g .Canthisreallybedonewithouthavingtherodscrapeonthe oor? Thisisaniceexampleofaquestionthatcanverynearlybe answeredbasedonlyonunits.Sincethethreevariables, m b and g ,allhavedifferentunits,theycan'tbeaddedorsubtracted. Theonlywaytocombinethemmathematicallyisbymultiplication ordivision.Multiplyingoneofthembyitselfisexponentiation,so ingeneralweexpectthattheanswermustbeoftheform p = Am j b k g l where A j k ,and l areunitlessconstants.Theresulthastohave unitsofkg m = s.Togetkilogramstotherstpower,weneed j =1, meterstotherstpowerrequires k + l =1, andsecondstothepower )]TJ/F39 10.9091 Tf 8.484 0 Td [(1implies l =1 = 2. Wend j =1, k =1 = 2,and l =1 = 2,sothesolutionmustbeofthe form p = Am p bg Notethatnophysicswasrequired! Considerationofunits,however,won'thelpustondtheunitlessconstant A .Let t bethetimetherodtakestofall,sothat = 2 gt 2 = b = 2.Iftherodisgoingtolandexactlyonitsside, thenthenumberofrevolutionsitcompleteswhileintheairmust be1/4,or3/4,or5/4,...,butallthepossibilitiesgreaterthan1/4 wouldcausetheheadoftherodtocollidewiththeoorprematurely.Therodmustthereforerotateataratethatwouldcause ittocompleteafullrotationinatime T =4 t ,andithasangular momentum L = = 6 mb 2 = T Themomentumlostbytheobjectstrikingtherodis p ,andby conservationofmomentum,thisistheamountofmomentum,in thehorizontaldirection,thattherodacquires.Inotherwords, therodwillyforwardalittle.However,thishasnoeffecton thesolutiontotheproblem.Moreimportantly,theobjectstriking therodlosesangularmomentum bp = 2,whichisalsotransferred totherod.Equatingthistotheexpressionabovefor L ,wend p = = 12 m p bg Section5.3TwoTheoremsAboutAngularMomentum 119

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n / Energy,momentum,and angularmomentumcanbetransferred.Theratesoftransferare calledpower,force,andtorque. o / Theplane'sfourengines producezerototaltorquebutnot zerototalforce. Finally,weneedtoknowwhetherthiscanreallybedonewithout havingthefootoftherodscrapeontheoor.Thegureshows thattheanswerisnoforthisrodofnitewidth,butitappears thattheanswerwouldbeyesforasufcientlythinrod.Thisis analyzedfurtherinhomeworkproblem28onpage141. DiscussionQuestion A Intheexampleofthecollidingasteroids,supposeplanetAwasmovingtowardthetopofthepage,atthesamespeedasthebottomasteroid. HowwouldplanetA'sastronomersdescribetheangularmomentaofthe asteroids?Wouldangularmomentumstillbeconserved? 5.4Torque:theRateofTransferofAngular Momentum Forcecanbeinterpretedastherateoftransferofmomentum.The equivalentinthecaseofangularmomentumiscalled torque rhymes withfork".Whereforcetellsushowhardwearepushingor pullingonsomething,torqueindicateshowhardwearetwistingon it.TorqueisrepresentedbytheGreeklettertau, ,andtherate ofchangeofanobject'sangularmomentumequalsthetotaltorque actingonit, total = L t Iftheangularmomentumdoesnotchangeataconstantrate,the totaltorqueequalstheslopeofthetangentlineonagraphof L versus t Aswithforceandmomentum,itoftenhappensthatangular momentumrecedesintothebackgroundandwefocusourintereston thetorques.Thetorque-focusedpointofviewisexempliedbythe factthatmanyscienticallyuntrainedbutmechanicallyaptpeople knowallabouttorque,butnoneofthemhaveheardofangular momentum.Carenthusiastseagerlycompareengines'torques,and thereisatoolcalledatorquewrenchwhichallowsonetoapplya desiredamountoftorquetoascrewandavoidovertighteningit. Torquedistinguishedfromforce Ofcourseaforceisnecessaryinordertocreateatorque|you can'ttwistascrewwithoutpushingonthewrench|butforceand torquearetwodierentthings.Onedistinctionbetweenthemis direction.Weusepositiveandnegativesignstorepresentforcesin thetwopossibledirectionsalongaline.Thedirectionofatorque, however,isclockwiseorcounterclockwise,notalineardirection. Theotherdierencebetweentorqueandforceisamatterof leverage.Agivenforceappliedatadoor'sknobwillchangethe door'sangularmomentumtwiceasrapidlyasthesameforceapplied halfwaybetweentheknobandthehinge.Thesameamountofforce producesdierentamountsoftorqueinthesetwocases. 120 Chapter5ConservationofAngularMomentum

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p / Thesimplephysicalsituationweusetoderiveanequation fortorque.Aforcethatpoints directlyinatoroutawayfromthe axisproducesneitherclockwise norcounterclockwiseangular momentum.Aforceintheperpendiculardirectiondoestransfer angularmomentum. Itispossibletohaveazerototaltorquewithanonzerototal force.Anairplanewithfourjetengines,o,wouldbedesignedso thattheirforcesarebalancedontheleftandright.Theirforcesare allinthesamedirection,buttheclockwisetorquesoftwoofthe enginesarecanceledbythecounterclockwisetorquesoftheother two,givingzerototaltorque. Converselywecanhavezerototalforceandnonzerototaltorque. Amerry-go-round'sengineneedstosupplyanonzerotorqueonit tobringituptospeed,butthereiszerototalforceonit.Ifthere wasnotzerototalforceonit,itscenterofmasswouldaccelerate! Relationshipbetweenforceandtorque Howdowecalculatetheamountoftorqueproducedbyagiven force?Sinceitdependsonleverage,weshouldexpectittodepend onthedistancebetweentheaxisandthepointofapplicationof theforce.We'llderiveanequationrelatingtorquetoforcefora particularverysimplesituation,andstatewithoutproofthatthe equationactuallyappliestoallsituations. Considerapointlikeobjectwhichisinitiallyatrestatadistance r fromtheaxiswehavechosenfordeningangularmomentum. Werstobservethataforcedirectlyinwardoroutward,alongthe lineconnectingtheaxistotheobject,doesnotimpartanyangular momentumtotheobject. Aforceperpendiculartothelineconnectingtheaxisandthe objectdoes,however,maketheobjectpickupangularmomentum. Newton'ssecondlawgives a = F m andassumingforsimplicitythattheforceisconstant,theconstant accelerationequation a = v= t allowsustondthevelocitythe objectacquiresafteratime t v = F t m Wearetryingtorelateforcetoachangeinangularmomentum,so wemultiplybothsidesoftheequationby mr togive m vr = F tr L = F tr Dividingby t givesthetorque: L t = Fr = Fr Section5.4Torque:theRateofTransferofAngularMomentum 121

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q / Thegeometricrelationships referedtointherelationship betweenforceandtorque. Ifaforceactsatanangleotherthan0or90 withrespecttothe linejoiningtheobjectandtheaxis,itwouldbeonlythecomponent oftheforceperpendiculartothelinethatwouldproduceatorque, = F ? r Althoughthisresultwasprovedunderasimpliedsetofcircumstances,itismoregenerallyvalid: relationshipbetweenforceandtorque Therateatwhichaforcetransfersangularmomentumtoan object,i.e.,thetorqueproducedbytheforce,isgivenby j j = r j F ? j where r isthedistancefromtheaxistothepointofapplicationoftheforce,and F ? isthecomponentoftheforcethat isperpendiculartothelinejoiningtheaxistothepointof application. Theequationisstatedwithabsolutevaluesignsbecausethe positiveandnegativesignsofforceandtorqueindicatedierent things,sothereisnousefulrelationshipbetweenthem.Thesign ofthetorquemustbefoundbyphysicalinspectionofthecaseat hand. Fromtheequation,weseethattheunitsoftorquecanbewrittenasnewtonsmultipliedbymeters.Metrictorquewrenchesare calibratedinN m,butAmericanonesusefoot-pounds,whichisalso aunitofdistancemultipliedbyaunitofforce.Weknowfromour studyofmechanicalworkthatnewtonsmultipliedbymetersequal joules,buttorqueisacompletelydierentquantityfromwork,and nobodywritestorqueswithunitsofjoules,eventhoughitwouldbe technicallycorrect. self-checkA Comparethemagnitudesandsignsofthefourtorquesshowninthe gure. Answer,p.166 122 Chapter5ConservationofAngularMomentum

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r / Thequantity r ? Howtorquedependsonthedirectionoftheforceexample9 Howcanthetorqueappliedtothewrenchinthegurebeexpressedintermsof r j F j ,andtheangle ? Theforcevectorandits F ? componentformthehypotenuse andonelegofarighttriangle, andtheinteriorangleoppositeto F ? equals .Theabsolutevalue of F ? canthusbeexpressedas F ? = j F j sin leadingto j j = r j F j sin Sometimestorquecanbemoreneatlyvisualizedintermsofthe quantity r ? showningurer,whichgivesusathirdwayofexpressingtherelationshipbetweentorqueandforce: j j = r ? j F j Ofcourseyouwouldnotwanttogoandmemorizeallthree equationsfortorque.Startingfromanyoneofthemyoucouldeasily derivetheothertwousingtrigonometry.Familiarizingyourselfwith themcanhoweverclueyouintoeasieravenuesofattackoncertain problems. Thetorqueduetogravity Upuntilnowwe'vebeenthinkingintermsofaforcethatacts atasinglepointonanobject,suchastheforceofyourhandonthe wrench.Thisisofcourseanapproximation,andforanextremely realisticcalculationofyourhand'storqueonthewrenchyoumight needtoaddupthetorquesexertedbyeachsquaremillimeterwhere yourskintouchesthewrench.Thisisseldomnecessary.Butin thecaseofagravitationalforce,thereisneveranysinglepointat whichtheforceisapplied.Ourplanetisexertingaseparatetugon everybrickintheLeaningTowerofPisa,andthetotalgravitational torqueonthetoweristhesumofthetorquescontributedbyallthe littleforces.Luckilythereisatrickthatallowsustoavoidsuch amassivecalculation.Itturnsoutthatforpurposesofcomputing thetotalgravitationaltorqueonanobject,youcangettheright answerbyjustpretendingthatthewholegravitationalforceactsat theobject'scenterofmass. Section5.4Torque:theRateofTransferofAngularMomentum 123

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t / Example11. s / Example10. Gravitationaltorqueonanoutstretchedarmexample10 Yourarmhasamassof3.0kg,anditscenterofmassis30 cmfromyourshoulder.Whatisthegravitationaltorqueonyour armwhenitisstretchedouthorizontallytooneside,takingthe shouldertobetheaxis? Thetotalgravitationalforceactingonyourarmis j F j =.0kg.8m = s 2 =29N. Forthepurposeofcalculatingthegravitationaltorque,wecan treattheforceasifitactedatthearm'scenterofmass.Theforce isstraightdown,whichisperpendiculartothelineconnectingthe shouldertothecenterofmass,so F ? = j F j =29N. Continuingtopretendthattheforceactsatthecenterofthearm, r equals30cm=0.30m,sothetorqueis = rF ? =9N m. Cowtippingexample11 In2005,Dr.MargoLillieandhergraduatestudentTracyBoechlerpublishedastudyclaimingtodebunkcowtipping.Theirclaim wasbasedonananalysisofthetorquesthatwouldberequired totipacow,whichshowedthatonepersonwouldn'tbeableto makeenoughtorquetodoit.Alivelydiscussionensuedonthe popularwebsiteslashdot.orgnewsfornerds,stuffthatmattersconcerningthevalidityofthestudy.Personally,Ihadalwaysassumedthatcow-tippingwasagroupsportanyway,butas aphysicist,Ialsohadsomequibbleswiththeircalculation.Here's myownanalysis. Therearethreeforcesonthecow:theforceofgravity F W ,the ground'snormalforce F N ,andthetippers'force F A Assoonasthecow'slefthoovesontherightfromourpointof viewbreakcontactwiththeground,theground'sforceisbeing appliedonlytohoovesontheotherside.Wedon'tknowthe ground'sforce,andwedon'twanttondit.Thereforewetake theaxistobeatitspointofapplication,sothatitstorqueiszero. Forthepurposeofcomputingtorques,wecanpretendthatgravityactsatthecow'scenterofmass,whichI'veplacedalittle lowerthanthecenterofitstorso,sinceitslegsandheadalso havesomemass,andthelegsaremoremassivethanthehead andstickoutfarther,sotheylowerthec.m.morethanthehead raisesit.Theangle W betweentheverticalgravitationalforce andtheline r W isabout14 .AnestimatebyMattSemkeatthe UniversityofNebraska-Lincolngives20 ,whichisinthesame ballpark. 124 Chapter5ConservationofAngularMomentum

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Togeneratethemaximumpossibletorquewiththeleastpossible force,thetipperswanttopushatapointasfaraspossiblefrom theaxis,whichwillbetheshoulderontheotherside,andthey wanttopushata90degreeanglewithrespecttotheradiusline r A Whenthetippersarejustbarelyapplyingenoughforcetoraise thecow'shoovesononeside,thetotaltorquehastobejust slightlymorethanzero.Inreality,theywanttopushalotharder thanthishardenoughtoimpartalotofangularmomentumto thecowfairinashorttime,beforeitgetsmadandhurtsthem. We'rejusttryingtocalculatethebareminimumforcetheycan possiblyuse,whichisthequestionthatsciencecananswer.Settingthetotaltorqueequaltozero, N + W + A =0, andlettingcounterclockwisetorquesbepositive,wehave 0 )]TJ/F102 10.9091 Tf 10.909 0 Td [(mgr W sin W + F A r A sin90 =0 F A = r W r A mg sin W 1 1.5 kg.8m = s 2 sin14 =1100N. The680kggureforthetypicalmassofacowisduetoLillie andBoechler,whoareveterinarians,soIassumeit'sfairlyaccurate.Myestimateof1100Ncomesoutsignicantlylowerthan their1400Ngure,mainlybecausetheirincorrectplacementof thecenterofmassgives W =24 .Idon'tthink1100Nisan impossibleamountofforcetorequireofonebig,strongperson it'sequivalenttoliftingabout110kg,or240pounds,butgiven thatthetippersneedtoimpartalargeangularmomentumfairly quickly,it'sprobablytruethatseveralpeoplewouldberequired. Themainpracticalissuewithcowtippingisthatcowsgenerally sleeplyingdown.Fallingonitssidecanalsoseriouslyinjurea cow. Section5.4Torque:theRateofTransferofAngularMomentum 125

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DiscussionquestionB. DiscussionquestionE. DiscussionQuestions A Thisseriesofdiscussionquestionsdealswithpaststudents'incorrect reasoningaboutthefollowingproblem. Supposeacometisatthepointinitsorbitshowninthegure.The onlyforceonthecometisthesun'sgravitationalforce. Throughoutthequestion,denealltorquesandangularmomenta usingthesunastheaxis. Isthesunproducinganonzerotorqueonthecomet?Explain. Isthecomet'sangularmomentumincreasing,decreasing,or stayingthesame?Explain. Explainwhatiswrongwiththefollowinganswers.Insomecases,theansweriscorrect,butthereasoningleadinguptoitiswrong.aIncorrect answertopart:Yes,becausethesunisexertingaforceonthecomet, andthecometisacertaindistancefromthesun. bIncorrectanswertopart:No,becausethetorquescancelout. cIncorrectanswertopart:Increasing,becausethecometisspeedingup. B Whichclawhammerwouldmakeiteasiertogetthenailoutofthe woodifthesameforcewasappliedinthesamedirection? C Youwhirlarockoveryourheadontheendofastring,andgradually pullinthestring,eventuallycuttingtheradiusinhalf.Whathappensto therock'sangularmomentum?Whatchangesoccurinitsspeed,thetime requiredforonerevolution,anditsacceleration?Whymightthestring break? D Ahelicopterhas,inadditiontothehugefanbladesontop,asmaller propellermountedonthetailthatrotatesinaverticalplane.Why? E Thephotoshowsanamusementparkridewhosetwocarsrotatein oppositedirections.Whyisthisagooddesign? 126 Chapter5ConservationofAngularMomentum

