M2.6 Energy, Work and Power.ppt

theseiconsindicatethatteacher snotesorusefulwebaddressesareavailableinthenotespage thisiconindicatestheslidecontainsactivitiescreatedinflash theseactivitiesarenoteditable formoredetailedinstructions seethegettingstartedpresentation boardworksltd2006 1of46 as levelmaths mechanics2forocr m2 6energy workandpower contents boardworksltd2006 2of46 energy energyworkconservationofenergywork energyanddissipativeforcespowerexamination stylequestions mechanicalenergy energyismeasuredinjoulesorkilojoules representedbyjorkj mechanicalenergyisenergythataparticlehasduetoeitheritsmotionoritsposition energyexistsinmanyforms theseincludeelectricalenergy chemicalenergy heatenergyandlightenergy mechanicalenergyistheenergythatwillbeconsideredinm2 itcomesintwoforms kineticandpotentialenergy energyisameasureofaparticlescapacitytodowork kineticandpotentialenergy kineticenergykineticenergyistheenergyaparticlepossessesduetoitsmotion kineticenergyisdefinedaswheremismassinkgandvisspeedinms 1 gravitationalpotentialenergygravitationalpotentialenergyistheenergyaparticlepossessesduetoitsposition itisstoredenergy changeingravitationalpotentialenergyisdefinedaswheremismassinkg gisaccelerationduetogravityandhistheverticaldistancetravelledinmetres kineticenergy findthekineticenergyofthefollowing aparticleofmass0 5kgtravellingat8ms 1acarofmass900kgtravellingat15ms 1abulletofmass20gtravellingat500ms 1 k e injoules m v2 0 5 82 16 k e injoules m v2 900 152 101250 k e injoules m v2 0 02 5002 2500 findthegravitationalpotentialenergyofthefollowing statingwhetheritisagainorlossingravitationalpotentialenergy gravitationalpotentialenergy g p e injoules m g h 2 9 8 8 156 8thisisagaining p e g p e injoules m g h 500 9 8 50 245000thisisalossofg p e g p e injoules m g h 80 9 8 75 58800thisisagaining p e aparticleofmass2kgraisedaverticaldistanceof8maliftofmass500kgdescending50mverticallyamanofmass80kgclimbingaverticaldistanceof75m energyquestion1 question1 findthegaininkineticenergyofacarofmass800kgasitacceleratesfrom10ms 1to25ms 1 initialk e m v2 800 102 40000 finalk e m v2 800 252 250000 thereforethegaininkineticenergyofthecaris210000j energyquestion2 question2 achildofmass30kgslidesdownaslideoflength4minclinedatanangleof30 tothehorizontal bymodellingthechildasaparticleandtheslideasasmoothinclinedplane findthelossofgravitationalpotentialenergyofthechild g p e m g h 30 9 8 4sin30 588 thereforethelossing p e is588j contents boardworksltd2006 9of46 work energyworkconservationofenergywork energyanddissipativeforcespowerexamination stylequestions work workcanbedefinedasforce distancei e theworkdonebyaconstantforceistheforce innewtons multipliedbythedistancemoved inmetres inthedirectionoftheforce ifthedirectionoftheforceisdifferenttothedirectionofmotionthen workquestion1 question1 findtheworkdonebyaforceof10nactingonaparticlethatmoves3minthedirectionoftheforce work force distance 10 3 30 thereforetheworkdonebytheforceis30j workquestion2 question2 aparticleofmass2kgispulled3malongasmoothhorizontalsurfacebyaforceof8nactingatanangleof60 tothehorizontal findtheworkdonebytheforce workdone fdcos 8 3 cos60 12 thereforeworkdonebytheforceis12j workquestion3 question3 astoneofmass50gfallsfromthetopofacliffofheight75m findtheworkdonebygravityasthestonefalls workdone force distance 0 05 9 8 75 36 75 thereforetheworkdonebygravityis36 75j workquestion4 question4 aparticleofmass5kgispulled8muparoughplaneinclinedatanangleof60 tothehorizontal thecoefficientoffrictionbetweentheplaneandtheparticleis0 3 findtheworkdoneagainstfriction