《物理双语教学课件》Chapter 4 Work and Energy 功和能.docx

《物理双语教学课件》Chapter 4 Work and Energy 功和能.docx

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《物理双语教学课件》Chapter4WorkandEnergy功和能

Chapter4WorkandEnergy

Theconceptofenergyisoneofthemostimportantintheworldofscience.Ineverydayusage,thetermenergyhastodowiththecostoffuelfortransportationandheating,electricityforlightsandappliances,andthefoodsweconsume.However,theseideasdonotreallydefineenergy.Theytellusonlythatfuelsareneededtodoajobandthatthosefuelsprovideuswithsomethingwecallenergy.

EnergyispresentintheUniverseinavarietyofforms,includingmechanicalenergy,chemicalenergy,electromagneticenergy,heatenergy,andnuclearenergy.Althoughenergycanbetransformedfromoneformtoanother,thetotalamountofenergyintheUniverseremainsthesame.Ifanisolatedsystemlosesenergyinsomeform,thenbytheprincipleofconservationofenergy,thesystemmustgainanequalamountofenergyinotherform.Thetransformationofenergyfromoneformintoanotherisanessentialpartofthestudyofphysics,chemistry,biology,geology,andastronomy.

Inthischapterweareconcernedonlywithmechanicalenergy.Weintroducetheconceptofkineticenergy,whichisdefinedastheenergyassociatedwithmotion,andtheconceptofpotentialenergy,theenergyassociatedwithposition.WeshallseethattheideasofworkandenergycanbeusedinplaceofNewton’slawtosolvecertainproblems.

4.1WorkandPower

1.WorkWdonebyaconstantforceisdefinedastheproductofthecomponentoftheforcealongthedirectionofdisplacementandthemagnitudeofthedisplacement.

Wheretheforcemakesanangleof

withdisplacement

TheSIunitofworkisthejoule（J）,namedforJamesPrescottJoule,anEnglishscientistofthe1800s.Itisderiveddirectlyfromtheunitsformassandvelocity:

1joule=1J=1（kg）（m/s2）（m）=1kgm2/s2

2.Workdonebyavariableforce

（1）.TheincrementofworkdWdoneontheparticlebyFduringthedisplacementdris

whereForceFisfunctionofitsposition.

（2）.TheworkWdonebyFwhiletheparticlemovesfromaninitialpositionatoafinalpositionbisthen

（3）.Weusethecomponentsof

and

toexpresstheforceanddisplacement,thenwehave

3.Workdonebymultipleforces:

Ifthereareseveralforcesactonaparticle,wecanreplaceFinaboveequationwiththenetforce

where

where

aretheindividualforces.Then

4.Power

（1）.Therateatwhichworkisdonebyaforceissaidtobethepowerduetotheforce.IfanamountofworkWisdoneinatimeinterval

byaforce,thentheaveragepowerduetotheforceis

.

（2）.TheinstantaneouspowerPistheinstantaneousrateofdoingwork,whichcanbewrittenas

.

（3）.TheSIunitofpoweristhejoulepersecond.Thisunitisusedsooftenthatithasaspecialname,thewatt（W）,afterJamesWatt,whogreatlyimprovedtherateatwhichsteamenginescoulddowork.

4.2KineticEnergyandWork-KineticEnergyTheorem

Energyisascalarquantitythatisassociatedwithastateofoneormoreobject.Thetermstateherehasitscommonmeaning:

itistheconditionofanobject.

1.KineticenergyKisassociatedwiththestateofmotionofanobject.Thefastertheobjectmoves,thegreaterisitskineticenergy.Foranobjectofmassmandwhosespeedviswellbelowthespeedoflight,wedefinekineticenergyas

TheSIunitofkineticenergyisthesameaswork—joule.

Aconvenientunitofenergyfordealingwithatomsorwithsubatomicparticlesistheelectron-volt（eV）.

1electron-volt=1eV=1.60x10–19J.

2.Work-kineticenergytheorem:

Ifaforcechangesthespeedofanobject,italsochangesthekineticenergyoftheobject.Ifthekineticenergyistheonlytypeofenergyoftheobjectbeingchangedbytheforce,thenthechangeinkineticenergyisequaltotheworkWdonebytheforce:

Here

istheinitialkineticenergy（=

）and

isthekineticenergy（

）aftertheworkisdone.

3.Wecanprovework-kineticenergytheoremasfollow:

（Numeratoranddenominatorofafraction）

4.3Workdonebyweightandbyaspringforce

1.Workdonebyweight:

Wecanfindthattheworkdonebyweightonaparticlebetweentwopointsdoesnotdependonthepathtakenbytheparticle.Ornomatterwhatpathwechoosetomovetheparticle,theworkdonebyitsweightisthesame.Inotherwordifwemoveaparticlearoundaclosedpath,theworkdonebyweightontheparticleiszero.

2.Workdoneonaparticle-likeobjectbyaparticulartypeofvariableforce,namely,springforce—theforceexertedbyaspring.

