《物理双语教学课件》Chapter 7 Rolling Torque and Angular Momentum 力矩与角动量Word文档格式.docx

《物理双语教学课件》Chapter 7 Rolling Torque and Angular Momentum 力矩与角动量Word文档格式.docx

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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

Likeallotherlinearquantities,linearmomentumhasitsangularcounterpart.Thefigureshowsaparticlewithlinearmomentump（=mv）locatedatpointPinthexyplane.TheangularmomentumlofthisparticlewithrespecttotheoriginOisavectorquantitydefinedas

where

isthepositionvectoroftheparticlewithrespecttoO.

2.TheSIunitofangularmomentumisthekilogram-meter-squarepersecond（

）,equivalenttothejoule-second（

3.Thedirectionoftheangularmomentumvectorcanbefoundtouseright-handrule,asshowninthefigure.

4.Themagnitudeoftheangularmomentumis

istheanglebetweenrandpwhenthesetwovectorsaretailtotail.

7.5Newton’ssecondlawinangularform

1.Wehaveseenenoughoftheparallelismbetweenlinearandangularquantitiestobeprettysurethatthereisalsoacloserelationbetweentorqueandangularmomentum.Itis

2.Thevectorsumofalltorquesactingonaparticleisequaltothetimerateofchangeoftheangularmomentumofthatparticle.Thetorqueandtheangularmomentumshouldbedefinedwithrespecttothesameorigin.

3.Proofoftheequation:

angularmomentumcanbewrittenas:

Differentiatingeachsidewithrespecttotimeyields

7.6TheAngularMomentumofASystemofParticles

Nowweturnourattentiontothemotionofasystemofparticleswithrespecttoanorigin.Notethat“asystemofparticles”includesarigidbodyasaspecialcase.

1.ThetotalangularmomentumLofasystemparticlesisthevectorsumoftheangularmomentaloftheparticles:

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

and

arethelinearmomentumandlinearspeedofthemasselement,and

istheanglebetween

and

3.Weareinterestedinthecomponentof

thatparalleltotherotationaxis,herethezaxis.Thatzcomponentis

4.Thezcomponentoftheangularmomentumfortherotatingrigidbodyasawholeisfoundbyaddingupthecontributionsofall