HW8-S10 Physics mastering physics solution

ManagethisAssignment: Print Version with Answers

HW8-S10Due: 1:00am on Thursday, March 18, 2010Note: To understand how points are awarded, read your instruc tor's Grading Policy.

ShootingaBlockupanIncline

Description:Shootanobjectupanincline(withfriction)usingaspring-gun.Useenergyconservationtofindthedistancetheobjecttravelsuptheincline.

Ablockofmass isplacedinasmooth-boredspringgunatthebottomoftheinclinesothatitcompressesthespringbyanamount .Thespringhasspringconstant .Theincline

makesanangle withthehorizontalandthecoefficientofkineticfrictionbetweentheblockandtheinclineis .Theblockisreleased,exitsthemuzzleofthegun,andslidesupan

inclineatotaldistance .

PartA

Find ,thedistancetraveledalongtheinclinebytheblockafteritexitsthegun.Ignorefrictionwhentheblockisinsidethegun.Also,assumethattheuncompressedspringisjustat

thetopofthegun(i.e.,theblockmovesadistance whileinsideofthegun).Use forthemagnitudeofaccelerationduetogravity.

HintA.1 Howtoapproachtheproblem

ThisisanexampleofaproblemthatwouldbeverydifficultusingonlyNewton'slawsandcalculus.Instead,usetheWork-EnergyTheorem: ,where is

thefinalenergy, istheinitialenergy,and istheworkdoneonthesystembyexternalforces.Letthegravitationalpotentialenergybezerobeforethespringisreleased.

Then, isthepotentialenergyduetothespring, isthepotentialenergyduetogravity,and istheworkdonebyfriction.Onceyou'vesetupthisequation

completely,solvefor .

HintA.2 Findtheinitialenergyoftheblock

Findtheinitialenergy oftheblock.Takethegravitationalpotentialenergytobezerobeforethespringisreleased.

HintA.2.1 Potentialenergyofacompressedspring

Recallthatthepotentialenergy ofaspringwithspringconstant compressedadistance is .

Expressyouranswerintermsofparametersgivenintheproblemintroduction.

ANSWER: =

HintA.3 Findtheworkdonebyfriction

Find ,theworkdonebyfrictionontheblock.

HintA.3.1 Howtocomputework

Theworkdonebyaforceactingalongthedirectionofmotionofanobjectisequaltothemagnitudeoftheforcetimesthedistanceoverwhichtheobjectmoves.Workisnegativeiftheforcedirectlyopposesthemotion.

Express intermsof , , , ,and .

ANSWER: =

HintA.4 Findthefinalenergyoftheblock

MasteringPhysics: Assignment Print View http://session.masteringphysics.com/myct/assignme...

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Findanexpressionforthefinalenergy oftheblock(theenergywhenithastraveledadistance uptheincline).Assumethatthegravitationalpotentialenergyoftheblockis

zerobeforethespringisreleasedandthattheblockmovesadistance insideofthegun.

HintA.4.1 Whatformdoestheenergytake?

Whentheblockstopsslidinguptheramp,allofitsenergyisintheformofgravitationalpotentialenergy.

Youranswershouldcontain and .

ANSWER: =

Expressthedistance intermsof , , , , ,and .

ANSWER: =

Fat:TheFuelofMigratingBirds

Description:Findthemaximumdistanceabirdcanflywithoutfeeding,giventheamountoffatconsumedintheflightandthebird'saveragespeedandpowerconsumption.Also,findhowmanygramsoffatahummingbirdneedstoflyacrosstheGulfofMexicowithoutfeeding.

Smallbirdscanmigrateoverlongdistanceswithoutfeeding,storingenergymostlyasfatratherthancarbohydrate.Fatisagoodformofenergystoragebecauseitprovidesthemostenergyperunitmass:1gramoffatprovidesabout9.4(food)Calories,comparedto4.2(food)Caloriesper1gramofcarbohydrate.RememberthatCaloriesassociatedwithfood,whicharealwayscapitalized,arenotexactlythesameascaloriesusedinphysicsorchemistry,eventhoughtheyhavethesamename.Morespecifically,onefoodCalorieisequalto1000caloriesofmechanicalworkor4186joules.Therefore,inthisproblemusetheconversionfactor .

PartA

Considerabirdthatfliesatanaveragespeedof10.7 andreleasesenergyfromitsbodyfatreservesatanaveragerateof3.70 (thisraterepresentsthepowerconsumptionofthe

bird).Assumethatthebirdconsumes offattoflyoveradistance withoutstoppingforfeeding.Howfarwillthebirdflybeforefeedingagain?


Fromtheaveragespeedofthebird,youcancalculatehowfarthebirdcanflywithoutstoppingifyouknowthedurationoftheflight.Todeterminethedurationoftheflight,firstfindtheamountofenergyavailablefromconverting4gramsoffat,andthenusethedefinitionofpower.

HintA.2 Findtheenergyusedduringtheflight

Howmuchenergy doesthebirdhaveavailablewhenitconverts4gramsoffat?

HintA.2.1 Convertingfatintoenergy

Asstatedintheintroductionofthisproblem,1gramoffatprovidesabout9.4(food)Calories.Alsokeepinmindthat .

Expressyouranswerinkilojoules.

ANSWER: =

HintA.3 Findthedurationoftheflight

Ifthebirdconsumesenergyatarateof3.70 ,howmanyhours canitflyusingtheenergysupplyprovidedby4gramsoffat?

HintA.3.1 Definitionofpower

Theaveragepower (measuredinwatts)istheratiooftheenergy transformedinthetimeinterval : Notethatpowermeasureseithertherateatwhichenergyis

transferred(ortransformed)ortherateatwhichworkisperformed.


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HintA.3.2 Power:units

Powerismeasuredinwatts.Onewattisequalto1joulepersecond(i.e., ).

Expressyouranswerinhours.

ANSWER:

=

HintA.4 Findthedistanceintermsofaveragevelocity

Whichofthefollowingexpressionsgivesthedistance traveledinthetimeinterval atanaveragespeed ?

ANSWER:

Nowusethisexpressiontofindthedistancetraveledbythebird.Makesurethattheunitsareconsistent!

Expressyouranswerinkilometers.

ANSWER:

=

PartB

Howmanygramsofcarbohydrate wouldthebirdhavetoconsumetotravelthesamedistance ?

HintB.1 Howtoapproachtheproblem

Asstatedintheintroductionofthisproblem,1gramoffatprovidesabout9.4Calories,while1gramofcarbohydrateprovides4.2Calories.

