work and energy - dr. james hedberg, ccny...
TRANSCRIPT
JamesPrescottJoule1818-1889
K
Ech
Eth
Usystem
environment
heat, work
Work and Energy
1.Introduction2.Work3.KineticEnergy4.PotentialEnergy5.ConservationofMechanicalEnergy6.Ex:TheLooptheLoop7.ConservativeandNon-conservativeForces8.Power9.VariableandConstantForces
Introduction
Thisiswordthatmeansalotofthingsdependingonthecontext:
1.EnergyConsumptionofaHousehold2.EnergyDrinks3.Auras&SpiritualEnergy4.RenewableEnergy
Energy: basics
Energyisascalarquantity.IthasSIunitsof:
TheunitiscalledaJoule,afterMr.Joule1Joule=1Newton×1meter
Thedifferentenergiesofasystemcanbetransferredbetweeneachother.
Work
Again,wemighthavewordproblemshere.Forus,Workwillbedefinedastheprocessoftransferringenergybetweenawell-definedsystemandtheenvironment.Theenergymaygofromthesystemtothe
environment.Or,itmaygofromtheenvironmenttothesystem
An example of work
Aconstantforceisappliedthisboxasitmovesacrossthefloor.Theworkdoneisequalgivenby: .Thisisforthecaseofaconstantforcewhichisappliedinthesamedirectionastheboxismoving.
Butwhatiftheforceisnotintheexactsamedirection.
Energyisanotherexampleofawordthatweuseveryofteninregularspeech.However,inphysicsithasaspecificmeaning.
kg ⋅ /m2 s2
W = Fd
PHY 207 - energy - J. Hedberg - 2017
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Nowwehavetoconsiderthecomponentoftheforcewhichdoespointinthedirectionofthedistancetraveled.
Whatistheworkbeingdoneonthebagbythepersoncarryingit?
Quick Question 1Iswingaballaroundmyheadatconstantspeedinacirclewithcircumference3m.Whatistheworkdoneontheballbythe10Ntensionforceinthestringduringonerevolutionoftheball?
Kinetic Energy
Wesaidtherewemanydifferentformsenergycantake(chemical,thermal,potential,etc)KineticEnergyistheenergyassociatedwithmovingobjects.
Someexamplekineticenergies
1.Antwalking: J2.Personwalking: J3.Bullet:5000J4.Car@100kph~ J5.Fasttrain~ J
Quick Question 2Twoballsofequalsizearedroppedfromthesameheightfromtheroofofabuilding.Oneballhastwicethemassoftheother.Whentheballsreachtheground,howdothekineticenergiesofthetwoballscompare?(assumenoairresistance)
Work and Kinetic Energy
W = d = F⋅ d = Fd cosθF∥
a)30Jb)20Jc)10Jd)0.333Je)0J
KE = m12
v2
1 × 10−8
1 × 10−5
5 × 105
1 × 1010
a)Thelighteronehasonefourthasmuchkineticenergyastheotherdoes.b)Thelighteronehasonehalfasmuchkineticenergyastheotherdoes.c)Thelighteronehasthesamekineticenergyastheotherdoes.d)Thelighteronehastwiceasmuchkineticenergyastheotherdoes.e)Thelighteronehasfourtimesasmuchkineticenergyastheotherdoes.
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final position
initial position
final position
initial position
Sincetheworkisrelatedtotheamountofenergyeitherenteringorleavingasystem,wecanestablisharelationshipbetweentheworkdoneonanobjectandthekineticenergy.
Potential Energy
Wecanstoreenergyinagravitationalsystembyseparatingthetwoobjects.Thecat,whenliftedofftheground,hasagravitationalpotentialenergy.
PotentialEnergyisonlydependentontheheight!Itdoesn’tmatterthepathanobjecttakeswhileclimbinghigherawayfromtheearth.Anyroutewhichendsatthesamepointwillimpartthesameamountofgravitationalpotentialenergytotheobject.
