合力所做的功等于分力所做功的代数和

Chapter4WorkandEnergyWork:Everydaymeaning:anyactivitythatrequiresmuscularormentaleffort.Inphysics:theworkdoneonanobjectbyaforceinavectordisplacementistheirinnerproduct(scalar)—thesamemagnitudeofforce:positive,mines,zeroworkmeternewtonjoulerFwFrcosworkalDifferentiFdrrdFdw（元功）nnn21)2()1()2()1(2)2()1(1)2()1(21)2()1()(Work:varyingforcesoralongacurve合力所做的功等于分力所做功的代数和。TmgFExample:FindtheworkdonebyFwhenparticlemisslowlyliftedtothepresentposition.Solution1:0GTFFi0GTFAAAA0TAGFAASolution2:cossinTmgTFdsFsdFAcos)cos1(sin)(costan00mgRdmgRRdmgATmgFtanmgF（1）平面直角坐标系(planarorthogonalcoordinatesystem))0ˆˆ,1ˆˆˆˆ(,ˆˆ,ˆˆjijjiiwherejdyidxrdjFiFFyx.dyFdxFdwyxxyOxr0r1yFrs0s1FsWorkindifferentcoordinatesystems（2）平面“自然”坐标系(planarintrinsicsystem))0ˆˆ,1ˆˆˆˆ(ntnnttwhere.ˆ)ˆˆ(dsFtdsnFtFdwtntr1rOr0F(,)（3）极坐标系(planarpolarsystem))0ˆˆ,1ˆˆˆˆ(jijjiiwherePower(功率)Everydaymeaning:energyorforceInphysics:thetimerateatwhichworkisdone)(poweraveragetwNpower)neous(instantanlim0vFdtrdFdtdwtwNt力的功率等于力与受力点速度的标积1J/s)(1Wond1joule/sec1wattSummaryonwork功的性质(1)Generallyspeaking,workisdependentonthepath.功是过程量，一般与路径有关。(2)Workisascalarwithmagnitudeandsigns.功是标量，但有正负。(3)Workdonebyresultantofallforcesequals合力的功为各分力的功的代数和。n21Casestudy1:Workdonebygravitation引力的功两个质点之间在万有引力作用下相对运动时，以M所在处为原点,M指向m的方向为矢径r的正方向。m受的引力方向与矢径方向相反。m在M的万有引力的作用下从a点运动到b点，万有引力的功:rderGMmrdFWbarba2cosredrdrdr)11(2babarrGMmdrrGMmWGravitationalwork:independentofthepathofthebody;dependsonlyonthestartingandendingpoints引力的功：与路径无关，仅仅取决于起始和终点的位置Casestudy2:WorkdonebyelasticforceskOa'Fb'FkOacbOa'FkkxF)(2122baxxbaxxkkxdxWbaba)(21)(212222bccaxxxxbcaxxkxxkkxdxkxdxWbccabcakOa'Fb'FkOa)(2122baxxk0Workdonebyelasticforces:弹力的功independentofthepathofthebody;dependsonlyonthestartingandendingpointsRuleofsigns:正负号00abWexternalforcedoworkbaWelasticforcedoworkW0,外力做功，W0,弹力做功“Work”canbestoredandrecovered.“功”可以储存和释放Conservativeandnonconservativeforces:保守力和非保守力Workdonebyaconservativeforce保守力的功：(1)Reversible,“work”canbestoredina“BANK”;可逆，功可以被储存(2)Independentofthepathofthebody;与路径无关(3)Zeroworkforclosedpath.闭合路径做功为零ABL1L20cosdsFrdFBABArdFrdF21等价Energyisdefinedastheabilitytodowork.Workisdefinedasthetransferofenergy.Thewaterissaidtopossesskineticenergysinceitismoving.Itgetsthisenergybecauseitisfallingthroughagravitationalfield.Potentialenergy(势能)Energyassociatedwiththepositionofasystem.Storedinasystem,laterrecovered.与相互作用物体的位置有关的能量。WorkbyconservativeforcespotentialenergyVVVrdFrdFWBABABABA21BAABWVThenVIf0势能的增量等于保守力所做功的负值.(1)保守力做功势能Gravitationalpotentialenergy)11(babarrGMmWbaVV)(2122babaxxkWElasticpotentialenergybaVVrGMmV221kxV??c?mghVOAOAArdFWV或VVVWBABAUzkyjxiUgradU)((2)势能保守力.lim),(0xVxVFxFVzyxxx不变.gradVVF.,,zVFyVFxVFzyxgradVVVzkyjxikzVjyVixVkFjFiFFzyx)()(梯度：Example:findgravityfromgravitationalpotentialV=mgyjmgkzmgyjymgyixmgymgyF)()()()(Solution:(3)1DEnergydiagram一维势能曲线xU(x)Theelasticpotentialenergyforaspring弹簧振子的势能曲线U(x)xOAhypotheticalpotentialenergyfunction假想的势能曲线平衡条件0dxdUFStableequilibrium稳定平衡？Unstableequilibrium不稳定平衡？Stableequilibrium稳定平衡条件：00/22dxUddxdUU(x)xOAhypotheticalpotentialenergyfunction假想的势能曲线A(x1,y1)B(x2,y1)C(x2,y2)D(x1,y2)yxFxFxyFyFyDAFBCFxCDFABFyDAFyBCFxCDFxABFdyFdxFdyFdxFrdFyyxxyyxxyyxxADyDCxCByBAx)()()()()()()()()()()()(1221保守力的判据：0;0;0yFxFxFzFzFyFxyzxyzyxFxFxyFyFrdFyyxx)()()()(12210/)()(/)()()()()()(12121221yFxFxxFxFyyFyFyxyxFxFxyFyFSrdFxyyyxxyyxx保守力的判据：.0,0FcurlF即0;0;0yFxFxFzFzFyFxyzxyzkyAxAjxAzAizAyAkAjAiAzkyjxiAxyzxyzkyx)(zkyjxi标量积与矢量积332211BABABABAkBABAjBABAiBABABBBAAAkjiBA)()()(122131132332321321jikikjkjikkjjiiikkjji,,1,0Summaryonpotentialenergy(1)Workdonebyaconservativeforcecanberepresentedintermsofapotentialenergy(2)Potentialenergyisasharedpropertyofthesystem,notoneparticle.(3)ComponentsofforceareInvectorformVVVW0).(zVkyVjxVigradVF.,,zVFyVFxVFzyxKineticenergy(动能)EnergyassociatedwiththemotionisT=1/2mv2AmovingobjecthastheabilitytodoworkAscalarquantityT=1/2mv2)2()1()2()1(221)2()1(221221221)2()1()2()1()2()1()()(TdmvdzmymxmddzzmdyymdxxmdzFdyFdxFrdFzyx.12WTTWork-energytheorem:Theworkdonebythenetforceonaparticleequalsthechangeintheparticle’skineticenergy动能定理:运动质点的动能的增加等于其它物体对它所做的功.KineticenergyT=1/2mv2isalsotheabilitytodowork运动质点速度改变而所作出的功.)('22212121)2()1(221)2()1(21)2()1()2()1()2()1()2()1()2()1(mvmvmvrrmdrdrmdtrrmrdrmrdFrdfW