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u / Thewindmillsarenotclosed systems,butangularmomentum isbeingtransferredoutofthem atthesamerateitistransferred in,resultinginconstantangular momentum.Togetanideaof thehugescaleofthemodern windmillfarm,notethesizesof thetrucksandtrailers. 5.5Statics Equilibrium Therearemanycaseswhereasystemisnotclosedbutmaintains constantangularmomentum.Whenamerry-go-roundisrunningat constantangularmomentum,theengine'storqueisbeingcanceled bythetorqueduetofriction. Whenanobjecthasconstantmomentumandconstantangular momentum,wesaythatitisinequilibrium.Thisisascientic redenitionofthecommonEnglishword,sinceinordinaryspeech nobodywoulddescribeacarspinningoutonanicyroadasbeing inequilibrium. Verycommonly,however,weareinterestedincaseswhereanobjectisnotonlyinequilibriumbutalsoatrest,andthiscorresponds morecloselytotheusualmeaningoftheword.Treesandbridges havebeendesignedbyevolutionandengineerstostayatrest,and todosotheymusthavenotjustzerototalforceactingonthembut zerototaltorque.Itisnotenoughthattheydon'tfalldown,they alsomustnottipover.Staticsisthebranchofphysicsconcerned withproblemssuchasthese. Solvingstaticsproblemsisnowsimplyamatterofapplyingand combiningsomethingsyoualreadyknow: Youknowthebehaviorsofthevarioustypesofforces,for examplethatafrictionalforceisalwaysparalleltothesurface ofcontact. Youknowaboutvectoradditionofforces.Itisthevectorsum oftheforcesthatmustequalzerotoproduceequilibrium. Youknowabouttorque.Thetotaltorqueactingonanobject mustbezeroifitistobeinequilibrium. Youknowthatthechoiceofaxisisarbitrary,soyoucanmake achoiceofaxisthatmakestheproblemeasytosolve. Ingeneral,thistypeofproblemcouldinvolvefourequationsinfour unknowns:threeequationsthatsaytheforcecomponentsaddup tozero,andoneequationthatsaysthetotaltorqueiszero.Most casesyou'llencounterwillnotbethiscomplicated.Inthefollowing example,onlytheequationforzerototaltorqueisrequiredinorder togetananswer. Section5.5Statics 127

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v / Example12. Aagpoleexample12 A10-kgagpoleisbeingheldupbyalightweighthorizontal cable,andisproppedagainstthefootofawallasshowninthe gure.Ifthecableisonlycapableofsupportingatensionof70 N,howgreatcantheangle bewithoutbreakingthecable? Allthreeobjectsinthegurearesupposedtobeinequilibrium: thepole,thecable,andthewall.Whicheverofthethreeobjects wepicktoinvestigate,alltheforcesandtorquesonithaveto cancelout.Itisnotparticularlyhelpfultoanalyzetheforcesand torquesonthewall,sinceithasforcesonitfromthegroundthat arenotgivenandthatwedon'twanttond.Wecouldstudythe forcesandtorquesonthecable,butthatdoesn'tletususethe giveninformationaboutthepole.Theobjectweneedtoanalyze isthepole. Thepolehasthreeforcesonit,eachofwhichmayalsoresultin atorque:thegravitationalforce,thecable'sforce,and thewall'sforce. Wearefreetodeneanaxisofrotationatanypointwewish,and itishelpfultodeneittolieatthebottomendofthepole,since bythatdenitionthewall'sforceonthepoleisappliedat r =0 andthusmakesnotorqueonthepole.Thisisgood,becausewe don'tknowwhatthewall'sforceonthepoleis,andwearenot tryingtondit. Withthischoiceofaxis,therearetwononzerotorquesonthe pole,acounterclockwisetorquefromthecableandaclockwise torquefromgravity.Choosingtorepresentcounterclockwisetorques aspositivenumbers,andusingtheequation j j = r j F j sin ,we have r cable j F cable j sin cable )]TJ/F102 10.9091 Tf 10.91 0 Td [(r grav j F grav j sin grav =0. Alittlegeometrygives cable =90 )]TJ/F35 10.9091 Tf 10.909 0 Td [( and grav = ,so r cable j F cable j sin )]TJ/F35 10.9091 Tf 10.909 0 Td [( )]TJ/F102 10.9091 Tf 10.909 0 Td [(r grav j F grav j sin =0. Thegravitationalforcecanbeconsideredasactingatthepole's centerofmass,i.e.,atitsgeometricalcenter,so r cable istwice r grav ,andwecansimplifytheequationtoread 2 j F cable j sin )]TJ/F35 10.9091 Tf 10.909 0 Td [( )-222(j F grav j sin =0. Theseareallquantitiesweweregiven,exceptfor ,whichisthe anglewewanttond.Tosolvefor weneedtousethetrig identitysin )]TJ/F102 10.9091 Tf 10.909 0 Td [(x =cos x 2 j F cable j cos )-222(j F grav j sin =0, 128 Chapter5ConservationofAngularMomentum

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w / Example13. whichallowsustond tan =2 j F cable j j F grav j =tan )]TJ/F39 7.9701 Tf 6.586 0 Td [(1 2 j F cable j j F grav j =tan )]TJ/F39 7.9701 Tf 6.586 0 Td [(1 2 70N 98N =55 Art!example13 Theabstractsculptureshowningurewcontainsacubeof mass m andsidesoflength b .Thecuberestsontopofacylinder, whichisoff-centerbyadistance a .Findthetensioninthecable. Therearefourforcesonthecube:agravitationalforce mg ,the force F T fromthecable,theupwardnormalforcefromthecylinder, F N ,andthehorizontalstaticfrictionalforcefromthecylinder, F s Thetotalforceonthecubeintheverticaldirectioniszero: F N )]TJ/F102 10.9091 Tf 10.909 0 Td [(mg =0. Asouraxisfordeningtorques,it'sconvenienttochoosethepoint ofcontactbetweenthecubeandthecylinder,becausethenneither F s nor F N makesanytorque.Thecable'storqueiscounterclockwise,thetorqueduetogravityisclockwise.Lettingcounterclockwisetorquesbepositive,andusingtheconvenientequation = r ? F ,wendtheequationforthetotaltorque: bF T )]TJ/F102 10.9091 Tf 10.91 0 Td [(mga =0. Wecouldalsowritedowntheequationsayingthatthetotalhorizontalforceiszero,butthatwouldbringinthecylinder'sfrictional forceonthecube,whichwedon'tknowanddon'tneedtond.We alreadyhavetwoequationsinthetwounknowns F T and F N ,so there'snoneedtomakeitintothreeequationsinthreeunknowns. Solvingtherstequationfor F N = mg ,wethensubstituteintothe secondequationtoeliminate F N ,andsolvefor F T = a = b mg Asacheck,ourresultmakessensewhen a =0;thecubeis balancedonthecylinder,sothecablegoesslack. Section5.5Statics 129

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x / Stableandunstableequilibria. y / Thedancer'sequilibrium isunstable.Ifshedidn'tconstantlymaketinyadjustments, shewouldtipover. z / Example14. Stableandunstableequilibria Apencilbalanceduprightonitstipcouldtheoreticallybein equilibrium,butevenifitwasinitiallyperfectlybalanced,itwould toppleinresponsetotherstaircurrentorvibrationfromapassingtruck.Thepencilcanbeputinequilibrium,butnotinstable equilibrium.Thethingsaroundusthatwereallydoseestayingstill areallinstableequilibrium. Whyisoneequilibriumstableandanotherunstable?Trypushingyourownnosetotheleftortheright.Ifyoupushitamillimeter totheleft,yourheadrespondswithagentleforcetotheright,which keepsyournosefromyingoofyourface.Ifyoupushyournosea centimetertotheleft,yourface'sforceonyournosebecomesmuch stronger.Thedeningcharacteristicofastableequilibriumisthat thefarthertheobjectismovedawayfromequilibrium,thestronger theforceisthattriestobringitback. Theoppositeistrueforanunstableequilibrium.Inthetop gure,theballrestingontheroundhilltheoreticallyhaszerototal forceonitwhenitisexactlyatthetop.Butinrealitythetotal forcewillnotbeexactlyzero,andtheballwillbegintomoveoto oneside.Onceithasmoved,thenetforceontheballisgreaterthan itwas,anditacceleratesmorerapidly.Inanunstableequilibrium, thefarthertheobjectgetsfromequilibrium,thestrongertheforce thatpushesitfartherfromequilibrium. Thisideacanberephrasedintermsofenergy.Thedierence betweenthestableandunstableequilibriashowningurexisthat inthestableequilibrium,thepotentialenergyisataminimum,and movingtoeithersideofequilibriumwillincreaseit,whereasthe unstableequilibriumrepresentsamaximum. Notethatweareusingthetermstable"inaweakersensethan inordinaryspeech.Adominostandinguprightisstableinthesense weareusing,sinceitwillnotspontaneouslyfalloverinresponseto asneezefromacrosstheroomorthevibrationfromapassingtruck. Wewouldonlycallitunstableinthetechnicalsenseifitcouldbe toppledby any force,nomatterhowsmall.Ineverydayusage,of course,itwouldbeconsideredunstable,sincetheforcerequiredto toppleitissosmall. Anapplicationofcalculusexample14 NancyNeutronislivinginauraniumnucleusthatisundergoing ssion.Nancy'spotentialenergyasafunctionofpositioncanbe approximatedby PE = x 4 )]TJ/F102 10.9091 Tf 11.348 0 Td [(x 2 ,wherealltheunitsandnumericalconstantshavebeensuppressedforsimplicity.Usecalculus tolocatetheequilibriumpoints,anddeterminewhethertheyare stableorunstable. TheequilibriumpointsoccurwherethePEisataminimumor maximum,andminimaandmaximaoccurwherethederivative 130 Chapter5ConservationofAngularMomentum

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aa / Thebicepsmuscleexesthe arm. ab / Thetricepsextendsthe arm. whichequalsminustheforceonNancyiszero.Thisderivativeisd PE = d x =4 x 3 )]TJ/F39 10.9091 Tf 11.15 0 Td [(2 x ,andsettingitequaltozero,wehave x =0, 1 = p 2.Minimaoccurwherethesecondderivativeispositive,andmaximawhereitisnegative.Thesecondderivative is12 x 2 )]TJ/F39 10.9091 Tf 11.113 0 Td [(2,whichisnegativeat x =0unstableandpositiveat x = 1 = p 2stable.Interpretation:thegraphofthePEisshaped likearoundedletter`W,'withthetwotroughsrepresentingthetwo halvesofthesplittingnucleus.Nancyisgoingtohavetodecide whichhalfshewantstogowith. 5.6SimpleMachines:TheLever Althoughwehavediscussedsomesimplemachinessuchasthepulley,withouttheconceptoftorquewewerenotyetreadytoaddressthelever,whichisthemachinenatureusedindesigningliving things,almosttotheexclusionofallothers.Wecanspeculatewhat lifeonourplanetmighthavebeenlikeiflivingthingshadevolved wheels,gears,pulleys,andscrews.Theguresshowtwoexamples ofleverswithinyourarm.Dierentmusclesareusedtoexand extendthearm,becausemusclesworkonlybycontraction. Analyzingexampleaaphysically,therearetwoforcesthatdo work.Whenweliftaloadwithourbicepsmuscle,themuscledoes positivework,becauseitbringstheboneintheforearminthedirectionitismoving.Theload'sforceonthearmdoesnegativework, becausethearmmovesinthedirectionoppositetotheload'sforce. Thismakessense,becauseweexpectourarmtodopositiveworkon theload,sotheloadmustdoanequalamountofnegativeworkon thearm.Ifthebicepswasloweringaload,thesignsoftheworks wouldbereversed.Anymuscleiscapableofdoingeitherpositive ornegativework. Thereisalsoathirdforceontheforearm:theforceoftheupper arm'sboneexertedontheforearmattheelbowjointnotshown withanarrowinthegure.Thisforcedoesnowork,becausethe elbowjointisnotmoving. Becausetheelbowjointismotionless,itisnaturaltodeneour torquesusingthejointastheaxis.Thesituationnowbecomes quitesimple,becausetheupperarmbone'sforceexertedatthe elbowneitherdoesworknorcreatesatorque.Wecanignoreit completely.Inanyleverthereissuchapoint,calledthefulcrum. Ifwerestrictourselvestothecaseinwhichtheforearmrotates withconstantangularmomentum,thenweknowthatthetotal torqueontheforearmiszero, muscle + load =0. Ifwechoosetorepresentcounterclockwisetorquesaspositive,then themuscle'storqueispositive,andtheload'sisnegative.Interms Section5.6SimpleMachines:TheLever 131

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oftheirabsolutevalues, j muscle j = j load j Assumingforsimplicitythatbothforcesactatanglesof90 with respecttothelinesconnectingtheaxistothepointsatwhichthey act,theabsolutevaluesofthetorquesare r muscle F muscle = r load F arm where r muscle ,thedistancefromtheelbowjointtothebiceps'point ofinsertionontheforearm,isonlyafewcm,while r load mightbe30 cmorso.Theforceexertedbythemusclemustthereforebeabout tentimestheforceexertedbytheload.Wethusseethatthislever isaforcereducer.Ingeneral,alevermaybeusedeithertoincrease ortoreduceaforce. Whydidourarmsevolvesoastoreduceforce?Ingeneral, yourbodyisbuiltforcompactnessandmaximumspeedofmotion ratherthanmaximumforce.Thisisthemainanatomicaldierence betweenusandtheNeanderthalstheirbrainscoveredthesame rangeofsizesasthoseofmodernhumans,anditseemstohave workedforus. Aswithallmachines,theleverisincapableofchangingthe amountofmechanicalworkwecando.Aleverthatincreasesforce willalwaysreducemotion,andviceversa,leavingtheamountof workunchanged. Itisworthnotinghowsimpleandyethowpowerfulthisanalysis was.Itwassimplebecausewewerewellpreparedwiththeconcepts oftorqueandmechanicalwork.Inanatomytextbooks,whosereadersareassumednottoknowphysics,thereisusuallyalongand complicateddiscussionofthedierenttypesoflevers.Forexample, thebicepslever,aa,wouldbeclassiedasaclassIIIlever,sinceit hasthefulcrumandloadontheendsandthemuscle'sforceacting inthemiddle.Thetriceps,ab,iscalledaclassIlever,becausethe loadandmuscle'sforceareontheendsandthefulcrumisinthe middle.Howtiresome!Witharmgraspoftheconceptoftorque, werealizethatallsuchexamplescanbeanalyzedinmuchthesame way.Physicsisatitsbestwhenitletsusunderstandmanyapparentlycomplicatedphenomenaintermsofafewsimpleyetpowerful concepts. 132 Chapter5ConservationofAngularMomentum