Work and Potential Energy
Changingthepotentialenergyofanobjectrequireswork.
Inthecaseofgravity:"Thechangeinthegravitationalpotentialenergyofanobjectisequaltothenegativeoftheworkdonebythegravitationalforce"
Gravity does positive work
Inthiscaseweletanobjectdrop,andgravitydoespositivework.
Gravity does negative work
Inthiscase,weraiseanobject,andgravitydoesnegativework.
W = K − K = m − mEf E012
v2f
12
v20
= mghUg
ΔU = −W
Δ = −UG WG
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Quick Question 3Abeamisbeingraisedbyaconstructioncranewithanaccelerationupwards(+ydirection).Calltheworkdonebythecabletension: andtheworkdonebyGravity: .Whichofthefollowingistrue:
Quick Question 4Abeamisbeingloweredbyaconstructioncraneaconstantvelocity.Calltheworkdonebythecabletension: andtheworkdonebyGravity: .Whichofthefollowingistrue:
Work - Energy
Let'snowtakealookatsystemswhichhavebothkineticandpotentialenergy.Forexample:thisboxwhichislocatedatadistanceabovetheground.
Usually,we'llcalltheground .
Ifthesearetheonlytwotypesofenergyinthesystem,thenwehaveaveryspecialsituation.Wecandefinethetotalmechanicalenergyenergyofthesystemasthekinetic+potential.
Conservation of Mechanical Energy
Ifwedon'tworryaboutanyofthoseretardingforceslikefrictionorairresistance,thenwecansaythatthetotalmechanicalenergyofasystemremainsthesameastimeadvances.
or
Upondroppingaball,thetotalmechanicalenergyisconserved.
Knowingthisconservationlaw,wecanuseeither tofind ,or tofind .
WT WG
a) > 0 & > 0WT WG
b) > 0 & < 0WT WG
c) < 0 & > 0WT WG
d) < 0 & < 0WT WG
e) = 0 & = 0WT WG
WT WG
a) > 0 & > 0WT WG
b) > 0 & < 0WT WG
c) < 0 & > 0WT WG
d) < 0 & < 0WT WG
e) = 0 & = 0WT WG
h
h = 0
= KE + UEmechanical
=Emechi
Emechf
m + mg = m + mg12
v2i
hi
12
v2f
hf
= constant = KE +Emech Ugrav
h v v h
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h
1
2
m
2h
h
1
2
m
h = 0
h
-h
1
2
m
h = 0
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30ºL
L
h
Quick Question 5Theblockstartsfromrestatpoint1.Usingtheconservationofmechanicalenergy,determinethespeedoftheblockatpoint2.
Quick Question 6Theblockstartsfromrestatpoint1.Usingtheconservationofmechanicalenergy,determinethespeedoftheblockatpoint2.
Quick Question 7Theblockstartsfromrestatpoint1.Usingtheconservationofmechanicalenergy,determinethespeedoftheblockatpoint2.
A car on a hill.
Quick Question 8If =4meters,howhighistheball?Thatis,findthedistance .
Block on a ramp
a)v = (2gh)2
b)v =gh√
2m
c)v = 2gh− −−
√d)v = 0
a)v = (4gh)2
b)v =gh√
4m
c)v = 2gh− −−
√d)v = 4gh
− −−√
a)v = (4gh)2
b)v =gh√
4m
c)v = 2gh− −−
√d)v = 4gh
− −−√
L h
Example Problem #1:
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h
Here'sa5kgblockonaramp,where ,andh=20meters.Findtheworkdonebygravityandthefinalspeed.