Expressyouransweringrams

ANSWER: =

Thisismorethantwicetheamountoffatthatwasneeded!Inaddition,tostore1gramofcarbohydrate(intheformofglycogen,themostcommonformofanimalcarbohydrate)about3gramsofwaterareneeded.Therefore,ifenergywerestoredascarbohydrates,thebirdwouldneedtocarrymorethaneighttimesthefuelmasstoperformthesamemigratoryflight!

PartC

Fieldobservationssuggestthatamigratingruby-throatedhummingbirdcanflyacrosstheGulfofMexicoonanonstopflighttravelingadistanceofabout800 .Assumingthatthe

birdhasanaveragespeedof40.0 andanaveragepowerconsumptionof1.70 ,howmanygramsoffat doesaruby-throatedhummingbirdneedtoaccomplishthe

nonstopflightacrosstheGulfofMexico?

HintC.1 Howtoapproachtheproblem

InPartAyouweregiventheamountoffatconsumedovertheentireflightandwereaskedtocalculatethedistancetraveledbythemigratingbird.Nowyouneedtosolvethereverseproblem.Thatis,giventhedistancetraveled,calculatetheamountofenergyrequiredtoperformtheflight.Thus,applythesamemethodastheoneusedinpartA,onlyinreverse.Fromtheinformationondistanceandaveragespeed,calculatethedurationofthenonstopflight.Thenuseyourresultandthegivenpowerconsumptiontodeterminetheamountofenergyrequiredfortheflight.Finally,calculatehowmanygramsareneededtoprovidethatamountofenergy.

HintC.2 Findthedurationoftheflight

Howmanyhours willtheruby-throatedhummingbirdflytotraveladistanceof800 atanaveragespeedof40.0 ?

Expressyouranswerinhours

ANSWER: =


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HintC.3 Findtheenergyrequiredforthenonstopflight

Giventhatthehummingbirdconsumesenergyatanaveragerateof1.70 ,howmuchenergy willitrequireduringthenonstopflightovertheGulfofMexico?

HintC.3.1 Howtousepowerandtime

Rememberthatpowerisenergytransferred(ortransformed)perunittime.Makesureyouareusingthecorrectunitfortime.

Expressyouranswerinjoules.

ANSWER: =

Nowcalculatehowmanygramsoffatarerequiredtoprovidethisamountofenergy.Recallthat1gramoffatprovides9.4Caloriesofenergy.

Expressyouransweringrams.

ANSWER:

=

Consideringthatinnormalconditionsthemassofaruby-throatedhummingbirdisonly3or4grams,thebirdwillneedtoalmostdoubleitsbodymasstostoreenoughfattoperformthenonstopflight.

Problem11.11

Description:Thetworopesseeninthefigureareusedtolowera255kgpianodfromasecond-storywindowtotheground.(a)Howmuchworkisdonebyeachofthethreeforces?

Thetworopesseeninthefigureareusedtolowera255kgpiano6.30 fromasecond-storywindowtotheground.

PartA

Howmuchworkisdonebyeachofthethreeforces?

Enteryouranswersnumericallyseparatedbycommas.

ANSWER:

, , =

Problem11.68

Description:APorsche944Turbohasaratedenginepowerof217hp.30%ofthepowerislostinthedrivetrain,and70%reachesthewheels.Thetotalmassofthecaranddriverism,andtwo-thirdsoftheweightisoverthedrivewheels.(a)Whatisthemaximum...

APorsche944Turbohasaratedenginepowerof217 .30%ofthepowerislostinthedrivetrain,and70%reachesthewheels.Thetotalmassofthecaranddriveris1500 ,and

two-thirdsoftheweightisoverthedrivewheels.

PartA

WhatisthemaximumaccelerationofthePorscheonaconcretesurfacewhere ?

HintA.1 Hint


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Whatforcepushesthecarforward?

ANSWER: =

PartB

WhatisthespeedofthePorscheatmaximumpoweroutput?

ANSWER: =

PartC

IfthePorscheacceleratesat ,howlongdoesittakeuntilitreachesthemaximumpoweroutput?

ANSWER:

=

VisualizingRotation

Description:Selecttherotatingdisksinthemoviethatrotatewithspecificproperties(e.g.,constantnegativeangularacceleration).Usesapplets.

LearningGoal:Tobeabletoidentifysituationswithconstantangularvelocityorconstantangularaccelerationbywatchingmoviesoftherotations.

Recallthatangularvelocitymeasurestheanglethroughwhichanobjectturnsovertime.Ifadiskhasconstantangularvelocityanditmakesaquarterrevolutioninonesecond,thenitwillmakeanotherquarterrevolutionthenextsecond.Ifthediskturnsintheclockwisedirection,ithas,bydefinition,negativeangularvelocity.Themagnitudeoftheangularvelocityistheangularspeed.Thisapplet,whichshowsafewrotatingdisksandliststheirangularvelocities,shouldhelpyoutogetafeelforhowdifferentangularvelocitieslook.

Angularaccelerationmeasureshowtheangularvelocitychangesovertime.Ifadiskhasconstantangularvelocity,thenithaszeroangularacceleration.Ifadiskturnsaquarterrevolutiononesecondandahalfrevolutionthenextsecond,thenitsangularvelocityischanging,andsoithasanangularacceleration.Thisappletshowstwodisksandliststheirinitialangularvelocitiesandangularaccelerations.Thisshouldhelpyoutogetafeelforhowdifferentangularaccelerationslook.Justaswithlinearaccelerations,ifapositiveangularvelocitydecreases,thatindicatesanegativeangularacceleration.Ifanegativeangularvelocitybecomesmorenegative(i.e.,itsmagnitudeincreases),thatalsoindicatesanegativeangularacceleration.

Thisappletshowssixdisksrotatingwithconstantangularacceleration.Notwohavethesameinitialangularvelocityandangularacceleration.Toanswerthefollowingquestions,numberthedisksstartingfromthetop.Thatis,calltheyellowdisk"1"andgosequentiallydowntothereddisk,whichwillbe"6".Inthefollowingquestions,youwillbeaskedtodeterminewhetherthedisks'angularvelocitiesandaccelerationsarepositive,negative,orzero.Keepinmindthatangularvelocityisconsideredpositiveifrotationisinthecounterclockwisedirection.Angularaccelerationispositiveiftherotationisinthecounterclockwise(positive)directionandtheangularspeedisincreasing,orifrotationistheclockwise(negative)directionandtheangularspeedisdecreasing(thustheangularvelocityisbecominglessnegative).Negativeangularaccelerationisdefinedanalogously.

PartA

Whichofthediskshavepositiveinitialangularvelocity?