3.ConservativeforceandNon-conservativeforce:

Iftheworkdonebytheforceisindependentofthepaththeparticlemoves,theforceisaconservativeforce;otherwiseanon-conservativeforce.Weightandspring-forceareconservativeforces;frictionisanon-conservativeforce.

4.4Potentialenergy

PotentialenergyUisenergythatcanbeassociatedwiththeconfiguration（orarrangement）ofasystemofobjectsthatexertaforceononeanother.Iftheconfigurationofthesystemchanges,thenthepotentialenergyofthesystemalsochanges.

1.Weknowthatworkdonebyaconservativeforcehasnothingtodowiththepaththeparticletaken.Sowecanintroduceaquantitywhichisthefunctionofthestateofthesystemtoindicatethiskindofnatureforaconservativeforce.Wecallitpotential.

2.GravitationalPotentialEnergy

（1）.Theworkdonebyweightcanbeexpressedas:

where

isthechangeinthegravitationalpotentialenergy.Sincetheworkdonebyweighthasdefinitemagnitudefromaninitialpositiontoafinalposition,soonlyachange

ingravitationalpotentialenergyisphysicallyimportant.

（2）.However,tosimplifyacalculationoradiscussion,wecansaythatacertaingravitationalpotentialUisassociatedwithanygivenconfigurationofthesystem,withtheparticleatagivenheighth.Todoso,werewritetheaboveequationas:

Thenwetake

tobethegravitationalpotentialenergyofthesystemwhenitisinareferenceconfiguration,withtheparticleatareferencepoint

.Usually,weset

and

thenwehave

.Sothegravitationalpotentialenergyassociatedwithaparticle-Earthsystemdependsontheheighthoftheparticlerelativetothereferencepositionofhi=0,notthehorizontalposition.

3.ElasticPotentialEnergy

（1）.Similartoaparticle-Earthsystem,theworkdonebythespringforcecanberewrittenas

.

（2）.ToassociatedapotentialenergyUwithanygivenconfigurationofthesystem,withtheblockatpositionx,wesetthereferencepointfortheblockasxi=0,whichisalwaysattheequilibriumpositionoftheblock.AndwesetthecorrespondingelasticpotentialenergyofthesystemasUi=0.Thuswehave

.

Attention:

（1）.Potentialenergybelongstothewholesystem.

（2）.Themagnitudeofpotentialenergydependson

thechoiceofthereferencepoint.

4.5Work-EnergyTheoremandConservationofMechanicalEnergy

1.Work-kineticenergytheoremforoneparticle:

Wehave

2.Work-EnergyTheorem:

SupposewehaveparticlesofNinthesystemwediscussedandweusework-kineticenergytheoremforeachparticle,thenwehavetotalamountofNequationslikethose:

Wecandividetheforcesexertedoneveryparticleintoexternalforcesandinternalforces.Andtheinternalforcescanalsobeclassifiedasconservativeinternalforcesandnon-conservativeforces.Summingthetwosideofaboveequations,wewillhave:

Ortobeexactly,

Wealsoknowthattheworkdonebytheconservativeinternalforcecanbewrittenastheminusdifferenceofpotentialenergy.SowegettheWork-EnergyTheorem:

Theworkdonebyexternalforcesandnon-conservativeinternalforcesinagivensystemisexactlyequaltothedifferenceofitsMechanicalEnergy.

3.ConservationofMechanicalEnergy:

Whenonlyconservativeforcesactwithinasystem,thekineticenergyandpotentialenergycanchange.However,theirsum,themechanicalenergyEofthesystem,doesnotchange.

4.6ReadingaPotentialenergycurve

Consideraparticlethatispartofasysteminwhichaconservativeforceacts.Supposethattheparticleisconstrainedtomovealonganxaxiswhiletheconservativeforcedoesworkonit.

1.FindingtheForceAnalytically:

Forone-dimensionalmotion,theworkWdonebyaconservativeforcethatactsonaparticleastheparticlemoves,andthepotentialenergyhavetherelationasfollow

Socangettheforcefromthepotentialenergy

Wecan,forexample,checkthisresultbyputting

whichistheelasticpotentialenergyfunctionforaspringforce.Aboveequationyields

asexpected.

2.ThePotentialEnergyCurve:

ThefollowingfigureisaplotofapotentialenergyfunctionU（x）forasysteminwhichaparticleisinone-dimensionalmotionwhileaconservativeforceF（x）doesworkonit.WecaneasilyfindF（x）bytakingtheslopeoftheU（x）curveatvariouspoints.Fig.（b）isaplotofF（x）foundinthisway.

3.TuningPoint:

Sincethereisonlyconservativeforceactingontheparticle,thesystemwillremainconservationofitsmechanicalenergy.Sowehave

.Sincekineticenergy

isnotlessthanzero.Astheparticlemovesfrom

to

（Fig.a）,whentheparticlereaches

itskineticenergyiszero,meanwhiletheforceontheparticleispositive.Itmeanstheparticledoesnotremainat

butinsteadbeginstomovebacktotheright.Hence

isatuningpoint,aplacewhereK=0andtheparticlechangesdirectionofitsmotion.

4.EquilibriumPoints:

Neutralequilibrium

Unstableequilibrium

Stableequilibrium