《物理双语教学课件》Chapter 7 Rolling Torque and Angular Momentum 力矩与角动量.docx
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《物理双语教学课件》Chapter 7 Rolling Torque and Angular Momentum 力矩与角动量.docx
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《物理双语教学课件》Chapter7RollingTorqueandAngularMomentum力矩与角动量
Chapter7RollingTorque,andAngularMomentum
7.1Rolling
Whenabicyclemovesalongastraighttrack,thecenterofeachwheelmovesforwardinpuretranslation.Apointontherimofthewheel,however,tracesoutamorecomplexpath,asthefigureshows.Inwhatfollow,weanalyzethemotionofarollingwheelfirstbyviewingitasacombinationofapuretranslationandapurerotation,andthenbyviewingitasrotatingalong.
1.Rollingasrotationandtranslationcombined
(1)
Imaginethatyouarewatchingthewheelofabicycle,whichpassesyouatconstantspeedwhilerollingsmoothly,withoutslipping,alongastreet.Asshowninthefigure,thecenterofmassOofthewheelmovesforwardatconstantspeed
.ThepointPatwhichthewheelcontactsthestreetalsomovesforwardatspeed
sothatitalwaysremainsdirectlybelowO.
(2)Duringatimeintervalt,youseebothOandPmoveforwardbyadistances.Thebicycleriderseesthewheelrotatethroughanangle
aboutthecenterofthewheel,withthepointofthewheelthatwastouchingthestreetatthebeginningoftmovingthrougharclengths.Wehavetherelation
whereRistheradiusofthewheel.
(3)Thelinearspeed
ofthecenterofthewheelis
andtheangularspeed
ofthewheelaboutitscenteris
.Sodifferentiatingtheequationwithrespecttotimegivesus
.
(4)
Thefigureshowsthattherollingmotionofawheelisacombinationofpurelytranslationalandpurelyrotationalmotions.
(5)
Themotionofanyroundbodyrollingsmoothlyoverasurfacecanbeseparatedintopurelytranslationalandpurelyrotationalmotion.
2.Rollingaspurerotation
(1)Thefiguresuggestsanotherwaytolookattherollingmotionofawheel,namely,aspurerotationaboutanaxisthatalwaysextendsthroughthepointwherethewheelcontactsthestreetasthewheelmoves.Thatis,weconsidertherollingmotiontobepurerotationaboutanaxispassingthroughpointPandperpendiculartotheplaneofthefigure.Thevectorsinthenrepresenttheinstantaneousvelocitiesofpointsontherollingwheel.
(2)Theangularspeedaboutthisnewaxisthatastationaryobserverassigntoarollingbicyclewheelisthesameangularspeedthattheriderassignstothewheelasheorsheobservesitinpurerotationaboutanaxisthroughitscenterofmass.
(3)
Toverifythisanswer,let’suseittocalculatethelinearspeedofthetopofthewheelfromthepointofviewofastationaryobserver,asshowninthefigure.
3.TheKineticenergy:
Letusnowcalculatethekineticenergyoftherollingwheelasmeasuredbythestationaryobserver.
(1)IfweviewtherollingaspurerotationaboutanaxisthroughPinabovefigure,wehave
inwhich
istheangularspeedofthewheeland
istherotationalinertiaofthewheelabouttheaxisthroughP.
(2)Fromtheparallel-axistheorem,wehave
inwhichMisthemassofthewheeland
isitsrotationalinertiaaboutanaxisthroughitscenterofmass.
(3)Substitutingtherelationaboutitsrotationalinertia,weobtain
.
(4)Wecaninterpretthefirstofthesetermsasthekineticenergyassociatedwiththerotationofthewheelaboutanaxisthroughitscenterofmass,andthesecondtermasthekineticenergyassociatedwiththetranslationalmotionofthewheel.
4.Frictionandrolling
(1)Ifthewheelrollsatconstantspeed,ithasnotendencytoslideatthepointofcontactP,andthusthereisnofrictionalforceactingonthewheelthere.
(2)However,ifaforceactingonthewheel,changingthespeed
ofthecenterofthewheelortheangularspeed
aboutthecenter,thenthereisatendencyforthewheeltoslide,thefrictionalforceactsonthewheelatPtoopposethattendency.
(3)Untilthewheelactuallybeginstoslide,thefrictionalforceisastaticfrictionalforcefs.Ifthewheelbeginstoslide,thentheforceisakineticfrictionalforcefk.
7.2TheYo-Yo
1.
Ayo-yo,asshowninthefigure,isaphysicslabthatyoucanfitinyourpocket.Ifayo-yorollsdownitsstringforadistanceh,itlosespotentialenergyinamountmghbutgainskineticenergyinbothtranslationalandrotationalform.Whenitisclimbingbackup,itloseskineticenergyandregainspotentialenergy.
2.Letusanalyzethemotionoftheyo-yodirectlywithNewton’ssecondlaw.Theabovefigureshowsitsfree-bodydiagram,inwhichonlytheyo-yoaxleisshown.
(1)ApplyingNewton’ssecondlawinitslinearformyields
HereMisthemassoftheyo-yo,andTisthetensionintheyo-yo’sstring.