Writedownthenumbers,inorder,thatcorrespondtothedisk(s)thatyoubelievearecorrect,withoutcommasorspacesbetweenthem.Forexample,ifyouthinkthattheyellowdiskandthegraydiskarethecorrectones,thenyoushouldenter15.

ANSWER: 46

PartB

Whichofthediskshavenegativeinitialangularvelocity?


ANSWER: 25

PartC

Whichofthediskshavezeroangularacceleration?

Writedownthenumbers,inorder,thatcorrespondtothedisk(s)thatyoubelievearecorrect,withoutcommasorspacesbetweenthem.Forexample,ifyouthinkthatthe


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yellowdiskandthegraydiskarethecorrectones,thenyoushouldenter15.

ANSWER: 56

PartD

Whichofthediskshavepositiveangularacceleration?


ANSWER: 12

PartE

Whichofthefollowingcharacterizestheinitialangularvelocity andtheangularacceleration ofdisk4?

ANSWER:

WeightandWheel

Description:Ablockisattached(viaastring)toawheelattwodifferentradii.Determinewhetherthefinalangularvelocityofthewheelaftertheblockfallsagivenheightisgreaterforthelongerorshorterradius.

Considerabicyclewheelthatinitiallyisnotrotating.Ablockofmass isattachedtothewheelandisallowedtofalladistance .Assumethatthewheelhasamomentofinertia

aboutitsrotationaxis.

PartA

Considerthecasethatthestringtiedtotheblockisattachedtotheoutsideofthewheel,ataradius

.Find ,theangularspeedofthewheelaftertheblockhasfallenadistance ,forthiscase.

HintA.1 Howtoapproachthisproblem

Themoststraighforwardwaytosolvethisproblemistouseconservationofmechanicalenergy.Thetotalinitialenergyofthesystemisequaltothetotalfinalenergyofthesystem(wherethesystemconsistsofthewheelandtheblock).Inotherwords,

.

Where istheinitialenergyofthesystem, isthefinalenergyoftheblockand isthefinalenergyofthewheel.

HintA.2 Initialenergyofthesystem

Initially,thewheelisnotrotating.Theinitialenergyofthesystemconsistsofthegravitationalpotentialenergystoredintheblock,sinceitisnotmovingeither.Supposingthatthegravitiationalpotentialenergyoftheblockiszeroat"groundlevel,"findtheinitialenergyofthesystem.

ANSWER: =


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Whataboutthepotentialenergyforthewheel?Ifyouweretobeveryparticular,youwouldeitherneedtoassignsomepotentialenergyforthewheel,orchooseadifferentreferenceheightforit(locatedatit'scenter),suchthatit'spotentialenergywaszero.However,sinceyouknowthatthe(centerofmassofthe)wheeldoesnotmove,it'spotentialenergydoesnotchangeandassuchyoudon'treallyneedtocalculateit.Again,thisisbecauseitisthechangesispotentialenergyandnottheabsolutevalueofpotentialenergythatareimportant.

HintA.3 Finalenergyofblock

Findthefinalenergyoftheblock.

HintA.3.1 Finalvelocityoftheblock

Find ,themagnitudeofthefinalvelocityoftheblock.

Expressthevelocityintermsof andthefinalangularvelocityofthewheel, .

ANSWER: =

Expressthefinalenergyoftheblockintermsofgivenquantities(excluding )andtheunknownfinalangularvelocityofthewheel, .

ANSWER: =

Sincewearemeasuringpotentialsw.r.t.thegroundfortheblock,thefinalpotentialenergyoftheblockiszero.

HintA.4 Finalenergyofwheel

Findthefinalkineticenergyofthewheel.

Expressyouranswerintermsof (thewheel'smomentofinertia)and .

ANSWER: =


ANSWER:

=

PartB

Nowconsiderthecasethatthestringtiedtotheblockiswrappedaroundasmallerinsideaxleofthewheelofradius .

Find ,theangularspeedofthewheelaftertheblockhasfallenadistance ,forthiscase.

HintB.1 Similaritytopreviouspart

Thederivationof isexactlythesameasthederivationfor ,using insteadof .


ANSWER:

=

PartC


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Whichofthefollowingdescribestherelationshipbetween and ?

HintC.1 Howtoapproachthisquestion

Tofigureoutwhichangularvelocityisgreater( or ),youonlyneedtoconsidertheradiusdependenceoftheexpressionfor .Ignoringalloftheotherparameters,youshould

havefoundthat goesas1/radius(where"radius"referstowherethestringisattached,whichisnotnecessarilytheouterradiusofthewheel).Theproblemthenreducestofiguring

outwhichisgreater, or .

ANSWER:

Thisisrelatedtowhygearsarefoundontheinsideratherthantheoutsideofawheel.

Problem12.8

Description:Am_1ballandam_2ballareconnectedbyal-cm-long,massless,rigidrod.Theballsrotateabouttheircenterofmassatomega.(a)Whatisthespeedofthem_1ball?

A150 ballanda210 ballareconnectedbya31- -long,massless,rigidrod.Theballsrotateabouttheircenterofmassat130 .

PartA

Whatisthespeedofthe150 ball?

Expressyouranswerusingtwosignificantfigures.

ANSWER:

=

PowerDissipationPutsaDragonRacing

Description:Computethechangeintopspeedofacar,subjecttoformdrag,whenthemaximumpowerofthecar'sengineisincreasedby10%.

Thedominantformofdragexperiencedbyvehicles(bikes,cars,planes,etc.)atoperatingspeedsiscalledformdrag.Itincreasesquadraticallywithvelocity(essentiallybecausetheamountofairyourunintoincreaseswith andsodoestheamountofforceyoumustexertoneachsmallvolumeofair).Thus

,

where isthecross-sectionalareaofthevehicleand iscalledthecoefficientofdrag.

PartA

Consideravehiclemovingwithconstantvelocity .Findthepowerdissipatedbyformdrag.


Becausethevelocityofthecarisconstant,thedragforceisalsoconstant.Therefore,youcanusetheresultthatthepower providedbyaconstantforce toanobjectmovingwith

constantvelocity is .Becarefultoconsidertherelativedirectionofthedragforceandthevelocity.

Expressyouranswerintermsof , ,andspeed .

ANSWER: =

PartB

Acertaincarhasanenginethatprovidesamaximumpower .Supposethatthemaximumspeedofthecar, ,islimitedbyadragforceproportionaltothesquareofthespeed(asin

thepreviouspart).Thecarengineisnowmodified,sothatthenewpower is10percentgreaterthantheoriginalpower( .

