work and energy - dr. james hedberg, ccny...

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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