(2)ApplyingNewton’ssecondlawinangularformaboutanaxisthroughthecenterofmassyields
Where
istheradiusoftheyo-yoaxleandIistherotationalinertialoftheyo-yoaboutitscenteraxis.
(3)Thelinearaccelerationandangularaccelerationhaverelation
.SoAftereliminatingTinbothequationsweobtain
.Thusanidealyo-yorollsdownitsstringwithconstantacceleration.
7.3Torquerevisited
Inchapter6wedefinedtorque
forarigidbodythatcanrotatearoundafixedaxis,witheachparticleinthebodyforcedtomoveinapaththatisacircleaboutthataxis.Wenowexpandthedefinitionoftorquetoapplyittoanindividualparticlethatmovesalonganypathrelativetoafixedpointratherthanafixedaxis.Thepathneednolongerbeacircle.
1.ThefigureshowssuchaparticleatpointPinthexyplane.AsingleforceFinthatplaneactsontheparticle,
andtheparticle’spositionrelativetotheoriginOisgivenbypositionvectorr.Thetorque
actingontheparticlerelativetothefixedpointOisavectorquantitydefinedas
.
2.Discussthedirectionandmagnitudeof
(
).
7.4AngularMomentum
1.
Likeallotherlinearquantities,linearmomentumhasitsangularcounterpart.Thefigureshowsaparticlewithlinearmomentump(=mv)locatedatpointPinthexyplane.TheangularmomentumlofthisparticlewithrespecttotheoriginOisavectorquantitydefinedas
where
isthepositionvectoroftheparticlewithrespecttoO.
2.TheSIunitofangularmomentumisthekilogram-meter-squarepersecond(
),equivalenttothejoule-second(
).
3.Thedirectionoftheangularmomentumvectorcanbefoundtouseright-handrule,asshowninthefigure.
4.Themagnitudeoftheangularmomentumis
where
istheanglebetweenrandpwhenthesetwovectorsaretailtotail.
7.5Newton’ssecondlawinangularform
1.Wehaveseenenoughoftheparallelismbetweenlinearandangularquantitiestobeprettysurethatthereisalsoacloserelationbetweentorqueandangularmomentum.Itis
.
2.Thevectorsumofalltorquesactingonaparticleisequaltothetimerateofchangeoftheangularmomentumofthatparticle.Thetorqueandtheangularmomentumshouldbedefinedwithrespecttothesameorigin.
3.Proofoftheequation:
angularmomentumcanbewrittenas:
Differentiatingeachsidewithrespecttotimeyields
7.6TheAngularMomentumofASystemofParticles
Nowweturnourattentiontothemotionofasystemofparticleswithrespecttoanorigin.Notethat“asystemofparticles”includesarigidbodyasaspecialcase.
1.ThetotalangularmomentumLofasystemparticlesisthevectorsumoftheangularmomentaloftheparticles:
inwhich
labelstheparticles.
2.Withtime,theangularmomentaofindividualparticlemaychange,eitherbecauseofinteractionswithinthesystem(betweentheindividualparticles)orbecauseofinfluencesthatmayactonthesystemfromtheoutside.WecanfindthechangeinLasthesechangestakeplacebytakingthetimederivativeofaboveequation.Thus
.
3.Sometorquesareinternal,associatedwithforcesthattheparticleswithinthesystemexertononeanother;othertorquesareexternal,associatedwithforcesthatactfromoutsidethesystem.Theinternalforces,becauseofNewton’slawofactionandreaction,cancelinpair(givealittlemoreexplanation).So,toaddthetorques,weneedconsideronlythoseassociatedwithexternalforces.Thenaboveequationbecomes
ThisisNewton’ssecondlawforrotationinangularform,expressforasystemofparticles.Theequationhasmeaningonlyifthetorqueandangularmomentumvectorsarereferredtothesameorigin.Inaninertiareferenceframe,theequationcanbeappliedwithrespecttoanypoint.Inanacceleratingframe,itcanbeappliedonlywithrespecttothecenterofmassofthesystem.
7.7TheAngularMomentumofaRigidBodyRotatingaboutaFixedAxis
Wenextevaluatetheangularmomentumofasystemofparticlesthatformarigidbody,whichrotatesaboutafixedaxis.Figure(a)showssuchabody.Thefixedaxisofrotationisthezaxis,andthebodyrotatesaboutitwithconstantangularspeed
.Wewishtofindtheangularmomentumofthebodyabouttheaxisofrotation.
1.Wecanfindtheangularmomentabysummingthezcomponentsoftheangularmomentaofthemasselementsinthebody.Infigure(a),atypicalmasselement
ofthebodymovesaroundthezaxisinacircularpath.ThepositionofthemasselementsislocatedrelativetotheoriginObypositionvector
.Theradiusofthemasselement’scircularpathis
theperpendiculardistancebetweentheelementandzaxis.
2.Themagnitudeoftheangularmomentum
ofthismasselement,withrespecttoO,is
where
and
arethelinearmomentumandlinearspeedofthemasselement,and
istheanglebetween
and
.
3.Weareinterestedinthecomponentof
thatparalleltotherotationaxis,herethezaxis.Thatzcomponentis
.
4.Thezcomponentoftheangularmomentumfortherotatingrigidbodyasawholeisfoundbyaddingupthecontributionsofall
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