Assumethefollowing:

Thetopspeedislimitedbyairdrag.Themagnitudeoftheforceofairdragatthesespeedsisproportionaltothesquareofthespeed.

Bywhatpercentage, ,isthetopspeedofthecarincreased?


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HintB.1 Findtherelationshipbetweenspeedandpower

Ifthemagnitudeoftheair-dragforceisproportionaltothesquareofthecar'sspeed,howisthepowerdelivered, ,relatedtothespeed ?

ANSWER:

HintB.2 Howisthealgebradone?

Therelationshipbetweenthenewpowerandtheoldpoweris .Therelationshipbetweenthenewtopspeedandtheoldtopspeedcanbewrittenas ,

where isthepercentchangeintopspeed.Finally,power isrelatedtomaximumspeed bytheformula .

Whatis intermsof ?

HintB.2.1 Helpwithsomemath

Startingwiththerelationship

,

substituteintheexpressionsfor and intermsof and :

.

Then,dividethislastexpressionbytherelationship

.

Thisisageneralapproachtoscalingproblems.Theadvantageisthattheunknownconstantofproportionality(inthiscase )dividesout.

ANSWER:

Expressthepercentincreaseintopspeednumericallytotwosignificantfigures.

ANSWER:

=

%

You'llnotethatyouranswerisveryclosetoone-thirdofthepercentagebywhichthepowerwasincreased.Thisdependenceofsmallchangesoneachother,whenthequantitiesarerelatedbyproportionalitiesofexponents,iscommoninphysicsandoftenmakesausefulshortcutforestimations.

ATaleofTwoNutcrackers

Description:Statictorqueproblemwhichrequiresthebalancingoftorquestoshowhowaforcecanbeamplifiedbytheratioofthemomentarmofalever.

Thisproblemexploresthewaysthattorquecanbeusedineverydaylife.

Case1

Tocrackanutaforceofmagnitude (orgreater)mustbeappliedonbothsides,asshowninthefigure.Onecanseethata

nutcrackeronlyappliesthisforceatthepointinwhichitcontactsthenut(atadistance fromthenutcrackerpivot).Inthis

problemthenutisplacedinanutcrackerandequalforcesofmagnitude areappliedtoeachend,directedperpendicularto

thehandle,atadistance fromthepivot.Thefrictionalforces

betweenthenutandthenutcrackerareequalandlargeenoughthatthenutdoesn'tshootoutofthenutcracker.


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PartA

Find ,themagnitudeoftheforceappliedtoeachsideofthenutcrackerrequiredtocrackthenut.

HintA.1 Sumoftorques

Thereisnoangularacceleration.Whatwillthesumofthetorquesbeaboutanypoint?

HintA.2 Sumoftorquesaboutaspecificpoint

Takethehingeasthepivotpoint.Notethattheforcesatthepivotpointgivenotorque.Sincetheforcesaresymmetricaboutthexaxiswecanjustconsiderthetoppieceofthenutcracker.Whatisthesumofthetorquesaboutthehinge generatedbyforcesonthetoppiece(both and )?Considercounterclockwise(right-handed)torquestobe

positive.

Answerintermsofgivenquantities.

ANSWER: =

Expresstheforceintermsof , ,and .

ANSWER: =

Case2

Thenutisnowplacedinanutcrackerwithonlyonelever,asshown,andagainfrictionkeepsthenutfromslipping.Thetop"jaw"(inblack)isfixedtoastationaryframesothatapersonjusthastoapplyaforcetothebottomlever.Assumethat

isdirectedperpendiculartothehandle.

PartB

Findthemagnitudeoftheforce requiredtocrackthenut.

HintB.1 Sumoftorques

Again,thesumofthetorquesaboutthehingewillbezero.Also,takingthehingeasapivotpointwillagaineliminatetorquesduetotheforcesbetweenthetwopiecesofthenutcracker.

Expressyouranswerintermsof , ,and .

ANSWER: =

PartC

Forthefirstnutcracker,twoappliedforcesofmagnitude wererequiredtocrackthenut,whereasforthesecond,onlyoneappliedforceofmagnitude wasrequired.Howwouldyou

explainthisdifference?

ANSWER: Inthesecondcasethenutcrackerhandleiseffectivelylongerandgeneratestwicethetorqueofthatinthefirstcase.


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Thereisanadditionalforceofmagnitude appliedtothenutbythefixedjawinthesecondcase.Thisjawisheldfixedbyexternalforces(suchasthe

forcesduetotheframe).Thenettorqueonthenutcrackerinthesecondcaseisnon-zero.

ARollingHollowSphere

Description:Forahollowsphericalshellrollingdownaslopewithoutslipping,calculateitsacceleration,thefrictionalforce,andtheminimumcoefficientoffrictiontopreventslipping.

Ahollowsphericalshellwithmass1.90 rollswithoutslippingdownaslopethatmakesanangleof40.0 withthehorizontal.

PartA

Findthemagnitudeoftheacceleration ofthecenterofmassofthesphericalshell.


First,drawadiagramofthesystem,includingallforcesactingonthesphere.Assumethatthepositivexdirectionpointsdownwardalongtheslopeandthepositiveydirectionpointsupwardnormaltotheslope,makingthepositivezdirectionoutofthescreen.Theangularspeedandangularaccelerationofthesphericalshellarenegative(clockwise)aroundthezaxis.Usingthiscoordinatesystem,determinethecomponentsofallforcesinthexandydirections,andsetupthecorrespondingNewtonianequationsforthetranslationalandrotationalmotionsoftheshell.Sincethereisnoslipping,usebothequationstocalculatetheaccelerationbysolvingtheangularmotionequationforthefrictionalforceintermsofthetranslationalaccelerationandthensubstitutingintothetranslationalmotionequation.

HintA.2 Translationalmotioninthexdirection

Inthexdirection,theonlyforcesthatareactingonthesphericalshellarethefrictionalforce,pointinguptheslope,andthecomponentoftheweightthatpointsdowntheslope.Becarefultousethecorrecttrigonometricfunction(sineorcosine)whenfindingthexcomponentoftheweightoftheshell.

HintA.3 Torqueonthesphericalshell

Sincetheonlyforceactingawayfromthecenterofthesphericalshellisthefriction,thiswillcausetheangularaccelerationofthespherebycreatingatorqueof ,where is

theradiusofthesphericalshell.

HintA.4 Momentofinertia

Themomentofinertiaofasphericalshellis ,where istheradiusofthesphericalshell.

HintA.5 Relationbetweenthetranslationalandangularaccelerations

Sincethesphericalshellrollsdowntheslopewithoutslipping,thetranslationalandrotationalspeedsoftheshellmustcanceleachotheratthesurfaceoftheslope,giving

,where istheradiusofthesphericalshell.Ifthederivativeistakenwithrespecttotime,itbecomesevidentthat .