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ac / The r )]TJ/F35 9.9626 Tf 9.76 0 Td [( representationofa curve. ad / Proofthatthetwoangleslabeled areinfactequal: Thedenitionofanellipseisthat thesumofthedistancesfrom thetwofocistaysconstant.Ifwe moveasmalldistance ` alongthe ellipse,thenonedistanceshrinks byanamount ` cos 1 ,whilethe othergrowsby ` cos 2 .These areequal,so 1 = 2 5.7 ? ProofofKepler'sEllipticalOrbitLaw Keplerdeterminedpurelyempiricallythattheplanets'orbitswere ellipses,withoutunderstandingtheunderlyingreasonintermsof physicallaw.Newton'sproofofthisfactbasedonhislawsofmotion andlawofgravitywasconsideredhiscrowningachievementboth byhimandbyhiscontemporaries,becauseitshowedthatthesame physicallawscouldbeusedtoanalyzeboththeheavensandthe earth.Newton'sproofwasverylengthy,butbyapplyingthemore recentconceptsofconservationofenergyandangularmomentum wecancarryouttheproofquitesimplyandsuccinctly,andwithout calculus. Thebasicideaoftheproofisthatwewanttodescribetheshape oftheplanet'sorbitwithanequation,andthenshowthatthisequationisexactlytheonethatrepresentsanellipse.Newton'soriginal proofhadtobeverycomplicatedbecauseitwasbaseddirectlyon hislawsofmotion,whichincludetimeasavariable.Tomakeany statementabouttheshapeoftheorbit,hehadtoeliminatetime fromhisequations,leavingonlyspacevariables.Butconservation lawstellusthatcertainthingsdon'tchangeovertime,sotheyhave alreadyhadtimeeliminatedfromthem. Therearemanywaysofrepresentingacurvebyanequation,of whichthemostfamiliaris y = ax + b foralineintwodimensions. Itwouldbeperfectlypossibletodescribeaplanet'sorbitusingan x )]TJ/F20 10.9091 Tf 11.505 0 Td [(y equationlikethis,butrememberthatweareapplyingconservationofangularmomentum,andthespacevariablesthatoccur intheequationforangularmomentumarethedistancefromthe axis, r ,andtheanglebetweenthevelocityvectorandthe r vector, whichwewillcall .Theplanetwillhave =90 whenitismoving perpendiculartothe r vector,i.e.,atthemomentswhenitisatits smallestorgreatestdistancesfromthesun.When islessthan 90 theplanetisapproachingthesun,andwhenitisgreaterthan 90 itisrecedingfromit.Describingacurvewithan r )]TJ/F20 10.9091 Tf 10.251 0 Td [( equation isliketellingadriverinaparkinglotacertainruleforwhatdirectiontosteerbasedonthedistancefromacertainstreetlightinthe middleofthelot. Theproofisbrokenintothethreepartsforeasierdigestion. Therstpartisasimpleandintuitivelyreasonablegeometricalfact aboutellipses,whoseproofwerelegatetothecaptionofguread; youwillnotbemissingmuchifyoumerelyabsorbtheresultwithout readingtheproof. Ifweuseoneofthetwofociofanellipseasanaxisfor deningthevariables r and ,thentheanglebetweenthetangent lineandthelinedrawntotheotherfocusisthesameas ,i.e.,the twoangleslabeled ingureadareinfactequal. Theothertwopartsformthemeatofourproof.Westatethe Section5.7 ? ProofofKepler'sEllipticalOrbitLaw 133

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ae / Proofofpart. resultsrstandthenprovethem. Aplanet,movingundertheinuenceofthesun'sgravity withlessthentheenergyrequiredtoescape,obeysanequationof theform sin = 1 p )]TJ/F20 10.9091 Tf 8.485 0 Td [(pr 2 + qr where p and q arepositiveconstantsthatdependontheplanet's energyandangularmomentum. Acurveisanellipseifandonlyifits r )]TJ/F20 10.9091 Tf 10.249 0 Td [( equationisofthe form sin = 1 p )]TJ/F20 10.9091 Tf 8.485 0 Td [(pr 2 + qr where p and q areconstantsthatdependonthesizeandshapeof theellipseand p isgreaterthanzero. Proofofpart Thecomponentoftheplanet'svelocityvectorthatisperpendiculartothe r vectoris v ? = v sin ,soconservationofangular momentumtellsusthat L = mrv sin isaconstant.Sincethe planet'smassisaconstant,thisisthesameasthecondition rv sin =constant. Conservationofenergygives 1 2 mv 2 )]TJ/F20 10.9091 Tf 12.104 7.38 Td [(GMm r =constant. Wesolvetherstequationfor v andplugintothesecondequation toeliminate v .Straightforwardalgebrathenleadstotheequation claimedabove,withtheconstant p beingpositivebecauseofour assumptionthattheplanet'senergyisinsucienttoescapefrom thesun,i.e.,itstotalenergyisnegative. Proofofpart Wedenethequantities d ,and s asshowninthegure.The lawofcosinesgives d 2 = r 2 + s 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 rs cos Using =180 )]TJ/F15 10.9091 Tf 8.65 0 Td [(2 andthetrigonometricidentitiescos )]TJ/F20 10.9091 Tf 8.65 0 Td [(x = )]TJ/F15 10.9091 Tf 10.303 0 Td [(cos x andcos2 x =1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2sin 2 x ,wecanrewritethisas d 2 = r 2 + s 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 rs )]TJ/F15 10.9091 Tf 5 -8.836 Td [(2sin 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 Straightforwardalgebratransformsthisinto sin = r r + s 2 )]TJ/F20 10.9091 Tf 10.909 0 Td [(d 2 4 rs Since r + s isconstant,thetopofthefractionisconstant,andthe denominatorcanberewrittenas4 rs =4 r constant )]TJ/F20 10.9091 Tf 11.325 0 Td [(r ,whichis equivalenttothedesiredform. 134 Chapter5ConservationofAngularMomentum

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Summary SelectedVocabulary angularmomentum........ ameasureofrotationalmotion;aconserved quantityforaclosedsystem axis........Anarbitrarilychosenpointusedinthedenitionofangularmomentum.Anyobjectwhose directionchangesrelativetotheaxisisconsideredtohaveangularmomentum.Nomatter whataxisischosen,theangularmomentumof aclosedsystemisconserved. torque......therateofchangeofangularmomentum;a numericalmeasureofaforce'sabilitytotwist onanobject equilibrium...astateinwhichanobject'smomentumand angularmomentumareconstant stableequilibriumoneinwhichaforcealwaysactstobringthe objectbacktoacertainpoint unstableequilibrium........ oneinwhichanydeviationoftheobjectfrom itsequilibriumpositionresultsinaforcepushingitevenfartheraway Notation L ..........angularmomentum t ..........torque Tthetimerequiredforarigidlyrotatingbodytocompleteone rotation OtherTerminologyandNotation period.......anameforthevariable T denedabove momentofinertia, I ....... theproportionalityconstantintheequation L =2 I=T Summary Angularmomentumisameasureofrotationalmotionwhichis conservedforaclosedsystem.Thisbookonlydiscussesangular momentumforrotationofmaterialobjectsintwodimensions.Not allrotationisrigidlikethatofawheeloraspinningtop.Anexample ofnonrigidrotationisacyclone,inwhichtheinnerpartstakeless timetocompletearevolutionthantheouterparts.Inordertodene ameasureofrotationalmotiongeneralenoughtoincludenonrigid rotation,wedenetheangularmomentumofasystembydividing itupintosmallparts,andaddingupalltheangularmomentaof thesmallparts,whichwethinkofastinyparticles.Wearbitrarily choosesomepointinspace,the axis ,andwesaythatanything thatchangesitsdirectionrelativetothatpointpossessesangular momentum.Theangularmomentumofasingleparticleis L = mv ? r where v ? isthecomponentofitsvelocityperpendiculartotheline Summary 135

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joiningittotheaxis,and r isitsdistancefromtheaxis.Positiveand negativesignsofangularmomentumareusedtoindicateclockwise andcounterclockwiserotation. The choiceofaxistheorem statesthatanyaxismaybeusedfor deningangularmomentum.Ifasystem'sangularmomentumis constantforonechoiceofaxis,thenitisalsoconstantforanyother choiceofaxis. The spintheorem statesthatanobject'sangularmomentum withrespecttosomeoutsideaxisAcanbefoundbyaddinguptwo parts: Therstpartistheobject'sangularmomentumfoundby usingitsowncenterofmassastheaxis,i.e.,theangularmomentum theobjecthasbecauseitisspinning. TheotherpartequalstheangularmomentumthattheobjectwouldhavewithrespecttotheaxisAifithadallitsmass concentratedatandmovingwithitscenterofmass. Torqueistherateofchangeofangularmomentum.Thetorque aforcecanproduceisameasureofitsabilitytotwistonanobject. Therelationshipbetweenforceandtorqueis j j = r j F ? j where r isthedistancefromtheaxistothepointwheretheforceis applied,and F ? isthecomponentoftheforceperpendiculartothe lineconnectingtheaxistothepointofapplication.Staticsproblems canbesolvedbysettingthetotalforceandtotaltorqueonanobject equaltozeroandsolvingfortheunknowns. 136 Chapter5ConservationofAngularMomentum

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Problem5. Problems Key p Acomputerizedanswercheckisavailableonline. R Aproblemthatrequirescalculus. ? Adicultproblem. 1 YouaretryingtoloosenastuckboltonyourRVusingabig wrenchthatis50cmlong.Ifyouhangfromthewrench,andyour massis55kg,whatisthemaximumtorqueyoucanexertonthe bolt? p 2 Aphysicaltherapistwantsherpatienttorehabilitatehisinjuredelbowbylayinghisarmatonatable,andthenliftinga2.1 kgmassbybendinghiselbow.Inthissituation,theweightis33 cmfromhiselbow.Hecallsherback,complainingthatithurtshim tograsptheweight.Heasksifhecanstrapabiggerweightonto hisarm,only17cmfromhiselbow.Howmuchmassshouldshe tellhimtousesothathewillbeexertingthesametorque?Heis raisinghisforearmitself,aswellastheweight. p 3 Anobjectthrownstraightupintheairismomentarilyatrest whenitreachesthetopofitsmotion.Doesthatmeanthatitisin equilibriumatthatpoint?Explain. 4 Anobjectisobservedtohaveconstantangularmomentum. Canyouconcludethatnotorquesareactingonit?Explain.[Based onaproblembySerwayandFaughn.] 5 Apersonofweight W standsontheballofonefoot.Find thetensioninthecalfmuscleandtheforceexertedbytheshinbones onthebonesofthefoot,intermsof W a ,and b .Forsimplicity, assumethatalltheforcesareat90-degreeanglestothefoot,i.e., neglecttheanglebetweenthefootandtheoor. 6 Twoobjectshavethesamemomentumvector.Assumethat theyarenotspinning;theyonlyhaveangularmomentumdueto theirmotionthroughspace.Canyouconcludethattheirangular momentaarethesame?Explain.[BasedonaproblembySerway andFaughn.] Problems 137

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Problem10. Problems8and9. Problem7. 7 Thesunturnsonitsaxisonceevery26.0days.Itsmassis 2.0 10 30 kganditsradiusis7.0 10 8 m.Assumeitisarigid sphereofuniformdensity. aWhatisthesun'sangularmomentum? p Inafewbillionyears,astrophysicistspredictthatthesunwilluse upallitssourcesofnuclearenergy,andwillcollapseintoaballof exotic,densematterknownasawhitedwarf.Assumethatitsradius becomes5.8 10 6 msimilartothesizeoftheEarth.Assumeit doesnotloseanymassbetweennowandthen.Don'tbefooledby thephoto,whichmakesitlooklikenearlyallofthestarwasthrown obytheexplosion.Thevisuallyprominentgascloudisactually thinnerthanthebestlaboratoryvacuumeveryproducedonearth. Certainlyalittlebitofmassisactuallylost,butitisnotatall unreasonabletomakeanapproximationofzerolossofmassaswe aredoing. bWhatwillitsangularmomentumbe? cHowlongwillittaketoturnonceonitsaxis? p 8 Auniformladderofmass m andlength L leansagainsta smoothwall,makinganangle q withrespecttotheground.Thedirt exertsanormalforceandafrictionalforceontheladder,producing aforcevectorwithmagnitude F 1 atanangle withrespecttothe ground.Sincethewallissmooth,itexertsonlyanormalforceon theladder;letitsmagnitudebe F 2 aExplainwhy mustbegreaterthan .Nomathisneeded. bChooseanynumericalvaluesyoulikefor m and L ,andshow thattheladdercanbeinequilibriumzerotorqueandzerototal forcevectorfor =45.00 and =63.43 9 Continuingthepreviousproblem,ndanequationfor in termsof ,andshowthat m and L donotenterintotheequation. Donotassumeanynumericalvaluesforanyofthevariables.You willneedthetrigidentitysin a )]TJ/F20 10.9091 Tf 11.336 0 Td [(b =sin a cos b )]TJ/F15 10.9091 Tf 11.336 0 Td [(sin b cos a .As anumericalcheckonyourresult,youmaywishtocheckthatthe anglesgiveninpart b ofthepreviousproblemsatisfyyourequation. ? 10 aFindtheminimumhorizontalforcewhich,appliedat theaxle,willpullawheeloverastep.Inventalgebrasymbolsfor whateverquantitiesyoundtoberelevant,andgiveyouranswer insymbolicform.[Hints:Therearefourforcesonthewheelat rst,butonlythreewhenitliftso.Normalforcesarealways perpendiculartothesurfaceofcontact.Notethatthecornerofthe stepcannotbeperfectlysharp,sothesurfaceofcontactforthis forcereallycoincideswiththesurfaceofthewheel.] bUnderwhatcircumstancesdoesyourresultbecomeinnite? Giveaphysicalinterpretation. 138 Chapter5ConservationofAngularMomentum

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Problem16. Problem17. 11 Ayo-yooftotalmass m consistsoftwosolidcylinders ofradius R ,connectedbyasmallspindleofnegligiblemassand radius r .Thetopofthestringisheldmotionlesswhilethestring unrollsfromthespindle.Showthattheaccelerationoftheyo-yo is g= + R 2 = 2 r 2 .[Hint:Theaccelerationandthetensioninthe stringareunknown.Use = L= t and F = ma todetermine thesetwounknowns.] ? 12 Aballisconnectedbyastringtoaverticalpost.Theballis setinhorizontalmotionsothatitstartswindingthestringaround thepost.Assumethatthemotionisconnedtoahorizontalplane, i.e.,ignoregravity.MichelleandAstridaretryingtopredictthe nalvelocityoftheballwhenitreachesthepost.Michellesays thataccordingtoconservationofangularmomentum,theballhas tospeedupasitapproachesthepost.Astridsaysthataccordingto conservationofenergy,theballhastokeepaconstantspeed.Who isright?[Hint:Howisthisdierentfromthecasewhereyouwhirl arockinacircleonastringandgraduallypullinthestring?] 13 Inthe1950's,seriousarticlesbeganappearinginmagazines like Life predictingthatworlddominationwouldbeachievedbythe nationthatcouldputnuclearbombsinorbitingspacestations,from whichtheycouldbedroppedatwill.Infactitcanbequitedicult togetanorbitingobjecttocomedown.Lettheobjecthaveenergy E = KE + PE andangularmomentum L .Assumethattheenergy isnegative,i.e.,theobjectismovingatlessthanescapevelocity. Showthatitcanneverreacharadiuslessthan r min = GMm 2 E )]TJ/F15 10.9091 Tf 8.485 0 Td [(1+ r 1+ 2 EL 2 G 2 M 2 m 3 [Notethatbothfactorsarenegative,givingapositiveresult.] 14 [Problem14hasbeendeleted.] 15 [Problem15hasbeendeleted.] ? 16 Twobarsoflength L areconnectedwithahingeandplaced onafrictionlesscylinderofradius r .aShowthattheangle shown inthegureisrelatedtotheunitlessratio r=L bytheequation r L = cos 2 2tan bDiscussthephysicalbehaviorofthisequationforverylargeand verysmallvaluesof r=L ? 17 Youwishtodeterminethemassofashipinabottlewithout takingitout.Showthatthiscanbedonewiththesetupshownin thegure,withascalesupportingthebottleatoneend,provided thatitispossibletotakereadingswiththeshipslidtotwodierent locations. Problems 139