findtheworkdoneagainstgravity workquestion4 a tofindtheworkdoneagainstthefrictionalforceitisfirstnecessarytofindthefrictionalforce resolvingperpendiculartotheplane r 5gcos60 2 5gf r 0 3 2 5g 7 35 thefrictionalforceis7 35n thedistancetravelledinthedirectionoppositetothefrictionalforceis8m workdoneagainstfriction 7 35 8 58 8j workquestion4 b tofindtheworkdoneagainstgravityitisnecessarytocalculatetheverticaldistancetravelled workdoneagainstgravity 5g 8sin60 339j 3s f sin60 h 8 h 8sin60 contents boardworksltd2006 17of46 conservationofenergy energyworkconservationofenergywork energyanddissipativeforcespowerexamination stylequestions conservationofenergy thetotalmechanicalenergyofaparticleremainsconstantiftheonlyforceactingontheparticleisgravity thetotalmechanicalenergyofaparticleisthesumofanykineticenergyandanypotentialenergy hence foraparticleonwhichtheonlyforceactingisgravity therefore conservationquestion1 question1 aparticleofmass5kgslidesdownasmoothslopeinclinedatanangleof30 tothehorizontal giventhattheparticlestartsfromrest findthedistancetravelledalongtheslopewhentheparticlehasreachedaspeedof4 9ms 1 sincetheonlyforceactingontheparticleisgravity energyisconserved initialk e 0 theparticleisatrest finalk e 5 4 92 60 025 gainink e 60 025j lossing p e 60 025j conservationquestion1 changeing p e 5 9 8 h 60 025 thereforethedistancemovedalongtheslopeis2 45m conservationquestion2 question2 aparticleofmass2kgisprojecteduparoughplaneinclinedatanangleof45 tothehorizontal thecoefficientoffrictionbetweentheparticleandtheplaneis0 15 iftheparticletravels5muptheplanebeforecomingtorest findthespeedwithwhichtheparticlewasprojected sincetheonlyforceactingontheparticleisgravity energyisconserved h 5sin45 thereforetheverticaldistancetravelledis3 54m 3s f tocalculatethegaining p e weneedtocalculatetheverticalheightgained conservationquestion2 gaining p e mgh 2 9 8 5sin45 69 3 3s f asenergyisconserved gaining p e lossink e initialk e 2 v2finalk e 0 particlecomestorest 2 v2 69 3 v 8 32 3s f thereforetheparticleisprojecteduptheplanewithaspeedof8 32ms 1 contents boardworksltd2006 23of46 work energyanddissipativeforces energyworkconservationofenergywork energyanddissipativeforcespowerexamination stylequestions work energyprinciple therelationshipbetweenworkandenergyisasimpleone thiscanbeseenfromthefollowingapplicationofnewton ssecondlaw aparticleofmassmkgismovedadistanceofdmhorizontallybyaconstanthorizontalforcefn lettheinitialspeedbeums 1andthefinalspeedbevms 1 work energyprinciple bynewton ssecondlaw f ma a f m sincetheforceisassumedconstant theaccelerationcanalsobeassumedconstant usingv2 u2 2asv2 u2 2 f m d v2 u2 mv2 mu2 fd work energyquestion1 question1 aparticleofmass3kgispulled6mbyahorizontalforceof10nacrossasmoothhorizontalsurface iftheparticlestartedfromrest finditsspeedwhenithastravelled6m workdone fd 10 6 60fd mv2 mu2 workdone mv2 mu2 60 3 v2 0 601 5v2 60 v 6 32 3s f thereforethespeedoftheparticlewhenithastravelled6mis6 32ms 1 work energyquestion2 question2 findthehorizontalforcethatcausesaparticleofmass2 5kgtoincreasespeedfrom2ms 1to5ms 1overadistanceof10monasmoothhorizontalsurface workdone changeinkineticenergyinitialk e 2 5 22 5finalk e 2 5 52 31 25 changeinkineticenergy 26 25 workdone fd 10f 26 25 f 2 625 thereforetherequiredforceis2 625n workanddissipativeforces workdoneagainstadissipativeforceisequaltothelossofthetotalmechanicalenergyofthesystem adissipativeforceisaforcewhichcausesaparticletoloseenergy frictionandairresistanceareexamplesofdissipativeforces thismeansthattheprincipleofconservationnolongerapplies