Takethefree-fallaccelerationtobe =9.80 .

ANSWER: =

PartB

Findthemagnitudeofthefrictionalforceactingonthesphericalshell.


AsinPartA,youshouldfirstdrawadiagramofthesystem,includingallforcesactingonthesphere.Assumethatthepositivexdirectionpointsdownwardalongtheslopeandthatthepositiveydirectionpointsupwardnormaltotheslope,makingthepositivezdirectionoutofthescreen.Theangularspeedandangularaccelerationofthesphericalshellarenegative(clockwise)aroundthezaxis.Usingthiscoordinatesystem,determinethecomponentsofallforcesinthexandydirections,andsetupthecorrespondingNewtonianequationsforthetranslationalandrotationalmotionsoftheshell.Sincethereisnoslipping,usebothequationstogethertocalculatetheaccelerationbysolvingtheangularmotionequationforthetranslationalaccelerationintermsofthefrictionalforce,andthensubstitutingintothetranslationalmotionequation.

Takethefree-fallaccelerationtobe =9.80 .

ANSWER: =

Thefrictionalforcekeepsthesphericalshellstucktothesurfaceoftheslope,sothatthereisnoslippingasitrollsdown.Iftherewerenofriction,theshellwouldsimplyslidedowntheslope,asarectangularboxmightdoonaninclined(frictionless)surface.

PartC


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Findtheminimumcoefficientoffriction neededtopreventthesphericalshellfromslippingasitrollsdowntheslope.


Sincethefrictionalforcewascalculatedinthepreviouspart,usetheequationrelatingthefrictionalforcetothenormalforce.Becarefulaboutthetrigonometricfunctionsusedtosolveforthecoefficientoffriction.

ANSWER: =

Problem12.25

Description:Anobject'smomentofinertiaisI.Itsangularvelocityisincreasingattherateofalpha.(a)Whatisthetorqueontheobject?

Anobject'smomentofinertiais2.00 .Itsangularvelocityisincreasingattherateof3.00 .

PartA

Whatisthetorqueontheobject?

ANSWER: Nm

Problem12.74

Description:Ahollowsphereisrollingalongahorizontalflooratvwhenitcomestoaalphaincline.(a)Howfaruptheinclinedoesitrollbeforereversingdirection?

Ahollowsphereisrollingalongahorizontalfloorat7.00 whenitcomestoa37.0 incline.

PartA

Howfaruptheinclinedoesitrollbeforereversingdirection?

ANSWER:

m

Problem12.83

Description:Am_bwoodblockhangsfromthebottomofam_r,l-m-longrod.Theblockandrodformapendulumthatswingsonafrictionlesspivotatthetopendoftherod.Ambulletisfiredintotheblock,whereitsticks,causingthependulumtoswingouttoa...

A2.3 woodblockhangsfromthebottomofa1.3 ,1.0- -longrod.Theblockandrodformapendulumthatswingsonafrictionlesspivotatthetopendoftherod.A14

bulletisfiredintotheblock,whereitsticks,causingthependulumtoswingouttoa33 angle.

PartA

Whatwasthespeedofthebullet?Youcantreatthewoodblockasaparticle.


ANSWER:

=

WorkRaisinganElevator

Description:Usereadoutsofworkandenergyinanappletofanelevatorbeingraisedtodeterminethemassoftheelevatorandthetensioninitscable.(usesapplet)

Lookatthisapplet.Itshowsanelevatorwithasmallinitialupwardvelocitybeingraisedbyacable.Thetensioninthecableisconstant.Theenergybargraphsaremarkedinintervalsof600 .

PartA


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Whatisthemass oftheelevator?Use forthemagnitudeoftheaccelerationofgravity.

HintA.1 Usingthegraphs

Thinkaboutwhichgraph(s)showenergiesthataredirectlyrelatedtothemassoftheelevator.Theremaybemorethanone.Youwouldliketogetthemostaccuratenumberyoucan,sochoosethegraphthatyoucanreadmostaccurately.

HintA.2 Neededformula

Recallthatthegravitationalpotentialenergy neartheearth'ssurfaceisgivenby ,where isthemassoftheobject, isthemagnitudeofthegravitationalacceleration,

and istheheightabovetheground.

Expressyouranswerinkilogramstotwosignificantfigures.

ANSWER: =

PartB

Findthemagnitudeofthetension inthecable.Becertainthatthemethodyouareusingwillbeaccuratetotwosignificantfigures.


Inthepreviouspart,youcouldusethegraphofpotentialenergytodeterminethemasstotwosignificantfigures,becausewhentheelevatorstopped,thetopofthepotentialenergybarlayrightononeofthegridlines.Inthisproblem,youcouldusethegraphofworktofindthetension,butsinceitliessomewherebetweenthegridlines,itisunlikelythatyoucoulddeterminethetensiontothenecessaryaccuracy.However,itisagoodwaytogetanestimatewithwhichtocheckyouranswer.

Thenumericaldatagiveninthewindowbeneaththegraphsdohavetwosignificantfiguresofaccuracy,andthustheycouldbeusedincombinationwiththedatainthegraphofthefinalenergytogetamoreaccuratevaluefortheworkdoneontheelevator.Recall,infact,thattheworkdoneontheelevatorbythetensionmustequalthechangeinmechanicalenergyofthesystem.

HintB.2 Findthechangeinmechanicalenergy

FromtheinformationgivenintheappletandtheinformationfoundinPartA,determinethechangeinthetotalmechanicalenergyofthesystem .

HintB.2.1 Findtheinitialmechanicalenergy

Assumingthatthepotentialenergyoftheelevatorattheinstantwhenyourunthesimulationiszero,whatistheinitialmechanicalenergy ofthesystem?

HintB.2.1.1 Definitionofmechanicalenergy

Recallthatthemechanicalenergyofasystemisdefinedasthesumofkineticenergyandpotentialenergy,

.

Notethat,attheinstantwhenyourunthesimulation,thepotentialenergy oftheelevatoriszero.Thus,thetotalinitialmechanicalenergyofthesystemissimplygivenbythe

initialkineticenergyoftheelevator ,whichcanbeevaluatedfromtheinformationaboutthemassoftheelevatorfoundinPartA,andtheinformationaboutthe

initialspeedoftheelevatorgiveninthewindowbeneaththebargraphsintheapplet.

Expressyouranswerinjoulestotwosignificantfigures.