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Problem20. Problem23. 18 Twoatomswillinteractviaelectricalforcesbetweentheir protonsandelectrons.OnefairlygoodapproximationtothepotentialenergyistheLennard-Jonespotential, PE r = k a r 12 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 a r 6 where r isthecenter-to-centerdistancebetweentheatoms. Showthatathereisanequilibriumpointat r = a ,btheequilibriumisstable,andctheenergyrequiredtobringtheatoms fromtheirequilibriumseparationtoinnityis k .[Hints:Therst twopartscanbedonemoreeasilybysetting a =1,sincethevalue of a onlychangesthedistancescale.Onewaytodopartbisby graphing.] R 19 SupposethatwelivedinauniverseinwhichNewton'slaw ofgravitygaveforcesproportionalto r )]TJ/F18 7.9701 Tf 6.586 0 Td [(7 ratherthan r )]TJ/F18 7.9701 Tf 6.587 0 Td [(2 .Which, ifany,ofKepler'slawswouldstillbetrue?Whichwouldbecompletelyfalse?Whichwouldbedierent,butinawaythatcouldbe calculatedwithstraightforwardalgebra? 20 Thegureshowsscaledrawingofapairofpliersbeing usedtocrackanut,withanappropriatelyreducedcentimetergrid. Warning:donotattemptthisathome;itisbadmanners.Ifthe forcerequiredtocrackthenutis300N,estimatetheforcerequired oftheperson'shand. Solution,p.170 21 Showthatasphereofradius R thatisrollingwithoutslipping hasangularmomentumandmomentumintheratio L=p = = 5 R 22 Supposeabowlingballisinitiallythrownsothatithasno angularmomentumatall,i.e.,itisinitiallyjustslidingdownthe lane.Eventuallykineticfrictionwillgetitspinningfastenoughso thatitisrollingwithoutslipping.Showthatthenalvelocityofthe ballequals5/7ofitsinitialvelocity.[Hint:You'llneedtheresultof problem21.] 23 Therodinthegureissupportedbythengerandthe string. aFindthetension, T ,inthestring,andtheforce, F ,fromthe nger,intermsof m b L ,and g p bCommentonthecases b = L and b = L= 2. cAreanyvaluesof b unphysical? 24 Twohorizontaltreebranchesonthesametreehaveequal diameters,butonebranchistwiceaslongastheother.Givea quantitativecomparisonofthetorqueswherethebranchesjointhe trunk.[ThankstoBongKang.] 25 aAlicesaysCathy'sbodyhaszeromomentum,butBob saysCathy'smomentumisnonzero.Nobodyislyingormakinga mistake.Howisthispossible?Giveaconcreteexample. 140 Chapter5ConservationofAngularMomentum

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Problem27. bAliceandBobagreethatDong'sbodyhasnonzeromomentum, butdisagreeaboutDong'sangularmomentum,whichAlicesaysis zero,andBobsaysisnonzero.Explain. 26 Penguinsareplayfulanimals.TuxthePenguininventsanew gameusinganaturalcirculardepressionintheice.Hewaddlesat topspeedtowardthecrater,aimingototheside,andthenhops intotheairandlandsonhisbellyjustinsideitslip.Hethenbellysurfs,movinginacirclearoundtherim.Theiceisfrictionless,so hisspeedisconstant.IsTux'sangularmomentumzero,ornonzero? Whataboutthetotaltorqueactingonhim?Takethecenterofthe cratertobetheaxis.Explainyouranswers. 27 Makearoughestimateofthemechanicaladvantageofthe levershowninthegure.Inotherwords,foragivenamountof forceappliedonthehandle,howmanytimesgreateristheresulting forceonthecork? 28 Inexample8onpage119,provethatiftherodissuciently thin,itcanbetoppledwithoutscrapingontheoor. Solution,p.170 ? 29 Amasslessrodoflength ` hasweights,eachofmass m ,attachedtoitsends.Therodisinitiallyputinahorizontalposition, andlaidonano-centerfulcrumlocatedatadistance b fromthe rod'scenter.Therodwilltopple.aCalculatethetotalgravitationaltorqueontheroddirectly,byaddingthetwotorques.b Verifythatthisgivesthesameresultaswouldhavebeenobtained bytakingtheentiregravitationalforceasactingatthecenterof mass. 30 Askilledmotorcyclistcanrideuparamp,ythroughthe air,andlandonanotherramp.Whywoulditbeusefulfortherider tospeeduporslowdownthebackwheelwhileintheair? Problems 141

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ChapterA Thermodynamics Thischapterisoptional,andshouldprobablybeomittedfromatwosemestersurveycourse.Itcanbecoveredatanytimeafterchapter 3. InadevelopingcountrylikeChina,arefrigeratoristhemarkof afamilythathasarrivedinthemiddleclass,andacaristheultimatesymbolofwealth.Bothoftheseare heatengines :devicesfor convertingbetweenheatandotherformsofenergy.Unfortunately fortheChinese,neitherisaveryecientdevice.Burningfossilfuels hasmadeChina'sbigcitiesthemostpollutedontheplanet,and thecountry'stotalenergysupplyisn'tsucienttosupportAmericanlevelsofenergyconsumptionbymorethanasmallfraction ofChina'spopulation.Couldwesomehowmanipulateenergyina moreecientway? Conservationofenergyisastatementthatthetotalamountof energyisconstantatalltimes,whichencouragesustobelievethat anyenergytransformationcanbeundone|indeed,thelawsof physicsyou'velearnedsofardon'tevendistinguishthepastfrom thefuture.Ifyougetinacaranddrivearoundtheblock,the neteectistoconsumesomeoftheenergyyoupaidforatthe gasstation,usingittoheattheneighborhood.Therewouldnot seemtobeanyfundamentalphysicalprincipletopreventyoufrom 143

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recapturingallthatheatandusingitagainthenexttimeyouwant togoforadrive.Moremodestly,whydon'tengineersdesignacar enginesothatitrecapturestheheatenergythatwouldotherwise bewastedviatheradiatorandtheexhaust? Hardexperience,however,hasshownthatdesignersofmoreand moreecientenginesrunintoabrickwallatacertainpoint.The generatorsthattheelectriccompanyusestoproduceenergyatan oil-fueledplantareindeedmuchmoreecientthanacarengine,but evenifoneiswillingtoacceptadevicethatisverylarge,expensive, andcomplex,itturnsouttobeimpossibletomakeaperfectlyecientheatengine|notjustimpossiblewithpresent-daytechnology, butimpossibleduetoasetoffundamentalphysicalprinciplesknown asthescienceof thermodynamics .Andthermodynamicsisn'tjusta peskysetofconstraintsonheatengines.Withoutthermodynamics, thereisnowaytoexplainthedirectionoftime'sarrow|whywe canrememberthepastbutnotthefuture,andwhyit'seasierto breakHumptyDumptythantoputhimbacktogetheragain. A.1PressureandTemperature Whenweheatanobject,wespeedupthemind-bogglinglycomplex randommotionofitsmolecules.Onemethodfortamingcomplexity istheconservationlaws,sincetheytellusthatcertainthingsmust remainconstantregardlessofwhatprocessisgoingon.Indeed, thelawofconservationofenergyisalsoknownastherstlawof thermodynamics. Butasalludedtointheintroductiontothischapter,conservationofenergybyitselfisnotpowerfulenoughtoexplaincertain empiricalfactsaboutheat.Asecondwaytosidestepthecomplexityofheatistoignoreheat'satomicnatureandconcentrateon quantitiesliketemperatureandpressurethattellusaboutasystem'spropertiesasawhole.Thisapproachiscalledmacroscopicin contrasttothemicroscopicmethodofattack.PressureandtemperaturewerefairlywellunderstoodintheageofNewtonandGalileo, hundredsofyearsbeforetherewasanyrmevidencethatatoms andmoleculesevenexisted. Unliketheconservedquantitiessuchasmass,energy,momentum,andangularmomentum,neitherpressurenortemperatureis additive.Twocupsofcoeehavetwicetheheatenergyofasingle cup,buttheydonothavetwicethetemperature.Likewise,the painfulpressureonyoureardrumsatthebottomofapoolisnot aectedifyouinsertorremoveapartitionbetweenthetwohalves ofthepool. Pressure Werestrictourselvestoadiscussionofpressureinuidsatrest andinequilibrium.Inphysics,thetermuid"isusedtomean 144 ChapterAThermodynamics

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a / Asimplepressuregauge consistsofacylinderopenatone end,withapistonandaspring inside.Thedepthtowhichthe springisdepressedisameasure ofthepressure.Todeterminethe absolutepressure,theairneeds tobepumpedoutoftheinteriorof thegauge,sothatthereisnoair pressureactingoutwardonthe piston.Inmanypracticalgauges, thebackofthepistonisopento theatmosphere,sothepressure thegaugeregistersequalsthe pressureoftheuidminusthe pressureoftheatmosphere. eitheragasoraliquid.Theimportantfeatureofauidcanbe demonstratedbycomparingwithacubeofjelloonaplate.The jelloisasolid.Ifyoushaketheplatefromsidetoside,thejellowill respondbyshearing,i.e.,byslantingitssides,butitwilltendto springbackintoitsoriginalshape.Asolidcansustainshearforces, butauidcannot.Auiddoesnotresistachangeinshapeunless itinvolvesachangeinvolume. Ifyou'reatthebottomofapool,youcan'trelievethepainin yourearsbyturningyourhead.Thewater'sforceonyoureardrum isalwaysthesame,andisalwaysperpendiculartothesurfacewhere theeardrumcontactsthewater.Ifyourearisontheeastsideof yourhead,thewater'sforceistothewest.Ifyoukeepyourhead inthesamespotwhileturningaroundsoyourearisonthenorth, theforcewillstillbethesameinmagnitude,anditwillchange itsdirectionsothatitisstillperpendiculartotheeardrum:south. Thisshowsthatpressurehasnodirectioninspace,i.e.,itisascalar. Thedirectionoftheforceisdeterminedbytheorientationofthe surfaceonwhichthepressureacts,notbythepressureitself.A uidowingoverasurfacecanalsoexertfrictionalforces,which areparalleltothesurface,butthepresentdiscussionisrestricted touidsatrest. Experimentsalsoshowthatauid'sforceonasurfaceisproportionaltothesurfacearea.Thevastforceofthewaterbehind adam,forexample,inproportiontothedam'sgreatsurfacearea. Thebottomofthedamexperiencesahigherproportionofitsforce. Basedontheseexperimentalresults,itappearsthattheuseful waytodenepressureisasfollows.Thepressureofauidata givenpointisdenedas F ? =A ,where A istheareaofasmallsurface insertedintheuidatthatpoint,and F ? isthecomponentofthe uid'sforceonthesurfacewhichisperpendiculartothesurface. Thisisessentiallyhowapressuregaugeworks.Thereasonthat thesurfacemustbesmallissothattherewillnotbeanysignicant dierentinpressurebetweenonepartofitandanotherpart.The SIunitsofpressureareevidentlyN = m 2 ,andthiscombinationcan beabbreviatedasthepascal,1Pa=1N = m 2 .Thepascalturnsout tobeaninconvenientlysmallunit,socartires,forexample,have recommendedpressuresimprintedontheminunitsofkilopascals. PressureinU.S.unitsexample1 InU.S.units,theunitofforceisthepound,andtheunitofdistance istheinch.Theunitofpressureisthereforepoundspersquare inch,orp.s.i.Notethatthepoundisnotaunitofmass. SectionA.1PressureandTemperature 145

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AtmosphericpressureinU.S.andmetricunitsexample2 AgurethatmanypeopleintheU.S.rememberisthatatmosphericpressureisabout15poundspersquareinch.Whatis thisinmetricunits? 15lb 1in 2 = 68N .0254m 2 =1.0 10 5 N = m 2 =100kPa Onlypressuredierencesarenormallysignicant. Ifyouspendenoughtimeonanairplane,thepaininyourears subsides.Thisisbecauseyourbodyhasgraduallybeenabletoadmitmoreairintothecavitybehindtheeardrum.Oncethepressure insideisequalizedwiththepressureoutside,theinwardandoutwardforcesonyoureardrumscancelout,andthereisnophysical sensationtotellyouthatanythingunusualisgoingon.Forthis reason,itisnormallyonlypressuredierencesthathaveanyphysicalsignicance.Thusdeep-seashareperfectlyhealthyintheir habitatbecausetheirbodieshaveenoughinternalpressuretocancel thepressurefromthewaterinwhichtheylive;iftheyarecaughtin anetandbroughttothesurfacerapidly,theyexplodebecausetheir internalpressureissomuchgreaterthanthelowpressureoutside. Gettingkilledbyapoolpumpexample3 Myhousehasapool,whichImaintainmyself.Apoolalways needstohaveitswatercirculatedthroughalterforseveralhours adayinordertokeepitclean.Thelterisalargebarrelwitha strongclampthatholdsthetopandbottomhalvestogether.My lterhasaprominentwarninglabelthatwarnsmenottotryto opentheclampswhilethepumpison,anditshowsacartoon ofapersonbeingstruckbythetophalfofthepump.Thecrosssectionalareaofthelterbarrelis0.25m 2 .Likemostpressure gauges,theoneonmypoolpumpactuallyreadsthedifferencein pressurebetweenthepressureinsidethepumpandatmospheric pressure.Thegaugereads90kPa.Whatistheforcethatis tryingtopopopenthelter? Ifthegaugetoldustheabsolutepressureofthewaterinside, we'dhavetondtheforceofthewaterpushingoutwardandthe forceoftheairpushinginward,andsubtractinordertondthe totalforce.Sinceairsurroundsusallthetime,wewouldhaveto dosuchasubtractioneverytimewewantedtocalculateanything usefulbasedonthegauge'sreading.Themanufacturersofthe gaugedecidedtosaveusfromallthisworkbymakingitreadthe differenceinpressurebetweeninsideandoutside,soallwehave todoismultiplythegaugereadingbythecross-sectionalareaof 146 ChapterAThermodynamics

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b / Thisdoesn'thappen.If pressurecouldvaryhorizontally inequilibrium,thecubeofwater wouldacceleratehorizontally. Thisisacontradiction,since weassumedtheuidwasin equilibrium. c / Thisdoeshappen.The sumoftheforcesfromthe surroundingpartsoftheuidis upward,cancelingthedownward forceofgravity. d / Thepressureisthesame atallthepointsmarkedwithdots. thelter: F = PA = 10 3 N = m 2 .25m 2 =22000N That'salotofforce! Thewordsuction"andotherrelatedwordscontainahidden misunderstandingrelatedtothispointaboutpressuredierences. Whenyousuckwaterupthroughastraw,thereisnothinginyour mouththatisattractingthewaterupward.Theforcethatliftsthe waterisfromthepressureofthewaterinthecup.Bycreatinga partialvacuuminyourmouth,youdecreasedtheair'sdownward forceonthewatersothatitnolongerexactlycanceledtheupward force. Variationofpressurewithdepth Thepressurewithinauidinequilibriumcanonlydependon depth,duetogravity.Ifthepressurecouldvaryfromsidetoside, thenapieceoftheuidinbetween,b,wouldbesubjecttounequal forcesfromthepartsoftheuidonitstwosides.Butuidsdonot exhibitshearforces,sotherewouldbenootherforcethatcouldkeep thispieceofuidfromaccelerating.Thiscontradictstheassumption thattheuidwasinequilibrium. self-checkA Howdoesthisprooffailforsolids? Answer,p.166 Tondthevariationwithdepth,weconsidertheverticalforces actingonatiny,imaginarycubeoftheuidhavingheight y and areasd A onthetopandbottom.Usingpositivenumbersforupward forces,wehave P bottom A )]TJ/F20 10.9091 Tf 10.909 0 Td [(P top A )]TJ/F20 10.9091 Tf 10.909 0 Td [(F g =0. Theweightoftheuidis F g = mg = Vg = A yg ,where is thedensityoftheuid,sothedierenceinpressureis P = )]TJ/F20 10.9091 Tf 8.485 0 Td [(g y .[variationinpressurewithdepthfor auidofdensity inequilibrium; positive y isup.] Thefactorof explainswhywenoticethedierenceinpressure whendiving3mdowninapool,butnotwhengoingdown3m ofstairs.Notealsothattheequationonlytellsusthedierencein pressure,nottheabsolutepressure.Thepressureatthesurfaceof aswimmingpoolequalstheatmosphericpressure,notzero,even thoughthedepthiszeroatthesurface.Thebloodinyourbody doesnotevenhaveanuppersurface. SectionA.1PressureandTemperature 147

