浙大物理英文版Rotational Dynamics

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2020-02-26

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Chapter9RotationalDynamics12020/2/27phypzq@zj165.com9-1TorqueTorqueandrotationChapter9RotationalDynamics22020/2/27phypzq@zj165.comFrTorqueasavectorChapter9RotationalDynamics32020/2/27phypzq@zj165.comThetorqueaboutzaxis:)(initiiiiFFRFRiititiizrFFRsinChapter9RotationalDynamics42020/2/27phypzq@zj165.comThemagnitudeofτisrFsinθ,anditsdirectionisdeterminedbyright-handrule.Chapter9RotationalDynamics52020/2/27phypzq@zj165.com9-2RotationalinertiaandNewton’ssecondlawChapter9RotationalDynamics62020/2/27phypzq@zj165.com1.RotationalinertiaofasingleparticleoxyFFsinθmrθτImrrFrmmaFT2sinsinI=mr2istherotationalinertiaoftheparticle.Therotationalinertiadependsonthemassoftheparticleandontheperpendiculardistancebetweentheparticleandtheaxisofrotation.Chapter9RotationalDynamics72020/2/27phypzq@zj165.comForrotationalmotionofasingleparticle,weobtain:τ=Iα2.Newton’ssecondlawforrotationLetusconsideramorecomplicatedrigidsystemconsistingofmanyparticles.iiiiamfFChapter9RotationalDynamics82020/2/27phypzq@zj165.comThetangentialcomponentformulaoftheequationis:FiT+fiT=miaiT=miriα2iiiiTiiTrmrfrFFiTri+fiTri=miri2αIrmrFextiiiiT;22iirmIChapter9RotationalDynamics92020/2/27phypzq@zj165.comHerethetorqueτext,therotationalinertiaIandtheangularaccelerationαareallaboutthezaxis.Notethat:τextisthesumofthetorquesduetoalltheexternalforces,isnotthetorqueofthesumofalltheexternalforces.IextThisistherotationalformofNewton’slaw,calledrotationaltheoremofrigidbodyaboutfixedaxisChapter9RotationalDynamics102020/2/27phypzq@zj165.com0iiicmrmr3.Theparallel-axistheorem222222)(MhIhrmmhrmhrmrmIcmiiiiiiiiyiRotationaboutanarbitraryaxisy2mhIICMChapter9RotationalDynamics112020/2/27phypzq@zj165.comRotationaboutanarbitraryaxiszxyiiiiiiiizIIymxmyxmrmIi)(22222Iz=Iy+IxChapter9RotationalDynamics122020/2/27phypzq@zj165.comIfthebodyisoneitasacontinuousdistributionofmatter,wecanimagineitdividedintoalargernumberofsmallmasselementsδm,mrmrInnmndδlim2209-3RotationalinertiaofsolidbodiesChapter9RotationalDynamics132020/2/27phypzq@zj165.comlrISrIVrIlmSmVmddddddddd222Chapter9RotationalDynamics142020/2/27phypzq@zj165.comExample:AthinrodwithmassM,itisuniformdensity,andlengthLrotateaboutzaxisthroughthecenterofmass.233222222222121])2()2[(31dddMLLLLMxLMxxxmxILLLLLLcChapter9RotationalDynamics152020/2/27phypzq@zj165.comxdxdmzIftheaxisthroughtheoneendoftherodinparallelwithaxisthroughthecenterofmass.230202023131dddMLLLMxLMxxxmxILLLWecansee:I=Ic+M(L/2)2Chapter9RotationalDynamics162020/2/27phypzq@zj165.comExample:Amerry-go-roundwithradiusR,andmassMrotateaboutzaxisperpendiculartotheplate.242022022142d2dMRRRMrrRMrmrIRRChapter9RotationalDynamics172020/2/27phypzq@zj165.comExample:Asphericalshellwiththeaxisacrossthecenter2022232dsin24sindMRθRRRMθRmrIπChapter9RotationalDynamics182020/2/27phypzq@zj165.comChapter9RotationalDynamics192020/2/27phypzq@zj165.com9-4TorqueduetogravityEachparticleinthebody,suchasmassmn,experienceagravitationalforcemngiicmrmMr1gmrgmriiicgThecenterofgravitycoincideswiththecenterofmass,ifweconsiderthegravitationalfieldisuniform.Chapter9RotationalDynamics202020/2/27phypzq@zj165.comTheforceontheentirebodyduetogravityis:grMgrmgmrgMgmFcmiiiii)(WrgMrcmcmThetoqueactedonabodyduetogravityaboutanypointisequaltothetoqueoftotalgravityactedattheCMaboutthispoint.Chapter9RotationalDynamics212020/2/27phypzq@zj165.comChapter9RotationalDynamics222020/2/27phypzq@zj165.comHowdeterminethepositionofcenterofgravityinaextendedobject?Chapter9RotationalDynamics232020/2/27phypzq@zj165.com9-5EquilibriumapplicationsofNewton’slawforrotation0,0:bodyanyForextextFChapter9RotationalDynamics242020/2/27phypzq@zj165.comm,Rm1m2Example:Finetheaccelerationoftheblocks(m1m2),thetensionintheropeandtheangularaccelerationofthepulley.T1T2mgTm1gm2gT1T21a2aChapter9RotationalDynamics252020/2/27phypzq@zj165.comIRTTamgmTamTgm)(212222111122121mRIRaaT1T2mgTm1gm2gT1T21a2aChapter9RotationalDynamics262020/2/27phypzq@zj165.comgmmmmmmTgmmmmmmTRgmmmmmgmmmmma22112121212121212122222222Chapter9RotationalDynamics272020/2/27phypzq@zj165.comExercises193839Problems1314Chapter9RotationalDynamics282020/2/27phypzq@zj165.com9-6NonequilibriumapplicationsofNewton’slawforrotationzzextI,xxmaFAnalysissampleproblem9-10,detailpleaseseepage192!Chapter9RotationalDynamics292020/2/27phypzq@zj165.comChapter9RotationalDynamics302020/2/27phypzq@zj165.comExample:Arod,massmandlengthl,canrotateaboutitsaxisperpendiculartothepagesurfacethroughthepointo.Therodrotatesfromtherestwhenitishorizontaltotheangleθ.Find(1)α;(2)ω;(3)acm;(4)Nx,Ny.Solution:cos23g31cos2)1(2lmlIIlmgChapter9RotationalDynamics312020/2/27phypzq@zj165.comlglglgtsin3dcos23dcos23dddd)2(00sin232,cos432)3(2glaglacNcTChapter9RotationalDynamics322020/2/27phypzq@zj165.comsinsincoscoscossinCMofmotion)4(cNyxcTyxmamgNNmamgNN)cos43sin