ANSWER: =

Thetotalmechanicalenergyofthesystemcanbedeterminedfromthedataintheenergybargraphsgivenintheapplet,justasyoudidinPartAtofindthemassoftheelevator.

Expressyouranswerinjoulestotwosignificantfigures.

ANSWER: =

Sincethechangeinthetotalmechanicalenergyofthesystemmustequalthework donebythetension,youranswergivesamoreaccurateestimateof thanwhatyou

couldhavecalculatedfromthedataintheworkbargraphsintheapplet.Nowusetheinformationaboutthedistancemovedbytheelevatorgiveninthewindowbeneaththegraphstofindthetension.

Expressyouranswerinnewtonstotwosignificantfigures.

ANSWER: =


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Hill'sLawConceptualQuestion

Description:Conceptualquestionontherelationshipbetweenforce,velocity,andpower.

Imaginethatyou'reloadingapickuptruckwithbagsofgroceries.Younoticethatthesmallertheweightyouattempttolift,thequickeryoucanliftit.However,youalsonoticethatthereisalimittohowquicklyyoucanliftevenverysmallweights,andthataboveacertainweight,youcannolongerlifttheweightatall.Thedetailedrelationshipbetweenthecontractionvelocityofamuscle(thespeedwithwhichyoucanliftsomething)andtheweightyouareattemptingtolift,isknownasHillslaw.

PartA

Basedonthisdescription,whichofthefollowinggraphsofvelocityvs.forceisapossiblerepresentationofHillslaw?

HintA.1 Maximumweight

Hillslawstatesthatthereexistsamaximumforcethatamusclecanexert,andthusamaximumweightthatamusclecanlift.Onlyonegraphhasalimittothemaximumforcethatcanbeproduced.

ANSWER: AB

C

D

E

PartB

Thepowerexertedbyamuscleistheproductoftheforceexertedandthevelocityofcontraction.Theareaofwhichoftheseshadedregionsrepresentsthepowerexertedwhileaweightisliftedatmaximumspeed?


Inthisexample,poweristheproductoftheforceexertedbythemusclesandthecontractionvelocityofthemuscles.Inliftinganygivenweight,ourmuscleshaveasinglemaximumcontractingvelocity;weassume,inthiscase,thattheweightisliftedatthisvelocity.Inliftingagivenweightataconstantvelocity,yourmusclesexertaconstantforce.Therefore,inthisexample,boththeforceandthecontractionvelocityareconstantforagivenweight.Howcantheproductoftwoconstantnumbersbeshowngraphically?

ANSWER: AB

C

D

Noneoftheabove

ThepowerproducedbyamuscleisrepresentedbytheareaoftherectangleformedbythetwocoordinateaxesandthepointontheHillslawgraphrepresentingtheweightbeinglifted.Noticethatifyouliftaverylargeweight(nearthelimitofthemaximumforceyourmusclecanproduce),theareaofthis"longandskinny"rectanglecanbequitesmall.Ifyouliftaverysmallweight,theareaofthis"tallandskinny"rectanglecanalsobequitesmall.However,ifyouliftaweightnearthemiddleofyourweightliftingrange,theareaoftherectangle,andhencethepowerproducedbyyourmuscle,isamaximum.

TorquesonaSeesaw:ATutorial


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Description:Balancingseesaw:intuition,howitrelatestotorque,andfinallybalancingbyusinghorizontalforce.

LearningGoal:Tomaketheconnectionbetweenyourintuitiveunderstandingofaseesawandthestandardformalismfortorque.

Thisproblemdealswiththeconceptoftorque,the"twist"thatanoff-centerforceappliestoabodythattendstomakeitrotate.

Trytouseyourintuitiontoanswerthefollowingquestion.Ifyourintuitionfails,worktherestoftheproblemandreturnherewhenyoufeelthatyouaremorecomfortablewithtorques.

PartA

Amotherishelpingherchildren,ofunequalweight,tobalanceonaseesawsothattheywillbeabletomakeittiltbackandforthwithouttheheavierchildsimplysinkingtotheground.Giventhatherheavierchildofweight issittingadistance totheleftofthepivot,atwhatdistance mustsheplacehersecondchildofweight ontherightsideof

thepivottobalancetheseesaw?


Considerwhether increasesordecreasesaseachofthevariablesthatitdependson-- , ,and --ismadelargerorsmaller.


ANSWER: =

Thefigureshowstheseesawslightlytilted,aswillbethecasewheninuse.Thisdoesnotchangethetorquebalancebecausethehorizontaldistancesfromthepivottoeachchild(whicharecalledthemomentarmsfortheverticallydirectedweightandmustbeusedtocalculatethetorqueinsteadofthedistancealongtheseesaw)arereducedequally,sothesumofthetorquesiszeroatanyangle.Giventhatthetorqueiszeroatalltimes(exceptwhenoneorbothchildrenpushontheground),therewillgenerallybenoangularaccelerationoftheseesaw,andtheseesawwillrotateataconstantvelocitybetweenpushesfromthefeetofthechildrenontheground.

Nowconsiderthisproblemasamoreformalintroductiontotorque.Thetorqueofeachchildaboutthepivotpointistheproductofthechild'sweightandthedistanceofthechild(strictlyspeaking,thechild'scenterofmass)fromthepivot.Thesignofthetorqueispositiveifitwouldcauseacounterclockwiserotationoftheseesaw.Thedistanceismeasuredperpendiculartothelineofforceandiscalledthemomentarm.

Theconceptoftorquerequiresbothaforceandaspecificationofthepivotpoint,emphasizedbythefirstsubscriptonthetorque.

PartB

Find ,thetorqueaboutthepivotduetotheweight ofthesmallerchildontheseesaw.

Expressyouranswerintermsof and .

ANSWER: =

Thechildren'smotherwantstheseesawtobalance,whichmeansthattherecanbenoangularaccelerationaboutthepivot.Thebalancedseesawwillthenbeinequilibriumsinceithasneitherlinearaccelerationnorrotationalacceleration.

Forthelinearaccelerationtobezero,thevectorsumofforcesactingontheseesawandchildrenmustequalzero.

Fortheangularaccelerationtobezero,thesumofthetorquesaboutthepivotmustequalzero.Thiscanbewritten

,

where isthetorqueaboutthepivotduetothe thforce.

PartC

Determine ,thesumofthetorquesontheseesaw.Consideronlythetorquesexertedbythechildren.


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HintC.1 Torquefromtheweightoftheseesaw

Theseesawissymmetricaboutthepivot,andsothegravitationalforceontheseesawproducesnonettorque.Moregenerally,whendeterminingtorques,thegravitationalforceonanobjectinauniformgravitationalfieldcanbetakentoactatthecenterofmass.Herethecenterofmassisdirectlyabovethepivot,sotheweightoftheseesawhaszeromomentarmandproducesnotorqueaboutthepivot.