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e / Wehavetowaitforthe thermometertoequilibrateits temperaturewiththetemperature ofIrene'sarmpit. Pressureoflavaunderneathavolcanoexample4 Avolcanohasjustnishederupting,andapoolofmoltenlava islyingatrestinthecrater.Thelavahascomeupthroughan openinginsidethevolcanothatconnectstotheearth'smolten mantle.Thedensityofthelavais4.1g = cm 3 .Whatisthepressure inthelavaunderneaththebaseofthevolcano,3000mbelowthe surfaceofthepool? P = g y =.1g = cm 3 .8m = s 2 m =.1 10 6 g = m 3 .8m = s 2 m =.1 10 3 kg = m 3 .8m = s 2 m =1.2 10 8 N = m 2 =1.2 10 8 Pa Thisisthedifferencebetweenthepressurewewanttondand atmosphericpressureatthesurface.Thelatter,however,istiny comparedtothe P wejustcalculated,sowhatwe'vefoundis essentiallythepressure, P Atmosphericpressureexample5 Thisexampleusescalculus. Gases,unlikeliquids,arequitecompressible,andatagiventemperature,thedensityofagasisapproximatelyproportionalto thepressure.Theproportionalityconstantisdiscussedinsection A.2,butfornowlet'sjustcallit k = kP .Usingthisfact,wecan ndthevariationofatmosphericpressurewithaltitude,assuming constanttemperature: d P = )]TJ/F35 10.9091 Tf 8.484 0 Td [( g d y d P = )]TJ/F102 10.9091 Tf 8.484 0 Td [(kPg d y d P P = )]TJ/F102 10.9091 Tf 8.484 0 Td [(kg d y ln P = )]TJ/F102 10.9091 Tf 8.484 0 Td [(kgy +constant[integratingbothsides] P =constant e )]TJ/F102 7.9701 Tf 6.586 0 Td [(kgy [exponentiatingbothsides] Pressurefallsoffexponentiallywithheight.Thereisnosharp cutofftotheatmosphere,buttheexponentialgetsextremelysmall bythetimeyou'retenorahundredmilesup. Temperature Thermalequilibrium Weusethetermtemperaturecasually,butwhatisitexactly? Roughlyspeaking,temperatureisameasureofhowconcentrated theheatenergyisinanobject.Alarge,massiveobjectwithvery littleheatenergyinithasalowtemperature. 148 ChapterAThermodynamics

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f / Thermalequilibriumcan beprevented.Ottershaveacoat offurthattrapsairbubblesforinsulation.Ifaswimmingotterwas inthermalequilibriumwithcold water,itwouldbedead.Heatis stillconductedfromtheotter's bodytothewater,butmuch moreslowlythanitwouldbeina warm-bloodedanimalthatdidn't havethisspecialadaptation. g / Ahotairballoonisinated. Becauseofthermalexpansion, thehotairislessdensethan thesurroundingcoldair,and thereforeoatsasthecoldair dropsunderneathitandpushesit upoutoftheway. Butphysicsdealswithoperationaldenitions,i.e.,denitionsof howtomeasurethethinginquestion.Howdowemeasuretemperature?Onecommonfeatureofalltemperature-measuringdevices isthattheymustbeleftforawhileincontactwiththethingwhose temperatureisbeingmeasured.Whenyoutakeyourtemperature withafeverthermometer,youwaitforthemercuryinsidetocome uptothesametemperatureasyourbody.Thethermometeractuallytellsyouthetemperatureofitsownworkinguidinthis casethemercury.Ingeneral,theideaoftemperaturedependson theconceptofthermalequilibrium.Whenyoumixcoldeggsfrom therefrigeratorwithourthathasbeenatroomtemperature,they rapidlyreachacompromisetemperature.Whatdeterminesthis compromisetemperatureisconservationofenergy,andtheamount ofenergyrequiredtoheatorcooleachsubstancebyonedegree. Butwithoutevenhavingconstructedatemperaturescale,wecan seethattheimportantpointisthephenomenonofthermalequilibriumitself:twoobjectsleftincontactwillapproachthesame temperature.WealsoassumethatifobjectAisatthesametemperatureasobjectB,andBisatthesametemperatureasC,then AisatthesametemperatureasC.Thisstatementissometimes knownasthezerothlawofthermodynamics,socalledbecauseafter therst,second,andthirdlawshadbeendeveloped,itwasrealized thattherewasanotherlawthatwasevenmorefundamental. Thermalexpansion Thefamiliarmercurythermometeroperatesontheprinciplethat themercury,itsworkinguid,expandswhenheatedandcontracts whencooled.Ingeneral,allsubstancesexpandandcontractwith changesintemperature.Thezerothlawofthermodynamicsguaranteesthatwecanconstructacomparativescaleoftemperatures thatisindependentofwhattypeofthermometerweuse.Ifathermometergivesacertainreadingwhenit'sinthermalequilibrium withobjectA,andalsogivesthesamereadingforobjectB,then AandBmustbethesametemperature,regardlessofthedetailsof howthethermometersworks. Whataboutconstructingatemperaturescaleinwhichevery degreerepresentsanequalstepintemperature?TheCelsiusscale has0asthefreezingpointofwaterand100asitsboilingpoint.The hiddenassumptionbehindallthisisthatsincetwopointsdenea line,anytwothermometersthatagreeattwopointsmustagreeat allotherpoints.Inrealityifwecalibrateamercurythermometer andanalcoholthermometerinthisway,wewillndthatagraph ofonethermometer'sreadingversustheotherisnotaperfectly straight y = x line.Thesubtleinconsistencybecomesadrasticone whenwetrytoextendthetemperaturescalethroughthepoints wheremercuryandalcoholboilorfreeze.Gases,however,aremuch moreconsistentamongthemselvesintheirthermalexpansionthan solidsorliquids,andthenoblegaseslikeheliumandneonaremore SectionA.1PressureandTemperature 149

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h / Asimpliedversionofan idealgasthermometer.The wholeinstrumentisallowedto comeintothermalequilibrium withthesubstancewhosetemperatureistobemeasured,and themouthofthecylinderisleft opentostandardpressure.The volumeofthenoblegasgivesan indicationoftemperature. i / Thevolumeof1kgofneon gasasafunctionoftemperature atstandardpressure.Although neonwouldactuallycondense intoaliquidatsomepoint,extrapolatingthegraphtozerovolume givesthesametemperatureas foranyothergas:absolutezero. consistentwitheachotherthangasesingeneral.Continuingto searchforconsistency,wendthatnoblegasesaremoreconsistent witheachotherwhentheirpressureisverylow. Asanidealization,weimagineagasinwhichtheatomsinteract onlywiththesidesofthecontainer,notwitheachother.Sucha gasisperfectlynonreactiveasthenoblegasesverynearlyare,and nevercondensestoaliquidasthenoblegasesdoonlyatextremely lowtemperatures.Itsatomstakeupanegligiblefractionofthe availablevolume.Anygascanbemadetobehaveverymuchlike thisifthepressureisextremelylow,sothattheatomshardlyever encountereachother.Suchagasiscalledanidealgas,andwedene theCelsiusscaleintermsofthevolumeofthegasinathermometer whoseworkingsubstanceisanidealgasmaintainedataxedvery lowpressure,andwhichiscalibratedat0and100degreesaccording tothemeltingandboilingpointsofwater.TheCelsiusscaleisnot justacomparativescalebutanadditiveoneaswell:everystepin temperatureisequal,anditmakessensetosaythatthedierence intemperaturebetween18and28 Cisthesameasthedierence between48and58. Absolutezeroandthekelvinscale Wendthatifweextrapolateagraphofvolumeversustemperature,thevolumebecomeszeroatnearlythesametemperaturefor allgases:-273 C.Realgaseswillallcondenseintoliquidsatsome temperatureabovethis,butanidealgaswouldachievezerovolumeatthistemperature,knownasabsolutezero.Themostuseful temperaturescaleinscienticworkisonewhosezeroisdenedby absolutezero,ratherthanbysomearbitrarystandardlikethemeltingpointofwater.Theidealtemperaturescaleforscienticwork, calledtheKelvinscale,isthesameastheCelsiusscale,butshifted by273degreestomakeitszerocoincidewithabsolutezero.ScientistsusetheCelsiusscaleonlyforcomparisonsorwhenachange intemperatureisallthatisrequiredforacalculation.Onlyonthe Kelvinscaledoesitmakesensetodiscussratiosoftemperatures, e.g.,tosaythatonetemperatureistwiceashotasanother. Whichtemperaturescaletouseexample6 Youopenanastronomybookandencountertheequation lightemitted=constant T 4 forthelightemittedbyastarasafunctionofitssurfacetemperature.Whattemperaturescaleisimplied? Theequationtellsusthatdoublingthetemperatureresultsin theemissionof16timesasmuchlight.Sucharatioonlymakes senseiftheKelvinscaleisused. 150 ChapterAThermodynamics

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A.2MicroscopicDescriptionofanIdealGas Evidenceforthekinetictheory Whydoesmatterhavethethermalpropertiesitdoes?Thebasic answermustcomefromthefactthatmatterismadeofatoms.How, then,dotheatomsgiverisetothebulkpropertiesweobserve? Gases,whosethermalpropertiesaresosimple,oerthebestchance forustoconstructasimpleconnectionbetweenthemicroscopicand macroscopicworlds. Acrucialobservationisthatalthoughsolidsandliquidsare nearlyincompressible,gasescanbecompressed,aswhenweincreasetheamountofairinacar'stirewhilehardlyincreasingits volumeatall.Thismakesussuspectthattheatomsinasolidare packedshouldertoshoulder,whileagasismostlyvacuum,with largespacesbetweenmolecules.Mostliquidsandsolidshavedensitiesabout1000timesgreaterthanmostgases,soevidentlyeach moleculeinagasisseparatedfromitsnearestneighborsbyaspace somethinglike10timesthesizeofthemoleculesthemselves. Ifgasmoleculeshavenothingbutemptyspacebetweenthem, whydon'tthemoleculesintheroomaroundyoujustfalltothe oor?Theonlypossibleansweristhattheyareinrapidmotion, continuallyreboundingfromthewalls,oorandceiling.Inchapter 2,wehavealreadyseensomeoftheevidenceforthekinetictheory ofheat,whichstatesthatheatisthekineticenergyofrandomly movingmolecules.ThistheorywasproposedbyDanielBernoulli in1738,andmetwithconsiderableopposition,becausetherewas noprecedentforthiskindofperpetualmotion.Norubberball, howeverelastic,reboundsfromawallwithexactlyasmuchenergy asitoriginallyhad,nordoweeverobserveacollisionbetweenballs inwhichnoneofthekineticenergyatallisconvertedtoheatand sound.Theanalogyisafalseone,however.Arubberballconsists ofatoms,andwhenitisheatedinacollision,theheatisaform ofmotionofthoseatoms.Anindividualmolecule,however,cannot possessheat.Likewisesoundisaformofbulkmotionofmolecules, socollidingmoleculesinagascannotconverttheirkineticenergyto sound.Moleculescanindeedinducevibrationssuchassoundwaves whentheystrikethewallsofacontainer,butthevibrationsofthe wallsarejustaslikelytoimpartenergytoagasmoleculeasto takeenergyfromit.Indeed,thiskindofexchangeofenergyisthe mechanismbywhichthetemperaturesofthegasanditscontainer becomeequilibrated. Pressure,volume,andtemperature Agasexertspressureonthewallsofitscontainer,andinthe kinetictheoryweinterpretthisapparentlyconstantpressureasthe averaged-outresultofvastnumbersofcollisionsoccurringevery secondbetweenthegasmoleculesandthewalls.Theempirical SectionA.2MicroscopicDescriptionofanIdealGas 151

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factsaboutgasescanbesummarizedbytherelation PV / nT ,[idealgas] whichreallyonlyholdsexactlyforanidealgas.Here n isthenumber ofmoleculesinthesampleofgas. Volumerelatedtotemperatureexample7 Theproportionalityofvolumetotemperatureatxedpressure wasthebasisforourdenitionoftemperature. Pressurerelatedtotemperatureexample8 Pressureisproportionaltotemperaturewhenvolumeisheldconstant.Anexampleistheincreaseinpressureinacar'stireswhen thecarhasbeendrivenonthefreewayforawhileandthetires andairhavebecomehot. Wenowconnecttheseempiricalfactstothekinetictheoryof aclassicalidealgas.Forsimplicity,weassumethatthegasis monoatomici.e.,eachmoleculehasonlyoneatom,andthatit isconnedtoacubicalboxofvolume V ,with L beingthelength ofeachedgeand A theareaofanywall.Anatomwhosevelocity hasan x component v x willcollideregularlywiththeleft-handwall, travelingadistance2 L paralleltothe x axisbetweencollisionswith thatwall.Thetimebetweencollisionsis t =2 L=v x ,andineach collisionthe x componentoftheatom'smomentumisreversedfrom )]TJ/F20 10.9091 Tf 8.485 0 Td [(mv x to mv x .Thetotalforceonthewallis F = p x ,1 t 1 + p x ,2 t 2 + ::: [monoatomicidealgas], wheretheindices1,2, ::: refertotheindividualatoms.Substituting p x i =2 mv x i and t i =2 L=v x i ,wehave F = mv 2 x ,1 L + mv 2 x ,2 L + ::: [monoatomicidealgas]. Thequantity mv 2 x i istwicethecontributiontothekineticenergy fromthepartoftheatom'scenterofmassmotionthatisparallelto the x axis.Sincewe'reassumingamonoatomicgas,centerofmass motionistheonlytypeofmotionthatgivesrisetokineticenergy. Amorecomplexmoleculecouldrotateandvibrateaswell.Ifthe quantityinsidethesumincludedthe y and z components,itwould betwicethetotalkineticenergyofallthemolecules.Bysymmetry, itmustthereforeequal2/3ofthetotalkineticenergy,so F = 2 KE total 3 L [monoatomicidealgas]. Dividingby A andusing AL = V ,wehave P = 2 KE total 3 V [monoatomicidealgas]. 152 ChapterAThermodynamics

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Thiscanbeconnectedtotheempiricalrelation PV / nT ifwe multiplyby V onbothsidesandrewrite KE total as nKE av ,where KE av istheaveragekineticenergypermolecule: PV = 2 3 nKE av [monoatomicidealgas]. Forthersttimewehaveaninterpretationforthetemperature basedonamicroscopicdescriptionofmatter:inamonoatomicideal gas,thetemperatureisameasureoftheaveragekineticenergyper molecule.Theproportionalitybetweenthetwois KE av = = 2 kT wheretheconstantofproportionality k ,knownasBoltzmann'sconstant,hasanumericalvalueof1.38 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(23 J = K.IntermsofBoltzmann'sconstant,therelationshipamongthebulkquantitiesforan idealgasbecomes PV = nkT ,[idealgas] whichisknownastheidealgaslaw.AlthoughIwon'tproveithere, thisequationappliestoallidealgases,eventhoughthederivation assumedamonoatomicidealgasinacubicalbox.Youmayhave seenitwrittenelsewhereas PV = NRT ,where N = n=N A isthe numberofmolesofatoms, R = kN A ,and N A =6.0 10 23 ,called Avogadro'snumber,isessentiallythenumberofhydrogenatomsin 1gofhydrogen. Pressureinacartireexample9 Afterdrivingonthefreewayforawhile,theairinyourcar's tiresheatsupfrom10 Cto35 C.Howmuchdoesthepressure increase? Thetiresmayexpandalittle,butweassumethiseffectissmall, sothevolumeisnearlyconstant.Fromtheidealgaslaw,the ratioofthepressuresisthesameastheratiooftheabsolute temperatures, P 2 = P 1 = T 2 = T 1 =K = K =1.09, ora9%increase. Earth'ssenescenceexample10 MicrobesweretheonlylifeonEarthupuntiltherelativelyrecentadventofmulticellularlife,andarearguablystillthedominantformoflifeonourplanet.Furthermore,thesunhasbeen graduallyheatingupeversinceitrstformed,andthiscontinuing processwillsoonsooninthesenseofgeologicaltimeeliminatemulticellularlifeagain.Heat-induceddecreasesintheatmosphere'sCO 2 contentwillkilloffallcomplexplantswithinabout SectionA.2MicroscopicDescriptionofanIdealGas 153