θ = 25∘
Wewanttocalculatetheworkdonebygravity:
So,weneedtodecomposethetwovectorsintocomponents.Let'scallthehorizontaldirection+xtotheleft,andthevertical+yup.Thus,inunitvectornotion,theforceofgravitywillbe:
andthedisplacementwillbe:
Now,wecanexecutethedotproductusingunitvectors:
Since and thisdotproductsimplifiesto:
whichwillbetheworkdonebygravity.Noticehowthatissameasifwejustlettheboxfallfromadistance abovetheground.Thissituationoccursbecausetheforceofgravityisaconservativeforce,i.e.itispathindependent.
Tofindthespeedatthebottomoftheramp,wecanusethework-energytheoremwhichrelatestheworkdonetothechangeinkineticenergy:
Initially,theboxisatrest,sothereisnokineticenergy:
Thekineticenergyatthebottomoftherampwillbegivenby:
Thus,the willbe Settingthisequaltotheworkdonethatwecalculatedabove:
allowsustosolvefor :
Thisresultshouldlookfamiliar.
Notice:weuseregular,non-rotatedcoordinatesforthisproblem.Tryitwithrotatedaxesandseewhatyouget.Shoulditbedifferent?
W = ⋅ dFG
= −mgFG j
d = − hh
tan θi j
W = ⋅ d = [−mg ] ⋅ [ − h ]FG jh
tan θi j
⋅ = 1i i ⋅ = 0i j
W = mgh
h
ΔKE = Workdone
K = 0Ei
K = mEf
12
v2
ΔKE m12
v2
mgh = m12
v2
v
v = 2gh− −−
√
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r
v
vtop
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success failAtthetopoftheloop,thesumofforcespointingtowardsthegroundwillbegivenby:
Ex: The Loop the Loop
Wesawthatifthevelocityoftheobjectwaslargeenough,thenitwouldremaininsideincontactwiththeloop.
Asuccessfulloop-the-looplookslikethis.Theballremainsinphysicalcontactwiththeroadatalltimes.(Thisisanotherwayofsayingthenormalforceisnotzero.)
Quick Question 9Aftertheballloosescontactwiththeloop,whichofthefollowingdottedlinesshowsthemostlikelytrajectory?
Anunsuccessfulloop-the-looplookslikethis.Theballloosesphysicalcontactwiththeroadatalltimes.(Thisisanotherwayofsayingthenormalforcebecomeszero.)
Inthecasewherethevelocityisnotfastenoughtogettheballallthewayaroundtheloop,theballloosescontactwiththeroadnearthetop.
∑ = w + = maFground FN
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Sincethisiscircularmotion,wehave
Thuswecanwrite:
r
vtop
h
Quick Question 10Whatistheminimumvelocityatthetopoftheloopneededtocompletetheloop-the-loopforaregular,nofriction,slidingobject.
Quick Question 11Whatisanexpressionfor thatwouldleadtoaspeedsotheballmaintainscontactatalltimes?
Conservative and Non-conservative Forces
Mostoftheforceswehaveseencanbeclassifiedasnon-conservative.TheexceptionisGravity.
ma = mv2
r
+ = mmg weight
FN
v2
r
a)v > rg−−√b)v > (rg)2
c)v > rg√
d)v > ∞e)v > 2rg
−−−√
h
a)h > 5r
b)h > 3r
2c)h > 3r
d)h > 5r
2
Herearetwodefinitionsforaconservativeforce:
1.AforceisconservativeiftheworkitdoestomoveanobjectfrompointAtopointBdoesnotdependontheroutechosen(akathepath)togetfrompointAtopointB.
2.Aforceisconservativeifthereiszeronetworkdonewhilemovinganobjectaroundaclosedloop,thatis,aroundapaththathasthesamepointforthebeginningandtheend.
ExamplesofConservativeForcesGravitationalforceElasticspringforceElectricforce
andNon-conservativeforces:StaticandkineticfrictionalforcesAirresistanceTensionNormalforce
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h
ShownaretwopathsthataboxcantaketogetfrompointAtopointBIfwelookatthechangeinpotentialenergy, ,thenwe'llseethattheWorkdonebygravityontheboxineithercaseisequaltozero.And,foranyotherpathyoucanimagine,it'salsozero.