Expressyouranswerintermsof , , ,and .

ANSWER:

Good!Ifyoudidnotsolveforthedistance requiredtobalancetheseesawinPartA,dosonow.

Theequation appliestoanybodythatisnotrotationallyaccelerating.Combiningthisequationwith (whichappliestoanybodythatisnotaccelerating

linearly)givesapairofequationsthataresufficienttoformthebasisofstatics;theseorsimilar,oftenmorecomplicatedequationsgovernstructuresthatarenotaccelerating.Thetorqueequationisoftenofmoreutility,however,becauseyoucanchoosethepivotpointarbitrarily(oftensothatunknownforceshavenomomentarmandthereforecontributenotorque).Theartofapplyingtheseequationstolargeorcomplicatedstructuresconstitutesasignificantpartofmechanicalandcivilengineering.

Thechildwithweight hasanidenticaltwinalsoofweight .Thetwotwinsnowsitonthesamesideoftheseesaw,withonetwinadistance fromthepivotandtheothera

distance .

PartD

Whereshouldthemotherpositionthechildofweight tobalancetheseesawnow?

HintD.1 Balancingtheseesaw

Fortheseesawtobalance,thesumofthetorquesfromthethreechildren mustbezero.Whatisthesumofthetorques?

Expressyouranswerintermsof , , , ,and .

ANSWER:

Nowsolveyoutorqueequationfor .


ANSWER: =

Badnews!Whenthemotherfindsthedistance fromthepreviouspartitturnsouttobegreaterthan ,thedistancefromthepivottotheendoftheseesaw.Hence,evenwiththe

childofweight attheveryendoftheseesawthetwinsexertmoretorquethantheheavierchilddoes.Themothernowelectstobalancetheseesawbypushingsidewaysonan

ornament(showninred)thatisaheight abovethepivot.


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PartE

Withwhatforceintherightwardsdirection, ,shouldthemotherpush?Notethatifyouthinktheforceexertedbythemothershouldbetowardtheleft,yourfinalanswer should

benegative.

HintE.1 Signconventions

Itiseasytomakesignerrorsintorqueproblems,andexperienceshowsthatitisbettertousestandardconventions(herethat goestotheright,thedirectionofpositivex)thanto

changethedirectionofpositivetorqueordisplacementtosuityourconvenience.(Youarelikelytoforgetyourunconventionalchoiceatalaterpointintheproblem.)Inthiscase,yourintuitioncorrectlyexpectsthatthemothermustpushtothelefttomakethingsbalance;theequationswillconfirmthisbygivinganegativeresultfor .(Apositiveresult

wouldmeanthattheforcewouldbedirectedtotherightinthefigure.)

HintE.2 Torqueduetomother'spush

Thesumofallfourtorques(duetoeachofthethreechildrenplusthemother)mustbezero.

Whatisthetorque duetothemother'spush?

Remember,apositivetorquewillcausecounterclockwiserotationoftheseesaw.

Expressyouranswerintermsoftheunknownforce andtheheightatwhichitisapplied .

ANSWER: =

Expressyouranswerintermsof , , , , ,and .

ANSWER: =

Thisanswerwillnecessarilybenegative,becauseyouweretoldthattobalancetheseesawwiththetwinsontheright,thechildofweight hadtobe"beyondtheend"ofthe

seesaw.Therefore,when isthepositionofthechildofweight , willbelessthanzero.Hencethemothermustpushtotheleftasyou'dexpect.

ASculptureinEquilibrium

Description:Locatethecenterofmass(centerofgravity)ofanirregularlyshaped,balancedobject.

Theabstractsculptureshowninthefigurecanbeplacedonahorizontalsurfacewithouttippingover.Differentpartsofthesculpturemaybeholloworsolid,ormadeofdifferentmaterials,buttheartistisn'trevealinganysuchinformation.

PartA

FivelocationslabeledAthroughEareindicatedonthediagram.Whichofthese,ifany,isapossiblelocationoftheobject'scenterofmass(centerofgravity)?

HintA.1 Thetorque(s)onthesculpture

Thesculptureisinequilibrium.Thismeansthatthenettorquethatactsonthesculptureiszero.

Theonlyforcesthatactonthesculpturearethenormalforceactingonthe"foot"ofthesculptureandtheweightactingeffectivelyatthecenterofmass.Ifyouusethefootastheaxisofrotation,theweightsuppliestheonlytorqueonthesculpture.Thinkaboutwheretheweightmustbelocatedinorderforthistorquetobezeroandthesculpturetobeinequilibrium.

ANSWER: AB


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CD

E

cannotbedetermined

Foranobjecttobeinequilibriumundertheactionofitsownweight,thecenterofmassoftheobjectmustliedirectlyabovethebaseoftheobject.Thisensuresthatthetorqueduetotheweightoftheobjectwillbezeroaboutanaxispassingthroughapointinthebase.Theobjectsinthefigurehavetheircentersofmassmarkedbyadot.Theleftobjectisinequilibrium,sincethecenterofmassisaboveapointonitsbase.Therightobject,however,willtipoverclockwise,sinceitsweightsuppliesanunbalancedclockwisetorqueaboutanypointonitsbase.

TorqueMagnitudeRankingTask

Description:Rankthetorquesinducedbyvariousforcesactingonawrench.(rankingtask)

Thewrenchinthefigurehassixforcesofequalmagnitudeactingonit.

PartA

Ranktheseforces(AthroughF)onthebasisofthemagnitudeofthetorquetheyapplytothewrench,measuredaboutanaxiscenteredonthebolt.

HintA.1 Definitionoftorque

Torqueisameasureofthe"twist"thatanappliedforceexertsonanobject.Mathematically,torqueisdefinedas

,

where isthemagnitudeofthedisplacementvectorfromtherotationaxistothepointofapplicationoftheforceofmagnitude ,and istheanglebetweenthisdisplacementand

theappliedforce,asshowninthefigure.

Thedirectionofatorquecanbeeithercounterclockwise(asabove)orclockwise.Thisisdeterminedbythedirectiontheobjectwillrotateundertheactionoftheforce.

HintA.2 Maximumtorque


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Basedonthemathematicaldefinitionoftorque,torqueismaximizedwhentheforceislargeinmagnitude,locatedalargedistancefromtheaxisofinterest,andorientedperpendiculartothedisplacement ,whichisoftenreferredtoastheleverarmoftheforce.