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j / Aspacesuitexample11. 500millionyears,andalthoughsomeanimalsmaybeabletolive byeatingalgae,itwillonlybeanotherfewhundredmillionyears atmostuntiltheplanetiscompletelyheat-sterilized. Whyisthesungettingbrighter?Theonlythingthatkeepsastar likeoursunfromcollapsingduetoitsowngravityisthepressure ofitsgases.Thesun'senergycomesfromnuclearreactionsat itscore,andthenetresultofthesereactionsistofusehydrogen atomsintoheliumatoms.Ittakesfourhydrogenstomakeone helium,sothenumberofatomsinthesuniscontinuouslydecreasing.Since PV = nkT ,thiscausesadecreaseinpressure, whichmakesthecorecontract.Asthecorecontracts,collisions betweenhydrogenatomsbecomemorefrequent,andtherateof fusionreactionsincreases. Apiston,arefrigerator,andaspacesuitexample11 Bothsidesoftheequation PV = nkT haveunitsofenergy.Supposethepressureinacylinderofgaspushesapistonout,asin thepowerstrokeofanautomobileengine.Letthecross-sectional areaofthepistonandcylinderbe A ,andletthepistontravela smalldistance x .Thenthegas'sforceonthepiston F = PA doesanamountofmechanicalwork W = F x = PA x = P V where V isthechangeinvolume.Thisenergyhastocome fromsomewhere;itcomesfromcoolingthegas.Inacar,what thismeansisthatwe'reharvestingtheenergyreleasedbyburningthegasoline. Inarefrigerator,weusethesameprocesstocoolthegas,which thencoolsthefood. Inaspacesuit,thequantity P V representstheworktheastronauthastodobecausebendingherlimbschangesthevolume ofthesuit.Thesuitinatesunderpressurelikeaballoon,and doesn'twanttobend.Thismakesitverytiringtoworkforany signicantperiodoftime. 154 ChapterAThermodynamics

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k / Thetemperaturedifferencebetweenthehotandcold partsoftheaircanbeusedto extractmechanicalenergy,for examplewithafanbladethat spinsbecauseoftherisinghotair currents. l / Ifthetemperatureofthe airisrstallowedtobecome uniform,thennomechanical energycanbeextracted.The sameamountofheatenergy ispresent,butitisnolonger accessiblefordoingmechanical work. A.3Entropy Efciencyandgradesofenergy Someformsofenergyaremoreconvenientthanothersincertain situations.Youcan'trunaspring-poweredmechanicalclockona battery,andyoucan'trunabattery-poweredclockwithmechanical energy.However,thereisnofundamentalphysicalprinciplethat preventsyoufromconverting100%oftheelectricalenergyina batteryintomechanicalenergyorvice-versa.Moreecientmotors andgeneratorsarebeingdesignedeveryyear.Ingeneral,thelaws ofphysicspermitperfectlyecientconversionwithinabroadclass offormsofenergy. Heatisdierent.Frictiontendstoconvertotherformsofenergy intoheateveninthebestlubricatedmachines.Whenweslidea bookonatable,frictionbringsittoastopandconvertsallitskinetic energyintoheat,butweneverobservetheoppositeprocess,inwhich abookspontaneouslyconvertsheatenergyintomechanicalenergy andstartsmoving!Roughlyspeaking,heatisdierentbecauseitis disorganized.Scramblinganeggiseasy.Unscramblingitisharder. Wesummarizetheseobservationsbysayingthatheatisalower gradeofenergythanotherformssuchasmechanicalenergy. Ofcourseitispossibletoconvertheatintootherformsofenergy suchasmechanicalenergy,andthatiswhatacarenginedoeswith theheatcreatedbyexplodingtheair-gasolinemixture.Butacar engineisatremendouslyinecientdevice,andagreatdealofthe heatissimplywastedthroughtheradiatorandtheexhaust.Engineershaveneversucceededincreatingaperfectlyecientdevicefor convertingheatenergyintomechanicalenergy,andwenowknow thatthisisbecauseofadeeperphysicalprinciplethatisfarmore basicthanthedesignofanengine. Heatengines Heatmaybemoreusefulinsomeformsthaninother,i.e.,there aredierentgradesofheatenergy.Ingurek,thedierencein temperaturecanbeusedtoextractmechanicalworkwithafan blade.Thisprincipleisusedinpowerplants,wheresteamisheated byburningoilorbynuclearreactions,andthenallowedtoexpand throughaturbinewhichhascoolersteamontheotherside.On asmallerscale,thereisaChristmastoythatconsistsofasmall propellerspunbythehotairrisingfromasetofcandles,verymuch likethesetupshowninthegure. Ingurel,however,nomechanicalworkcanbeextractedbecausethereisnodierenceintemperature.Althoughtheairinl hasthesametotalamountofenergyastheairink,theheatinl isalowergradeofenergy,sincenoneofitisaccessiblefordoing mechanicalwork. SectionA.3Entropy 155

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m / Thebeginningoftherst expansionstroke,inwhichthe workinggasiskeptinthermal equilibriumwiththehotreservoir. n / Thebeginningofthesecondexpansionstroke,inwhich theworkinggasisthermally insulated.Theworkinggascools becauseitisdoingworkonthe pistonandthuslosingenergy. o / Thebeginningoftherst compressionstroke.Theworking gasbeginsthestrokeatthesame temperatureasthecoldreservoir, andremainsinthermalcontact withitthewholetime.Theengine doesnegativework. p / Thebeginningofthesecondcompressionstroke,inwhich mechanicalworkisabsorbed, heatingtheworkinggasbackup to T H Ingeneral,wedeneaheatengineasanydevicethattakesheat fromareservoirofhotmatter,extractssomeoftheheatenergytodo mechanicalwork,andexpelsalesseramountofheatintoareservoir ofcoldmatter.Theeciencyofaheatengineequalstheamountof usefulworkextracted, W ,dividedbytheamountofenergywehad topayforinordertoheatthehotreservoir.Thislatteramount ofheatisthesameastheamountofheattheengineextractsfrom thehigh-temperaturereservoir, Q H .Theletter Q isthestandard notationforatransferofheat.Byconservationofenergy,wehave Q H = W + Q L ,where Q L istheamountofheatexpelledintothe low-temperaturereservoir,sotheeciencyofaheatengine, W=Q H canberewrittenas eciency=1 )]TJ/F20 10.9091 Tf 13.01 7.38 Td [(Q L Q H .[eciencyofanyheatengine] Itturnsoutthatthereisaparticulartypeofheatengine,the Carnotengine,which,althoughnot100%ecient,ismoreecient thananyother.Thegradeofheatenergyinasystemcanthusbe unambiguouslydenedintermsoftheamountofheatenergyinit that cannot beextracted,evenbyaCarnotengine. Howcanwebuildthemostecientpossibleengine?Let'sstart withanunnecessarilyinecientenginelikeacarengineandsee howitcouldbeimproved.Theradiatorandexhaustexpelhot gases,whichisawasteofheatenergy.Thesegasesarecoolerthan theexplodedair-gasmixtureinsidethecylinder,buthotterthan theairthatsurroundsthecar.Wecouldthusimprovetheengine's eciencybyaddinganauxiliaryheatenginetoit,whichwould operatewiththerstengine'sexhaustasitshotreservoirandthe airasitscoldreservoir.Ingeneral,anyheatenginethatexpels heatatanintermediatetemperaturecanbemademoreecientby changingitsothatitexpelsheatonlyatthetemperatureofthe coldreservoir. Similarly,anyheatenginethatabsorbssomeenergyatanintermediatetemperaturecanbemademoreecientbyaddingan auxiliaryheatenginetoitwhichwilloperatebetweenthehotreservoirandthisintermediatetemperature. Basedonthesearguments,wedeneaCarnotengineasaheat enginethatabsorbsheatonlyfromthehotreservoirandexpelsit onlyintothecoldreservoir.Figuresm-pshowarealizationofa Carnotengineusingapistoninacylinderlledwithamonoatomic idealgas.Thisgas,knownastheworkinguid,isseparatefrom, butexchangesenergywith,thehotandcoldreservoirs.Itturnsout thatthisparticularCarnotenginehasaneciencygivenby eciency=1 )]TJ/F20 10.9091 Tf 13.01 7.38 Td [(T L T H ,[eciencyofaCarnotengine] where T L isthetemperatureofthecoldreservoirand T H isthe 156 ChapterAThermodynamics

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q / Entropycanbeunderstood usingthemetaphorofawater wheel.Lettingthewaterlevels equalizeislikelettingtheentropy maximize.Takingwaterfromthe highsideandputtingitintothe lowsideincreasestheentropy. Waterlevelsinthismetaphor correspondtotemperaturesin theactualdenitionofentropy. temperatureofthehotreservoir.Aproofofthisfactisgivenin mybook SimpleNature ,whichyoucandownloadforfree. Evenifyoudonotwishtodigintothedetailsoftheproof, thebasicreasonforthetemperaturedependenceisnotsohardto understand.Usefulmechanicalworkisdoneonstrokesmandn, inwhichthegasexpands.Themotionofthepistonisinthesame directionasthegas'sforceonthepiston,sopositiveworkisdone onthepiston.Instrokesoandp,however,thegasdoesnegative workonthepiston.Wewouldliketoavoidthisnegativework, butwemustdesigntheenginetoperformacompletecycle.Luckily thepressuresduringthecompressionstrokesarelowerthantheones duringtheexpansionstrokes,sotheenginedoesn'tundoallitswork witheverycycle.Theratiosofthepressuresareinproportionto theratiosofthetemperatures,soif T L is20%of T H ,theengineis 80%ecient. WehavealreadyprovedthatanyenginethatisnotaCarnot engineislessthanoptimallyecient,anditisalsotruethatall Carnotenginesoperatingbetweenagivenpairoftemperatures T H and T L havethesameeciency.ThusaCarnotengineisthemost ecientpossibleheatengine. Entropy Wewouldliketohavesomenumericalwayofmeasuringthe gradeofenergyinasystem.Wewantthisquantity,calledentropy, tohavethefollowingtwoproperties: Entropyisadditive.Whenwecombinetwosystemsand considerthemasone,theentropyofthecombinedsystemequals thesumoftheentropiesofthetwooriginalsystems.Quantities likemassandenergyalsohavethisproperty. TheentropyofasystemisnotchangedbyoperatingaCarnot enginewithinit. Itturnsouttobesimplerandmoreusefultodenechanges inentropythanabsoluteentropies.Supposeasanexamplethata systemcontainssomehotmatterandsomecoldmatter.Ithasa relativelyhighgradeofenergybecauseaheatenginecouldbeused toextractmechanicalworkfromit.Butifweallowthehotand coldpartstoequilibrateatsomelukewarmtemperature,thegrade ofenergyhasgottenworse.Thusputtingheatintoahotterarea ismoreusefulthanputtingitintoacoldarea.Motivatedbythese considerations,wedeneachangeinentropyasfollows: S = Q T [changeinentropywhenadding heat Q tomatterattemperature T ; S isnegativeifheatistakenout] Asystemwithahighergradeofenergyhasalowerentropy. SectionA.3Entropy 157

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Entropyisadditive.example12 Sincechangesinentropyaredenedbyanadditivequantityheat dividedbyanon-additiveonetemperature,entropyisadditive. Entropyisn'tchangedbyaCarnotengine.example13 Theefciencyofaheatengineisdenedby efciency=1 )]TJ/F102 10.9091 Tf 10.909 0 Td [(Q L = Q H andtheefciencyofaCarnotengineis efciency=1 )]TJ/F102 10.9091 Tf 10.909 0 Td [(T L = T H soforaCarnotenginewehave Q L = Q H = T L = T H ,whichcanbe rewrittenas Q L = T L = Q H = T H .Theentropylostbythehotreservoir isthereforethesameastheentropygainedbythecoldone. Entropyincreasesinheatconduction.example14 Whenahotobjectgivesupenergytoacoldone,conservation ofenergytellsusthattheamountofheatlostbythehotobject isthesameastheamountofheatgainedbythecoldone.The changeinentropyis )]TJ/F102 10.9091 Tf 8.485 0 Td [(Q = T H + Q = T L ,whichispositivebecause T L < T H Entropyisincreasedbyanon-Carnotengine.example15 Theefciencyofanon-Carnotengineislessthan1T L = T H so Q L = Q H > T L = T H and Q L = T L > Q H = T H .Thismeansthatthe entropyincreaseinthecoldreservoirisgreaterthantheentropy decreaseinthehotreservoir. Abookslidingtoastopexample16 Abookslidesacrossatableandcomestoastop.Onceitstops, allitskineticenergyhasbeentransformedintoheat.Asthebook andtableheatup,theirentropiesbothincrease,sothetotalentropyincreasesaswell. Examples14-16involvedclosedsystems,andinallofthemthe totalentropyeitherincreasedorstayedthesame.Itneverdecreased. Herearetwoexamplesofschemesfordecreasingtheentropyofa closedsystem,withexplanationsofwhytheydon'twork. Usingarefrigeratortodecreaseentropy?example17 Arefrigeratortakesheatfromacoldareaanddumpsitintoa hotarea.Doesthisleadtoanetdecreaseintheentropyof aclosedsystem?CouldyoumakeaCarnotenginemoreefcientbyrunningarefrigeratortocoolitslow-temperaturereservoirandejectheatintoitshigh-temperaturereservoir? No.Theheatthatcomesoffoftheradiatorcoilsonthe backofyourkitchenfridgeisagreatdealmorethantheheatthe fridgeremovesfrominside;thedifferenceiswhatitcoststorun yourfridge.Theheatradiatedfromthecoilsissomuchmore thantheheatremovedfromtheinsidethattheincreaseinthe 158 ChapterAThermodynamics

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entropyoftheairintheroomisgreaterthanthedecreaseofthe entropyinsidethefridge.Themostefcientrefrigeratorisactually aCarnotenginerunninginreverse,whichleadstoneitheran increasenoradecreaseinentropy. No.ThemostefcientrefrigeratorisareversedCarnotengine.YouwillnotachieveanythingbyrunningoneCarnotengine inreverseandanotherforward.Theywilljustcanceleachother out. Maxwell'sdaemonexample18 PhysicistJamesClerkMaxwellimaginedpairofneighboring rooms,theirairbeinginitiallyinthermalequilibrium,havingapartitionacrossthemiddlewithatinydoor.Aminisculedaemonis postedatthedoorwithalittleping-pongpaddle,andhisdutyisto trytobuildupfaster-movingairmoleculesinroomBandslower onesinroomA.Forinstance,whenafastmoleculeisheaded throughthedoor,goingfromAtoB,heletsitby,butwhena slowerthanaveragemoleculetriesthesamething,hehitsitback intoroomA.Wouldthisdecreasethetotalentropyofthepairof rooms? No.Thedaemonneedstoeat,andwecanthinkofhisbody asalittleheatengine.Hismetabolismislessefcientthana Carnotengine,soheendsupincreasingtheentropyratherthan decreasingit. Observationsuchastheseleadtothefollowinghypothesis,known asthesecondlawofthermodynamics: Theentropyofaclosedsystemalwaysincreases,oratbeststays thesame: S 0. Atpresentmyargumentstosupportthisstatementmayseem lessthanconvincing,sincetheyhavesomuchtodowithobscure factsaboutheatengines.AmoresatisfyingandfundamentalexplanationforthecontinualincreaseinentropywasachievedbyLudwig Boltzmann,andyoumaywishtolearnmoreaboutBoltzmann's ideasfrommybook SimpleNature ,whichyoucandownloadfor free.Briey,Boltzmannrealizedthatentropywasameasureofrandomnessattheatomiclevel,andrandomnessdoesn'tspontaneously changeintoorganization. Toemphasizethefundamentalanduniversalnatureofthesecondlaw,hereareafewexamples. Entropyandevolutionexample19 Afavoriteargumentofmanycreationistswhodon'tbelieveinevolutionisthatevolutionwouldviolatethesecondlawofthermodynamics:thedeathanddecayofalivingthingreleasesheatas whenacompostheapgetshotandlessenstheamountofenergyavailablefordoingusefulwork,whilethereverseprocess, SectionA.3Entropy 159