Movingaboxalong2pathsinthepresenceoffriction.Thesituationisverydifferentifweaskabouttheworkdoneagainstfrictionduringthesetwopaths.Path2willrequiremoreworkthanpath1.
Andso,forceslikefrictionandairresistanceareconsiderednon-conservative.Wecanhoweveraccountfortheseforcesinouranalysisofcertainsystems.
Ifthechangeinenergyisnon-zero,bewteentheinitialandfinaltimesofamechanicalsystem,thentheremustbesomenon-conservativeforcesdoingworkduringthemotion.
Now,wehavearampthathassomefriction.
Howcanwedescribethemotionofaboxonthisramp?
ΔUgrav
W = −Δ = mg(h− h) = 0UG
Hereisanexampleofaconservativeforceinteractingwithabox.IfweslidetheboxfrompointAtoBalongpath1,wecancalculatetheworkdonebyfiguringouttheworkdonebygravitytogofromthegrounduptotheapex,andthenbackdowntotheground.Addingthesetwoworkstogetherwillresultinzero,sinceonthewayup,gravityispointingagainstthedisplacementvector:[ )]andonthewaydown,gravitywillbeinthesamedirectionasthedisplacementvector:[ ].Thus,thesetwovaluesareequalinmagnitudebutoppositeinsignandsowhenthatareaddedtogetherwillbeequaltozero.Inthecaseofpath2,theworkdonebygravitywillbezeroalso,sincethereisnochangeinheightalongthepath.Thus,nomatterhowwechosetogofromtheAtoB,thenetworkdonebygravitywillbezero.Thisisthedefinitionofaconservativeforce.
d cos(FG 180∘
d cos(FG 0∘
Now,ifyouwanttomovetheboxfromAtoB,butareaskingabouttheworkdonebyfrictionontheboxduringthemotion,theshorterpath(path1)willresultinlesswork.Sincetheforceoffrictionisalwaysoppositetothedirectionofmotion,therewillneverbethecaseofcancelingoutlikewehadinthegravitationalcase.Path2willleadtomoreworkbeingdonebykineticfriction,whichisthereforeconsideredanon-conservativeforce.
= −WNC Ef E0
Example Problem #2:
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Achild,startingfromrest,slidesdownaslide.Onthewaydown,akineticfrictionalforce(anonconservativeforce)actsonher.Thechildhasamassof50.0kg,andtheheightoftheslideis18m.Ifthekineticfrictionalforcedoes Joulesofwork,howfastisthechildgoingatthebottomoftheslide?
Iftheobjectslidesdownahillwithanelevationof10m,wherewillitstop?Considertheslopedpartfrictionlessandtheroughpartatthebottomhavinga equalto0.3.
10 m µ = 0
µ = .3
Power
Again,here'sanotherwordwehavetobeverycarefulwith.Wemightbeusedtocallinganythingthatseemsstrong'morepowerful'.Forushowever,we'llneedtorestrictouruseofthewordpowertodescribehowfastaprocessconvertsenergy.
TheSIunitofpoweriscalledthewatt:
AunitofpowerintheUSCustomarysystemishorsepower:1hp=746WUnitsof[power-time]canalsobeusedtoexpressenergy(e.g.kilowatt-hour)
Weshouldbeabletounderstandthesemysteriousdocumentsnow.HowmanyJoulesofenergydidIuse?HowmanypushupswouldIhavetodotogeneratethatmuchenergy?
Example Problem #3:
6.50 × 103
Example Problem #4:
μk
Power = P =changeinenergy
time
1watt = 1joule/second = 1kg ⋅ /m2 s3
1kWh = (1000W)(3600s) = 3.6 × J106
Example Problem #5:
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F
∆x
F
∆x
F
∆x
F
∆x
Iusedtoworkoutonarowingmachine.EachtimeIpulledontherowingbar(whichsimulatestheoars),itmovedadistanceof1.1minatimeof1.6s.Therewasalittledisplaythatshowedmypoweranditsaid90Watts.HowlargewastheforcethatIappliedtotherowingbar?