asthetotalmechanicalenergyoftheparticleisnotconstant workanddissipativeforces example aparticleofmass3kgismovingdownaroughslopeinclinedatanangleof30 tothehorizontal theparticlepassesapointaat3ms 1andapointb2mdowntheslopeat4ms 1 calculatetheworkdoneagainstfriction gainink e 3 42 3 32 10 5j lossing p e 3 g 1 29 4j sin30 h 2 lossofenergy 29 4 10 5 18 9 workdoneagainstfriction 18 9j h 2sin30 1 contents boardworksltd2006 30of46 power energyworkconservationofenergywork energyanddissipativeforcespowerexamination stylequestions power powerisdefinedastherateofdoingwork withrespecttotime thes i unitofpoweristhewatt whichisarateofworkof1joulepersecond thewattisarelativelysmallunit youwillmorecommonlymeetthelargerunit thekilowatt 1kilowatt abbreviatedto1kw 1000watts 1000joulespersecond poweriscalculatedby power consideraforcefactingforasmalltimeintervaltoverasmalldistanced workdone f d so power so thepowerofanengineormachinemovingataspeedvsubjecttoadrivingforcefisfv power fv powerquestion1 question1 acarexertsadrivingforceof1500n ifthecaristravellingataconstantspeedof35ms 1 calculatethepoweroftheengine power inwatts fv 1500 35 52500 thereforethepowerproducedbythecarengineis52 5kw powerquestion2 question2 theengineofacarisgeneratingpowerof18kw findthedrivingforceproducedwhenthecaristravellingataspeedof a 12ms 1b 20ms 1c 30ms 1 forceproduced 18000 12 1500nb forceproduced 18000 20 900nc forceproduced 18000 30 600n power force velocity so powerquestion3 question3 acyclististravellingalongahorizontalroadataconstantspeedof7ms 1 ifthecyclistexperiencesatotalresistancetomotionof20n findthepowerproducedbythecyclist sincethespeedisconstantthereisnoacceleration therefore drivingforce resistanceforce drivingforce 20nvelocity 7ms 1 power 20 7 140w contents boardworksltd2006 36of46 examination stylequestions energyworkconservationofenergywork energyanddissipativeforcespowerexamination stylequestions examquestion1 examination stylequestion1 acarofmass1000kgmovesalongastraight horizontalroad thecarengineisworkingataconstantrateof45kwandthetotalresistancetothemotionofthecaris500n a findtheaccelerationofthecarwhenitsspeedis15ms 1 thecarcomestoahillwhichisinclinedatanangleof tothehorizontal wheresin 0 1 theresistancetothemotionofthecarisunchanged b findthemaximumspeedofthecarupthehillwhenitisworkingatarateof45kw examquestion1 power force velocity applyingnewton ssecondlaw f ma3000 500 1000a1000a 2500 a 2 5 thereforetheaccelerationofthecarwhenitistravellingat15ms 1is2 5ms 2 a tocalculatetheaccelerationitisfirstnecessarytocalculatethedrivingforce so examquestion1 applyingnewton ssecondlawuptheslope d 500 1000gsin 0d 500 1000g 0 1 1480 thedrivingforceofthecaris1480n power force velocity b beforecalculatingthemaximumspeedofthecaritisfirstnecessarytocalculatethedrivingforce themaximumspeedofthecaris30 4ms 1 examquestion2 examination stylequestion2 astoneofmass0 5kgisslidingdownaroughplaneinclinedatanangleof15 tothehorizontal thestonepassesapointawithaspeedof10ms 1andapointbwithaspeedof6ms 1 a findthelossofmechanicalenergyofthestoneasitmovesfrompointatopointb b calculatethecoefficientoffrictionbetweenthestoneandtheplane examquestion2 a lossofenergyofthestone lossofk e lossofg p e initialk e 0 5 102finalk e 0 5 62lossofk e 25 9 16j lossofg p e 0 5 9 8 10sin15 12 7 3s f 16 12 7 28 7 h 10sin15 thereforethelossofenergyofthestoneis28 7j examquestion2 r 0 5gcos15 4 73 3s f b tocalculatethecoefficientoffriction itisnecessarytocalculateboththenormalcontactforceandthefrictionalforce thecoefficientoffrictionis0 61 lossofenergyofthestonefoundinparta isequaltotheworkdone