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theemergenceoflifefromnonlivingmatter,wouldrequireadecreaseinentropy.Theirargumentisfaulty,sincethesecondlaw onlyappliestoclosedsystems,andtheearthisnotaclosedsystem.Theearthiscontinuouslyreceivingenergyfromthesun. Theheatdeathoftheuniverseexample20 Victorianphilosophersrealizedthatlivingthingshadlowentropy, asdiscussedinexample19,andspentalotoftimeworrying abouttheheatdeathoftheuniverse:eventuallytheuniverse wouldhavetobecomeahigh-entropy,lukewarmsoup,withno lifeororganizedmotionofanykind.Fortunately?,wenow knowagreatmanyotherthingsthatwillmaketheuniverseinhospitabletolifelongbeforeitsentropyismaximized.Lifeon earth,forinstance,willendwhenthesunevolvesintoagiantstar andvaporizesourplanet. Hawkingradiationexample21 Anyprocessthatcoulddestroyheatorconvertitintonothingbutmechanicalworkwouldleadtoareductioninentropy. Blackholesaresupermassivestarswhosegravityissostrong thatnothing,notevenlight,canescapefromthemonceitgets withinaboundaryknownastheeventhorizon.Blackholesare commonlyobservedtosuckhotgasintothem.Doesthisleadto areductionintheentropyoftheuniverse?Ofcourseonecould arguethattheentropyisstillthereinsidetheblackhole,butbeing abletohideentropythereamountstothesamethingasbeing abletodestroyentropy. ThephysicistStevenHawkingwasbotheredbythisquestion,and nallyrealizedthatalthoughtheactualstuffthatentersablack holeislostforever,theblackholewillgraduallyloseenergyinthe formoflightemittedfromjustoutsidetheeventhorizon.Thislight endsupreintroducingtheoriginalentropybackintotheuniverse atlarge. 160 ChapterAThermodynamics

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Problems Key p Acomputerizedanswercheckisavailableonline. R Aproblemthatrequirescalculus. ? Adicultproblem. 1 aShowthatunderconditionsofstandardpressureandtemperature,thevolumeofasampleofanidealgasdependsonlyon thenumberofmoleculesinit. bOnemoleisdenedas6.0 10 23 atoms.Findthevolumeofone moleofanidealgas,inunitsofliters,atstandardtemperatureand pressure Cand101kPa. p 2 Agasinacylinderexpandsitsvolumebyanamount V pushingoutapiston.Showthattheworkdonebythegasonthe pistonisgivenby W = P V 3 aAheliumatomcontains2protons,2electrons,and2 neutrons.Findthemassofaheliumatom. p bFindthenumberofatomsin1kgofhelium. p cHeliumgasismonoatomic.Findtheamountofheatneeded toraisethetemperatureof1kgofheliumby1degreeC.Thisis knownashelium'sheatcapacityatconstantvolume. p 4 Refrigerators,airconditioners,andheatpumpsareheat enginesthatworkinreverse.Youputinmechanicalwork,and ittheeectistotakeheatoutofacoolerreservoiranddeposit heatinawarmerone: Q L + W = Q H .Aswiththeheatengines discussedpreviously,theeciencyisdenedastheenergytransfer youwant Q L forarefrigeratororairconditioner, Q H foraheat pumpdividedbytheenergytransferyoupayfor W Ecienciesaresupposedtobeunitless,buttheeciencyofanair conditionerisnormallygivenintermsofanEERratingoramore complexversioncalledanSEER.TheEERisdenedas Q L =W ,but expressedinthebarbaricunitsofofBtu/watt-hour.AtypicalEER ratingforaresidentialairconditionerisabout10Btu/watt-hour, correspondingtoaneciencyofabout3.Thestandardtemperaturesusedfortestinganairconditioner'seciencyare80 F C insideand95 F Coutside. aWhatwouldbetheEERratingofareversedCarnotengineused asanairconditioner? p bIfyourana3-kWresidentialairconditioner,withaneciency of3,foronehour,whatwouldbetheeectonthetotalentropy oftheuniverse?Isyouranswerconsistentwiththesecondlawof thermodynamics? p 5 aEstimatethepressureatthecenteroftheEarth,assuming itisofconstantdensitythroughout.Usethetechniqueofexample 5onpage148.Notethat g isnotconstantwithrespecttodepth Problems 161

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|itequals Gmr=b 3 for r ,thedistancefromthecenter,lessthan b theearth'sradius. 1 Stateyourresultintermsof G m ,and b bShowthatyouranswerfrompartahastherightunitsforpressure. cEvaluatetheresultnumerically. p dGiventhattheearth'satmosphereisontheorderofonethousandththethicknessoftheearth'sradius,andthatthedensityof theearthisseveralthousandtimesgreaterthanthedensityofthe loweratmosphere,checkthatyourresultisofareasonableorderof magnitude. R 6 aDeterminetheratiobetweentheescapevelocitiesfromthe surfacesoftheearthandthemoon. p bThetemperatureduringthelunardaytimegetsuptoabout 130 C.Intheextremelythinalmostnonexistentlunaratmosphere, estimatehowthetypicalvelocityofamoleculewouldcomparewith thatofthesametypeofmoleculeintheearth'satmosphere.Assumethattheearth'satmospherehasatemperatureof0 C. p cSupposeyouweretogotothemoonandreleasesomeuorocarbongas,withmolecularformulaC n F 2 n +2 .Estimatewhatis thesmallestuorocarbonmoleculelowest n whosetypicalvelocity wouldbelowerthanthatofanN 2 moleculeonearthinproportion tothemoon'slowerescapevelocity.Themoonwouldbeableto retainanatmospheremadeofthesemolecules. p 7 Mostoftheatomsintheuniverseareintheformofgasthat isnotpartofanystarorgalaxy:theintergalacticmediumIGM. TheIGMconsistsofabout10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(5 atomspercubiccentimeter,with atypicaltemperatureofabout10 3 K.Theseare,insomesense,the densityandtemperatureoftheuniversenotcountinglight,orthe exoticparticlesknownasdarkmatter".Calculatethepressureof theuniverseor,speakingmorecarefully,thetypicalpressuredue totheIGM. p 8 Asampleofgasisenclosedinasealedchamber.Thegas consistsofmolecules,whicharethensplitinhalfthroughsome processsuchasexposuretoultravioletlight,orpassinganelectric sparkthroughthegas.Thegasreturnstothermalequilibriumwith thesurroundingroom.Howdoesitspressurenowcomparewithits pressurebeforethemoleculesweresplit? 9 ThegureshowsademonstrationperformedbyOttovon GuerickeforEmperorFerdinandIII,inwhichtwoteamsofhorses failedtopullapartapairofhemispheresfromwhichtheairhad beenevacuated.aWhatobjectmakestheforcethatholdsthe 1 Derivation:Theshelltheoremtellsusthatthegravitationaleldat r isthe sameasifallthemassexistingatgreaterdepthswasconcentratedattheearth's center.Sincevolumescaleslikethethirdpowerofdistance,thisconstitutesa fraction r=b 3 oftheearth'smass,sotheeldis Gm=r 2 r=b 3 = Gmr=b 3 162 ChapterAThermodynamics

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Problem9. hemispherestogether?bThehemispheresareinamuseumin Berlin,andhaveadiameterof65cm.Whatistheamountofforce holdingthemtogether?Hint:Theanswerwouldbethesameif theywerecylindersorpieplatesratherthenhemispheres. 10 Evenwhenresting,thehumanbodyneedstodoacertain amountofmechanicalworktokeeptheheartbeating.Thisquantity isdiculttodeneandmeasurewithhighprecision,andalsodependsontheindividualandherlevelofactivity,butit'sestimated tobeabout1to5watts.Supposeweconsiderthehumanbody asnothingmorethanapump.Apersonwhoisjustlyinginbed alldayneedsabout1000kcal/dayworthoffoodtostayalive.a Estimatetheperson'sthermodynamiceciencyasapump,andb comparewiththemaximumpossibleeciencyimposedbythelaws ofthermodynamicsforaheatengineoperatingacrossthedierence betweenabodytemperatureof37 Candanambienttemperature of22 C.cInterpretyouranswer. Answer,p.167 Problems 163

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Appendix1:Exercises Exercise5A:Torque Equipment: rulerswithholesinthem springscalestwopergroup Whileonepersonholdsthepencilwhichformstheaxlefortheruler,theothermembersofthe grouppullonthescaleandtakereadings.Ineachcase,calculatethetotaltorqueontheruler, andndoutwhetheritequalszerotoroughlywithintheaccuracyoftheexperiment.Finish thecalculationsforeachpartbeforemovingontothenextone.

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Appendix2:PhotoCredits Exceptasspecicallynotedbeloworinaparentheticalcreditinthecaptionofagure,alltheillustrationsin thisbookareundermyowncopyright,andarecopyleftlicensedunderthesamelicenseastherestofthebook. Insomecasesit'sclearfromthedatethatthegureispublicdomain,butIdon'tknowthenameofthe artistorphotographer;Iwouldbegratefultoanyonewhocouldhelpmetogivepropercredit.Ihaveassumed thatimagesthatcomefromU.S.governmentwebpagesarecopyright-free,sinceproductsoffederalagenciesfall intothepublicdomain.I'veincludedsomepublic-domainpaintings;photographicreproductionsofthemarenot copyrightableintheU.S.BridgemanArtLibrary,Ltd.v.CorelCorp.,36F.Supp.2d191,S.D.N.Y.1999. WhenPSSCPhysics"isgivenasacredit,itindicatesthatthegureisfromthersteditionofthetextbook entitledPhysics,bythePhysicalScienceStudyCommittee.Theearlyeditionsofthesebooksneverhadtheir copyrightsrenewed,andarenowthereforeinthepublicdomain.Thereisalsoablanketpermissiongivenin thelaterPSSCCollegePhysicsedition,whichstatesonthecopyrightpagethatThematerialstakenfromthe originalandsecondeditionsandtheAdvancedTopicsofPSSCPHYSICSincludedinthistextwillbeavailable toallpublishersforuseinEnglishafterDecember31,1970,andintranslationsafterDecember31,1975." CreditstoMillikanandGalerefertothetextbooksPracticalPhysicsandElementsofPhysics. Botharepublicdomain.The1927versiondidnothaveitscopyrightrenewed.Sinceitispossiblethatsomeof theillustrationsinthe1927versionhadtheircopyrightsrenewedandarestillundercopyright,Ihaveonlyused themwhenitwasclearthattheywereoriginallytakenfrompublicdomainsources. Inafewcases,Ihavemadeuseofimagesunderthefairusedoctrine.However,Iamnotalawyer,andthe lawsonfairusearevague,soyoushouldnotassumethatit'slegalforyoutousetheseimages.Inparticular, fairuselawmaygiveyoulessleewaythanitgivesme,becauseI'musingtheimagesforeducationalpurposes, andgivingthebookawayforfree.Likewise,ifthephotocreditsayscourtesyof...,"thatmeansthecopyright ownergavemepermissiontouseit,butthatdoesn'tmeanyouhavepermissiontouseit. Cover Sun: ImagefromtheSOHOspaceprobe,NASA/EuropeanSpaceAgency,notcopyrighted. Cover HurricaneAndrew: NationalOceanicandAtmosphericAdministration,1992,notcopyrighted. 13 Jupiter: ImagesfromtheHubbleSpaceTelescope,NASA,notcopyrighted. 18 HooverDam: U.S.Departmentofthe Interior,BureauofReclamation,LowerColoradoRegion,notcopyrighted. 33 Hydraulicram: Millikanand Gale,1920. 35 Bonre,grapes: GFDLlicensed,byWikipediauserFir0002. 38 Skaterinpool: Courtesyof J.D.Rogge,www.sonic.net/ shawn. 43 Plutoniumpellet: U.S.DepartmentofEnergy,publicdomain.. 46 Skateboarderontopofpipe: OulaLehtinen,WikimediaCommons,GFDL1.2. 90 Jupiter: Uncopyrightedimage fromtheVoyagerprobe.Lineartbytheauthor. 51 Basebalpitch: WikipediauserRickDikeman,GFDL1.2. 56 BreakingTrail: ArtbyWalterE.Bohl.ImagecourtesyoftheUniversityofMichiganMuseumofArt/Schoolof InformationandLibraryStudies. 79 DeepSpace1engine: NASA. 80 NucleusofHalley'scomet: NASA,not copyrighted. 87 Chadwick'sapparatus: Redrawnfromthepublic-domaingureinChadwick'soriginalpaper. 88 Wrench: PSSCPhysics. 107 Tornado: NOAAPhotoLibrary,NOAACentralLibrary;OAR/ERL/National SevereStormsLaboratoryNSSL;public-domainproductoftheU.S.government. 108 Longjump: Thomas Eakins,publicdomain. 117 Diver: PSSCPhysics. 115 Pendulum: PSSCPhysics. 124 Cow: Drawn bytheauthor,fromaGFDL-licensedphotooncommons.wikimedia.orgbyuserB.navez.. 127 Old-fashioned windmill: Photobytheauthor. 127 Modernwindmillfarm,Tehachapi,CA: U.S.DepartmentofEnergy,not copyrighted. 130 Ballerina: RickDikeman,1981,GFDL1.2license,www.gnu.org/copyleft/fdl.html,fromthe WikipediaarticleonballetretouchedbyB.Crowell. 138 Whitedwarf: ImageofNGC2440fromtheHubble SpaceTelescope,H.BondandR.Ciardullo. 149 Otters: DmitryAzovtsev,CreativeCommonsAttribution License,wikipedia.org. 149 Hotairballoon: RandyOostdyk,GFDLlicensed. 154 Spacesuit: JawedKarim, GFDL1.2license. 163 Magdeburgspheres: MillikanandGale,ElementsofPhysics,1927,reproducedfromthe coverofMagdeburg'sbook.

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Appendix3:HintsandSolutions AnswerstoSelf-Checks AnswerstoSelf-ChecksforChapter1 Page22,self-checkA: Aspring-loadedtoyguncancauseabullettomove,sothespring iscapableofstoringenergyandthenconvertingitintokineticenergy.Theamountofenergy storedinthespringrelatestotheamountofcompression,whichcanbemeasuredwitharuler. AnswerstoSelf-ChecksforChapter2 Page42,self-checkA: Bothballsstartfromthesameheightandendatthesameheight,so theyhavethesame y .Thisimpliesthattheirlossesinpotentialenergyarethesame,sothey mustbothhavegainedthesameamountofkineticenergy. AnswerstoSelf-ChecksforChapter3 Page50,self-checkA: Workisdenedasthetransferofenergy,solikeenergyitisascalar withunitsofjoules. Page53,self-checkB: Wheneverenergyistransferredoutofthespring,thesameamount hastobetransferredintotheball,andviceversa.Asthespringcompresses,theballisdoing positiveworkonthespringgivingupitsKEandtransferringenergyintothespringasPE, andasitdecompressestheballisdoingnegativeworkextractingenergy. Page56,self-checkC: aNo.Thepackismovingatconstantvelocity,soitskineticenergy isstayingthesame.Itisonlymovinghorizontally,soitsgravitationalpotentialenergyisalso stayingthesame.Noenergytransferisoccurring.bNo.Thehorse'supwardforceonthe packformsa90-degreeanglewiththedirectionofmotion,socos =0,andnoworkisdone. Page58,self-checkD: Onlyinacanweuse Fd tocalculatework.Inbandc,theforce ischangingasthedistancechanges. AnswerstoSelf-ChecksforChapter5 Page122,self-checkA: 1,2,and4allhavethesamesigm,becausetheyaretryingtotwist thewrenchclockwise.Thesignoftorque3isoppositetothesignsoftheothers.Themagnitude oftorque3isthegreatest,sinceithasalarge r ,andtheforceisnearlyallperpendiculartothe wrench.Torques1and2arethesamebecausetheyhavethesamevaluesof r and F ? .Torque 4isthesmallest,duetoitssmall r AnswerstoSelf-ChecksforChapter5 Page147,self-checkA: Solidscanexertshearforces.Asolidcouldbeinanequilibriumin whichtheshearforceswerecancelingtheforcesduetounequalpressuresonthesidesofthe cube.