Agokartweighs1000kgandacceleratesat4m/s2for4seconds.Whatistheaveragepower?
Variable and Constant Forces
Inthemostbasicsituation,theforceappliedtoanobjectwasconstant.Whenthisisthecase,theWorkwasgivenby:
Thisisalsoequaltothe"areaunderthecurve"
Work done by a variable force
However,it'salsopossiblethattheworkwillvaryasafunctionofdistance.Inthiscase,theworkwillnotbegivenbythestandardform,because,theFisnotconstant.
However,theworkisstillequaltothe"areaunderthecurve"
Work done by a variable force
Wedon'thavethetools(i.e.integration)tobeabletodomuchwiththisatthislevelofphysics.
Work done by a variable force
Todealwiththesesituations,we'llhavetousesomeintegrationtechniques.Here'sanappliedforcethatvariesasthesquareofthedistance.Wecouldfigureouttheareaunderthecurve,whichdoesequalthework,butperformingtheintegral
Inthiscase:
Example Problem #6:
W = F⋅ d
W ≠ F⋅ d
W = F(x)dx∫ xf
xi
W = dx =∫ xf
xi
x2 13
x3∣∣∣xf
xi
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In the case of 3d:
Ifourforcesarevariableand3dimensional,(andperhapsnotpointeddirectlyinthedirectionofthedisplacement),thenwehavetousesomemoredevelopedvectorcalc:
Or,ifweintegrateoverthedistance:
Sinceworkisascalar,thenthesetermswilljustaddupwithoutanycomplications.
Recoverthework-kineticenergyequation?
Weshouldbeabletoshowthatthisgetsbacktokineticenergysomehow...
Basedontheabove:
Nowwecantakethederivativewithrespecttox:
Drawplotsofthegravitationalpotentialenergyandthegravitationalforceforanobjectasafunctionofheightabovetheearth.Consideronlysmalldistancesabovethesurfaceoftheearth.
Iftheforcefromanelasticbandwhenstretchedisgivenby ,whatwouldthechangeinpotentialenergybeforanobjectattachedtotherubberbandandpulledbackbyadistanced?(kisaconstantthatisdeterminedbytherubberbandmaterial)Plottheforceandasfunctionsofdistance.
dW = F⋅ dr = dx+ dy+ dzFx Fy Fz
W = dW = dx+ dy+ dz∫ rf
ri
∫ xf
xi
Fx ∫ yf
yi
Fy ∫ zf
zi
Fz
W = F(x)dx∫ xf
xi
ΔU(x) = −W = −FΔx
F(x) = −dU(x)
dx
Example Problem #7:
Example Problem #8:
F = −kx
Urubberband
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U
∆x
A
B
C
DE
F
0.5 1.0 1.5 2.0 2.5 3.0
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
0.5 1.0 1.5 2.0 2.5 3.0
-1.0
-0.5
0.0
0.5
1.0
Quick Question 12Hereisthepotentialenergyofanunknownasafunctionofdistancealongthex-axis.Wherewouldthatforcebethelargest,andpointedinthepositivedirection?
Adiatomicmoleculehastwoatoms(H2forexample).Ifthepotentialenergyofsuchasystemisgivenby:
Findtheequilibriumseparationofsuchamolecule.Thatis,findadistance thatwillleadtozeronetforceoneachatom.(AandBarejustconstants)
HereareplotsofthePotentialfunction,anditsderivative(i.e.force).
Force Microscopy
Moreinfo:IntrotoAFM
Emmy
EmmyNoetherwasatheoreticalphysicistwhodidpioneeringworkinthestudyofconservationprinciples.
Example Problem #9:
U = −A
r12
B
r6
r
U = −1
r12
1
r6
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23March1882–14April1935
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