《大學(xué)物理》英文課件11 Rolling,Torque,and Angular Momentum

?大學(xué)物理?英文課件11Rolling,Torque,andAngularMomentumRollingasTranslationandRotationCombinedLinearspeedofsmoothrollingmotion:Arclengthanddistance:TheKineticEnergyofRollingGoal:tocalculatethekineticenergyofrollingwheelofpurerotationaboutanaxisofpointPmeasuredbyastationaryobserver.TheForcesofRolling–FrictionandRollingAwheelrollshorizontallywithoutsliding(smoothrolling)whileacceleratingwithlinearaccelerationacom.AstaticfrictionalforcefsactsonthewheelatPopposingitstendencytoslide.acom:linearaccelerationα:angularaccelerationTheForcesofRolling–RollingDownaRampArounduniformbodyofmassMandradiusRrollsdownarampwithoutsliding(smoothrolling)atanangleθalongxaxis.SampleProblemProblem:auniformballofmassM=6.00kgandradiusRrollssmoothlyfromrestdownarampatangleθo.(a)Theballdescendsaverticalheighth=1.20mtoreachthebottomoftheramp.Whatisitsspeedatthebottom?(b)Whatarethemagnitudeanddirectionofthefrictionalforceontheballasitrollsdowntheramp?Solution:M=6.00kg,θo,h(a)SampleProblemProblem:auniformballofmassM=6.00kgandradiusRrollssmoothlyfromrestdownarampatangleθo.(a)Theballdescendsaverticalheighth=1.20mtoreachthebottomoftheramp.Whatisitsspeedatthebottom?(b)Whatarethemagnitudeanddirectionofthefrictionalforceontheballasitrollsdowntheramp?Solution:M=6.00kg,θo,h(b)TheYo-YoWhenaYo-Yorollsdownorupthestring,itskineticenergy(translationalform(1/2)Mv2comandrotationalform(1/2)Icomω2)andpotentialenergy(Mgh)exchange.TorqueRevisitedMagnitude:SampleProblemProblem:threeforces,eachofmagnitude2.0N,actsonaparticle.TheparticleisinthexzplaneatpointAgivenbypositionr=3.0m,θ=30o.ForceF1isparalleltothexaxis,F2isparalleltothezaxis,andF3isparalleltotheyaxis.WhatisthetorqueabouttheoriginOduetoeachforce?Solution:LinearMomentumLinearmomentumofaparticle:Thelawofconservationoflinearmomentum:Ifnonetexternalforceactsonasystemofparticles,thetotallinearmomentumcannotchange.

Note:particledoesnothavetorotatearoundO,but

onlywithrespecttoO

AngularMomentumAngularmomentum:SampleProblem–twoparticlesystemProblem:Twoparticlesmoveatconstantmomentumalonghorizontalpaths.Particle1withmomentump11andwillpass2.0mfromO.Particle2withp22andwillpass4.0mfromO.WhatarethemagnitudeanddirectionofthenetangularmomentumLaboutO?Solution:Newton’sSecondLawinAngularForm

Note:thetorqueandtheangularmomentumaredefinedwithrespecttothesamepoint.Forlinearmomentum:Forangularmomentum:SampleProblemProblem:apenguinofmassmfallsfromrestatpointA,ahorizontaldistanceDfromtheoriginOofanxyzcoordinatesystem.(a)WhatistheangularmomentumofthefallingpenguinaboutO?(b)WhatisthetorqueonthepenguinduetothegravitationalforceFgabouttheoriginO?Solution:(a)Directionofl?Directedintotheplanecomposedbyrandp.(b)TheAngularMomentumofaSystemofParticlesTheAngularMomentumofaRigidBodyRotatingAboutaFixedAxisI:rotationalinertiaω:angularvelocityL:angularmomentumLinearmomentumofaparticle:Thelawofconservationofangularmomentum:

-Ifthenetexternaltorqueactingonasystemiszero,theangularmomentumofthesystemremainsconstant,nomatterwhatchangestakeplacewithinthesystem.HoldforparticulardirectionHold

everywhereConservationofAngularMomentumThelawofconservationoflinearmomentumThreeimportantconservationlawsinnatureThelawofconservationofangularmomentumThelawofconservationofenergyExampleofConservationofAngularMomentumRolling,Torque,andAngularMomentum

SampleProblem–rotatingwheeldemoProblem:astudentsitonastoolthatcanrotatefreelyaboutaverticalaxis.Thestudent,initiallyatrest,isholdingabicyclewheelwhoserimisloadedwithleadandwhoserotationalinertiaIwh2.Thewheelisrotatingatanangularspeedωwhof3.9rev/sandcounterclockwiseseenfromoverhead.Theaxisofthewheelisvertical,andtheangularmomentumLwhofthewheelpointsverticallyupward.Thestudentnowinvertsthewheelsothatitisrotatingclockwise.Itsangularmomentumisnow–Lwh.Theinversionresultsinthestudent,thestool,andthewheel’scenterrotatingtogetherasacompositerigidbodyaboutthestool’srotationaxiswithrotationalinertialIb=6.8kg.m2.Withwhatangularspeedωbandinwhatdirectiondoesthecompositebodyrotateaftertheinversionofthewheel.SampleProblem–rotatingwheeldemoSolution:SampleProblem–cockroachondiskProblem:acockroachwithmassmmandradiusR.Thediskrotateslikeamerry-go-groundarounditscentralaxisatangularspeedωirR,butthenitcrawlsouttotherimofthedisk.Treatthecockroachasaparticle.Whatthenistheangularspeed?Solution:Precession(歲差〕ofaGyroscope〔陀螺儀〕Precessionrate:VariablesofTr