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AnswerstoSelectedHomeworkProblems SolutionsforChapterA Page163,problem10: a 2 )]TJ/F15 10.9091 Tf 11.89 0 Td [(10%b5%cThehighendforthebody'sactual eciencyishigherthanthelimitimposedbythelawsofthermodynamics.However,thehigh endofthe1-5wattrangequotedintheproblemprobablyincludeslargepeoplewhoaren'tjust lyingaround.Still,it'simpressivethatthehumanbodycomessoclosetothethermodynamic limit. SolutionstoSelectedHomeworkProblems SolutionsforChapter1 Page30,problem7: Aforceisaninteractionbetweentwoobjects,sowhilethebulletisin theair,thereisnoforce.Thereisonlyaforcewhilethebulletisincontactwiththebook. Thereisenergythewholetime,andthetotalamountdoesn'tchange.Thebullethassome kineticenergy,andtransferssomeofittothebookasheat,sound,andtheenergyrequiredto tearaholethroughthebook. Page31,problem8: aTheenergystoredinthegasolineisbeingchangedintoheatvia frictionalheating,andalsoprobablyintosoundandintoenergyofwaterwaves.Notethatthe kineticenergyofthepropellerandtheboatarenotchanging,sotheyarenotinvolvedinthe energytransformation.bThecrusingspeedwouldbegreaterbyafactorofthecuberootof 2,orabouta26%increase. Page31,problem9: Wedon'thaveactualmassesandvelocitiestoplugintotheequation, butthat'sOK.Wejusthavetoreasonintermsofratiosandproportionalities.Kineticenergy isproportionaltomassandtothesquareofvelocity,soB'skineticenergyequals .4J.77 = .34 2 =9.23J Page31,problem11: Roomtemperatureisabout20 C.Thefractionoftheenergythat actuallygoesintoheatingthewateris g = .24g C = J C )]TJ/F15 10.9091 Tf 10.909 0 Td [(20 C .25 10 3 J = ss =0.53 Soroughlyhalfoftheenergyiswasted.Thewastedenergymightbeinseveralforms:heating ofthecup,heatingoftheovenitself,orleakageofmicrowavesfromtheoven. SolutionsforChapter2 167

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Page45,problem5: E total i = E total f PE i +heat i = PE f + KE f +heat f 1 2 mv 2 = PE i )]TJ/F20 10.9091 Tf 10.91 0 Td [(PE f +heat i )]TJ/F15 10.9091 Tf 10.909 0 Td [(heat f = )]TJ/F15 10.9091 Tf 8.485 0 Td [( PE )]TJ/F15 10.9091 Tf 10.909 0 Td [(heat v = s 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [( PE )]TJ/F15 10.9091 Tf 10.91 0 Td [(heat m =6.4m = s Page46,problem7: Let betheanglebywhichhehasprogressedaroundthepipe.Conservationofenergygives E total i = E total f PE i = PE f + KE f Let'smake PE =0atthetop,so 0= mgr cos )]TJ/F15 10.9091 Tf 10.909 0 Td [(1+ 1 2 mv 2 Whileheisstillincontactwiththepipe,theradialcomponentofhisaccelerationis a r = v 2 r andmakinguseofthepreviousequationwend a r =2 g )]TJ/F15 10.9091 Tf 10.909 0 Td [(cos Therearetwoforcesonhim,anormalforcefromthepipeandadownwardgravitationforce fromtheearth.Atthemomentwhenhelosescontactwiththepipe,thenormalforceiszero, sotheradialcomponent, mg cos ,ofthegravitationalforcemustequal ma r mg cos =2 mg )]TJ/F15 10.9091 Tf 10.909 0 Td [(cos whichgives cos = 2 3 Theamountbywhichhehasdroppedis r )]TJ/F15 10.9091 Tf 10.909 0 Td [(cos ,whichequals r= 3atthismoment. Page46,problem9: aExample:Asonechildgoesupononesideofasee-saw,anotherchild ontheothersidecomesdown.bExample:Apoolballhitsanotherpoolball,andtransfers someKE. Page46,problem11: Supposetheriveris1mdeep,100mwide,andowsataspeed of10m/s,andthatthefallsare100mtall.In1second,thevolumeofwaterowingover thefallsis10 3 m 3 ,withamassof10 6 kg.Thepotentialenergyreleasedinonesecondis 6 kg g m=10 9 J,sothepoweris10 9 W.Atypicalhouseholdmighthave10hundredwattapplicancesturnedonatanygiventime,soitconsumesabout10 3 wattsontheaverage. Theplantcouldsupplyaaboutmillionhouseholdswithelectricity. 168 Appendix3:HintsandSolutions

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SolutionsforChapter3 Page73,problem18: No.Workdescribeshowenergywastransferredbysomeprocess.It isn'tameasurablepropertyofasystem. SolutionsforChapter4 Page103,problem8: Let m bethemassofthelittlepuckand M =2.3 m bethemassof thebigone.Allweneedtodoisndthedirectionofthetotalmomentumvectorbeforethe collision,becausethetotalmomentumvectoristhesameafterthecollision.Giventhetwo componentsofthemomentumvector p x = mv and p y = Mv ,thedirectionofthevectoris tan )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 p y =p x =23 counterclockwisefromthebigpuck'soriginaldirectionofmotion. Page104,problem11: Momentumisavector.Thetotalmomentumofthemoleculesis alwayszero,sincethemomentaindierentdirectionscancaloutontheaverage.Cooling changesindividualmolecularmomenta,butnotthetotal. Page104,problem14: Byconservationofmomentum,thetotalmomentaofthepiecesafter theexplosionisthesameasthemomentumofthereworkbeforetheexplosion.However,there isnolawofconservationofkineticenergy,onlyalawofconservationofenergy.Thechemical energyinthegunpowderisconvertedintoheatandkineticenergywhenitexplodes.Allwecan sayaboutthekineticenergyofthepiecesisthattheirtotalisgreaterthanthekineticenergy beforetheexplosion. Page104,problem15: aParticle i hadvelocity v i inthecenter-of-massframe,andhas velocity v i + u inthenewframe.Thetotalkineticenergyis 1 2 m 1 v 1 + u 2 + ::: where..."indicatesthatthesumcontinuesforalltheparticles.Rewritingthisintermsof thevectordotproduct,wehave 1 2 m 1 v 1 + u v 1 + u + ::: = 1 2 m 1 v 1 v 1 +2 u v 1 + u u + ::: Whenweaddupallthetermsliketherstone,weget K cm .Addingupallthetermslikethe thirdone,weget M j u j 2 = 2.Thetermslikethesecondtermcancelout: m 1 u v 1 + ::: = u m 1 v 1 + ::: wherethesuminbracketsequalsthetotalmomentuminthecenter-of-massframe,whichis zerobydenition. bChangingframesofreferencedoesn'tchangethedistancesbetweentheparticles,sothe potentialenergiesareallunaectedbythechangeofframesofreference.Supposethatina givenframeofreference,frame1,energyisconservedinsomeprocess:theinitialandnal energiesadduptobethesame.Firstlet'stransformtothecenter-of-massframe.Thepotential energiesareunaectedbythetransformation,andthetotalkineticenergyissimplyreduced bythequantity M j u 1 j 2 = 2,where u 1 isthevelocityofframe1relativetothecenterofmass. Subtractingthesameconstantfromtheinitialandnalenergiesstillleavesthemequal.Now wetransformtoframe2.Again,theeectissimplytochangetheinitialandnalenergiesby addingthesameconstant. Page105,problem16: Aconservationlawisaboutaddition:itsaysthatwhenyouaddup acertainthing,thetotalalwaysstaysthesame.Funkositywouldviolatetheadditivenatureof conservationlaws,becauseatwo-kilogrammasswouldhavetwiceasmuchfunkosityasapair ofone-kilogrammassesmovingatthesamespeed. 169

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SolutionsforChapter5 Page140,problem20: Thepliersarenotmoving,sotheirangularmomentumremains constantatzero,andthetotaltorqueonthemmustbezero.Notonlythat,buteachhalfofthe pliersmusthavezerototaltorqueonit.Thistellsusthatthemagnitudeofthetorqueatone endmustbethesameasthatattheotherend.Thedistancefromtheaxistothenutisabout2.5 cm,andthedistancefromtheaxistothecentersofthepalmandngersareabout8cm.The anglesarecloseenoughto90 thatwecanpretendthey're90degrees,consideringtherough natureoftheotherassumptionsandmeasurements.TheresultisN.5cm= F cm, or F =90N. Page141,problem28: Thefootoftherodismovinginacirclerelativetothecenterofthe rod,withspeed v = b=T ,andacceleration v 2 = b= 2= 2 = 8 g .Thisaccelerationisinitially upward,andisgreaterinmagnitudethan g ,sothefootoftherodwillliftowithoutdragging. Wecouldalsoworryaboutwhetherthefootoftherodwouldmakecontactwiththeooragain beforetherodnishesupatonitsback.Thisisaquestionthatcanbesettledbygraphing, orsimplybyinspectionofguremonpage119.Thekeyhereisthatthetwopartsofthe accelerationarebothindependentof m and b ,sotheresultisuniveral,anditdoessuceto checkagraphinasingleexample.Inpracticalterms,thistellsussomethingabouthowdicult thetrickistodo.Because 2 = 8=1.23isn'tmuchgreaterthanunity,ahitthatisjustalittle tooweakbyafactorof1.23 1 = 2 =1.11willcauseafairlyobviousqualitativechangeinthe results.Thisiseasilyobservedifyoutryitafewtimeswithapencil. 170 Appendix3:HintsandSolutions

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Index alchemists,13 angularmomentum choiceofaxistheorem,116 dened,109 denition,110 introductionto,107 relatedtoareasweptout,113 spintheorem,116 Avogadro'snumber,153 blackhole,160 Boltzmann'sconstant,153 Celsiusunit,150 centerofmass frameofreference,90 relatedtomomentum,88 Chadwick,James discoveryofneutron,85 choiceofaxistheorem,116 collision dened,83 conductionofheat distinguishedfromwork,50 correspondenceprinciple,17 dotproductoftwovectors,66 electricalforce inatoms,85 electron,85 element,chemical,14 ellipticalorbitlaw,133 energy gravitationalpotentialenergy,40 potential,38 equilibrium dened,127 fulcrum,131 gammaray,86 Hawkingradiation,160 heat asauid,36 asaformofkineticenergy,36 heatconduction distinguishedfromwork,50 heatengine,143 idealgaslaw,153 jouleunit,18 Joyce,James,36 kelvinunit,150 Kepler ellipticalorbitlaw,133 lawofequalareas,113 kineticenergy,23 comparedtomomentum,81 lever,131 momentum comparedtokineticenergy,81 dened,77 examplesinthreedimensions,94 oflight,80 rateofchangeof,91 relatedtocenterofmass,88 transferof,91 Neanderthals,132 neutron discoveryof,85 nucleus,85 particlezoo,35 pascal unit,145 perpetualmotionmachine,14 potentialenergy electrical,42 gravitational,40,63 nuclear,43 ofaspring,62 relatedtowork,62 power,25 pressure,144 proton,85

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quarks,36 rigidrotation dened,109 scalardotproduct,66 slingshoteect,90 spintheorem,116 spring potentialenergyof,62 workdoneby,62 statics,127 Stevin,Simon,17 temperature absolutezero,150 asameasureofenergyperatom,37 Celsius,150 Kelvin,150 macroscopicdenition,150 thermodynamics,37,144 rstlawof,144 secondlawof,159 zerothlawof,149 thermometer,150 torque dened,120 duetogravity,123 relationshiptoforce,121 wattunit,26 work calculatedwithcalculus,60 dened,50 distinguishedfromheatconduction,50 donebyaspring,62 donebyavaryingforce,57 inthreedimensions,55 positiveandnegative,53 relatedtopotentialenergy,62 work-kineticenergytheorem,65 172 Index

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Index 173

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174 Index

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Index 175

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176 Index

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Index 177

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UsefulData MetricPrexes M-mega-10 6 k-kilo-10 3 m-milli-10 )]TJ/F19 5.9776 Tf 5.756 0 Td [(3 -Greekmumicro-10 )]TJ/F19 5.9776 Tf 5.756 0 Td [(6 n-nano-10 )]TJ/F19 5.9776 Tf 5.756 0 Td [(9 p-pico-10 )]TJ/F19 5.9776 Tf 5.756 0 Td [(12 f-femto-10 )]TJ/F19 5.9776 Tf 5.756 0 Td [(15 Centi-,10 )]TJ/F19 5.9776 Tf 5.756 0 Td [(2 ,isusedonlyinthecentimeter. TheGreekAlphabet Aalpha Nnu Bbeta xi )-1633(gamma oOomicron delta pi Eepsilon Prho Zzeta sigma Heta Ttau theta Yupsilon Iiota phi Kkappa Xchi lambda psi Mmu omega FundamentalConstants gravitationalconstant G =6.67 10 )]TJ/F19 5.9776 Tf 5.756 0 Td [(11 N m 2 = kg 2 speedoflight c =3.00 10 8 m/s SubatomicParticles particlemasskgradiusfm electron9.109 10 )]TJ/F19 5.9776 Tf 5.756 0 Td [(31 0.01 proton1.673 10 )]TJ/F19 5.9776 Tf 5.756 0 Td [(27 1.1 neutron1.675 10 )]TJ/F19 5.9776 Tf 5.756 0 Td [(27 1.1 Theradiiofprotonsandneutronscanonlybegivenapproximately,sincetheyhavefuzzysurfaces.Forcomparison,a typicalatomisaboutamillionfminradius. NotationandUnits quantityunitsymbol distancemeter,m x x timesecond,s t t masskilogram,kg m densitykg = m 3 aream 2 squaremetersA volumem 3 cubicmetersV velocitym/s v accelerationm = s 2 a gravitationaleldJ = kg morm = s 2 g forcenewton,1N=1kg m = s 2 F pressure1Pa=1N = m 2 P energyjoule,J E powerwatt,1W=1J/s P momentumkg m = s p angularmomentumkg m 2 = sorJ s L torqueN m periods T Conversions Nonmetricunitsintermsofmetricones: 1inch=25.4mmbydenition 1pound-force=4.5newtonsofforce kg g =2.2pounds-force 1scienticcalorie=4.18J 1kcal=4.18 10 3 J 1gallon=3.78 10 3 cm 3 1horsepower=746W Whenspeakingoffoodenergy,thewordCalorie"isused tomean1kcal,i.e.,1000calories.Inwriting,thecapitalC maybeusedtoindicate1Calorie=1000calories. RelationshipsamongU.S.units: 1footft=12inches 1yardyd=3feet 1milemi=5280feet Earth,Moon,andSun bodymasskgradiuskmradiusoforbitkm earth5.97 10 24 6.4 10 3 1.49 10 8 moon7.35 10 22 1.7 10 3 3.84 10 5 sun1.99 10 30 7.0 10 5 | 178 Index

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Index 179




TRACE ROUTE

Total Execution Time: 71 Milliseconds

MILLISECOND   CLASS.METHODMESSAGE
0sobekcm_page_globals.constructor
0sobekcm_page_globals.constructorApplication State validated or built
0sobekcm_database.verify_item_lookup_object
0sobekcm_page_globals.constructorNavigation Object created from URI query string
0sobekcm_database.verify_item_lookup_object
0sobekcm_page_globals.display_itemRetrieving item or group information
0sobekcm_page_globals.get_entire_collection_hierarchyRetrieving hierarchy information
0sobekcm_assistant.get_entire_collection_hierarchy
0cached_data_manager.retrieve_item_aggregation
0cached_data_manager.retrieve_item_aggregationFound item aggregation on local cache
0item_aggregation_builder.get_item_aggregationFound 'all' item aggregation in cache
0system.web.ui.page.page_load (ufdc.page_load)
0sobekcm_page_globals.constructor.on_page_load
0html_echo_mainwriter.add_style_referencesAdding style references to HTML
0html_echo_mainwriter.add_text_to_pageReading the text from the file and echoing back to the output stream
71html_echo_mainwriter.add_text_to_pageFinished reading and writing the file