Rankfromlargesttosmallest.Torankitemsasequivalent,overlapthem.

ANSWER:

View

PivotedRodwithUnequalMasses

Description:Findthemomentofinertiaforarodwithtwomasses(treatedaspointparticles)attachedtoitsends.Thencomputetheangularaccelerationoftherodwhenitisreleasedfromrest.Thenfindthetorqueandinitialangularaccelerationaboutthepivotpoint.

Thefigureshowsasimplemodelofaseesaw.Theseconsistofaplank/rodofmass andlength allowedtopivotfreely

aboutitscenter(orcentralaxis),asshowninthediagram.Asmallsphereofmass isattachedtotheleftendoftherod,

andasmallsphereofmass isattachedtotherightend.Thespheresaresmallenoughthattheycanbeconsideredpoint

particles.Thegravitationalforceactsdownward.Themagnitudeoftheaccelerationduetogravityisequalto .

PartA

Whatisthemomentofinertia ofthisassemblyabouttheaxisthroughwhichitispivoted?


Themomentofinertiaoftheassemblyaboutthepivotisequaltothesumofthemomentsofinertiaofeachofthecomponentsoftheassemblyaboutthepivotpoint.Thatis,thetotalmomentofinertiaisequaltothemomentofinertiaoftherodplusthemomentofinertiaoftheparticleofmass plusthemomentofinertiaoftheparticleofmass ,all

measuredwithrespecttothepivotpoint.

HintA.2 Findthemomentofinertiaduetothesphereofmass

Whatisthemomentofinertiaoftheparticleofmass measuredaboutthepivotpoint?

HintA.2.1 Formulaformomentofinertia

Consideranobjectconsistingofparticleswithmasses .Let bethedistanceofthe thparticlefromtheaxisofrotation.Thenthemomentofinertia oftheobjectaboutthe

axisofrotationisgivenby

.

Expressyouranswerintermsofgivenquantities.

ANSWER: =

HintA.3 Findthemomentofinertiaduetothesphereofmass

Whatisthemomentofinertiaoftheparticleofmass measuredaboutthepivotpoint?

Expressyouranswerintermsofgivenquantities.

ANSWER: =

HintA.4 Findthemomentofinertiaoftherod


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Whatisthemomentofinertiaoftherodaboutthepivotpoint?

HintA.4.1 Generalformulaforthemomentofinertiaofarod

Considerarodoftotallength andmass ,pivotedaboutitscenter.(Inthisproblem, equals .)Whatisthemomentofinertiaoftherodaboutitspivotpoint?

ANSWER:

Express intermsof and .

ANSWER: =

Expressthemomentofinertiaintermsof , , ,and .Keepinmindthatthelengthoftherodis ,not .

ANSWER: =

PartB

Supposethattherodisheldatresthorizontallyandthenreleased.(Throughouttheremainderofthisproblem,youranswermayincludethesymbol ,themomentofinertiaofthe

assembly,whetherornotyouhaveansweredthefirstpartcorrectly.)

Whatistheangularacceleration oftherodimmediatelyafteritisreleased?


Theforcesactingonthesystem(spheresandrod)aretheweightsofthespheresandtherod,andthereactionforcefromthepivot.Findthetorqueduetoeachoftheseforcesaboutthepivotpointandaddthemwiththecorrectsigns.Finally,useNewton'ssecondlawforrotationalmotion: .

HintB.2 Findthetorqueduetothesphereofmass

Findthetorqueaboutthepivotduetothesphereofmass .

HintB.2.1 Formulafortorque

Thetorqueaboutthepivotpointduetoaforce is

,

where isthevectorfromthepivotpointtothepointwheretheforceisapplied.Theothersymbolshavetheirusualmeanings.Ifyouareusinganyofthelattertwo

expressions,youmustrememberthatiftheforcetendstocauseaclockwiserotation,youneedtoincludeanegativesigninyourexpressionsincethetorqueduetosuchaforceistakentobenegative(byconvention).

Expressyouranswerintermsofgivenquantities.Keepinmindthatthepositivedirectioniscounterclockwise.

ANSWER: =

HintB.3 Findthetorqueduetothesphereofmass

Findthetorqueaboutthepivotduetotheparticleofmass .

Expressyouranswerintermsofgivenquantities.Keepinmindthatthepositivedirectioniscounterclockwise.

ANSWER: =

HintB.4 Torqueduetoforcesactingontherod

Besidesthetwomasses,therearetwomoreforcestoconsider:thenormalforceactingatthepivotandthegravitationalforceactingontherod.Thenormalforceactsatthepivotpoint,soitsdistancefromthepivotpointiszero,andthusthisforcecontributeszerotorque.Thegravitationalforceactsattherod'scenterofmass,whichisalsoatthepivotpoint.Therefore,thetorqueduetothegravitationalforceaboutthepivotpointisalsozerofortherod.


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HintB.5 Relatingtheangularaccelerationtothenettorque

Letthenettorqueactingonthesystemaboutthepivotpointbedenotedby .Findanexpressionfor .

Expressyouranswerintermsofthesystem'smomentofinertia anditsresultingangularacceleration .(Use inyouranswer,nottheexpressionfor youfoundin

PartA.)

ANSWER: =

Takethecounterclockwisedirectiontobepositive.Express intermsofsomeorallofthevariables , , , , ,and .

ANSWER:

=

Substitutingfor ,thevalueobtainedinPartAyields

.

Alargeangularaccelerationisoftendesirable.Thiscanbeaccomplishedbymakingtheconnectingrodlightandshort(sinceboth and appearinthedenominatorofthe

expressionfor ).Foraseesaw,ontheotherhand, and areusuallychosentobeaslargeaspossible,whilemakingsurethatthe"rod"doesnotgettooheavyandunwieldy.

Thisensuresthattheangularaccelerationisquitelow.

Problem11.51

Description:Anmcrateispulledlupa30degree(s)inclinebyaropeangledalphaabovetheincline.ThetensionintheropeisTandthecrate'scoefficientofkineticfrictionontheinclineismu.(a)Howmuchworkisdonebytension,bygravity,andby...

An7.7 crateispulled5.2 upa30 inclinebyaropeangled19 abovetheincline.Thetensionintheropeis140 andthecrate'scoefficientofkineticfrictionontheinclineis

0.26.

PartA

Howmuchworkisdonebytension,bygravity,andbythenormalforce?

Expressyouranswersusingtwosignificantfigures.Enteryouranswersnumericallyseparatedbycommas.

ANSWER:

, , =

PartB

Whatistheincreaseinthermalenergyofthecrateandincline?


ANSWER: =

Where'stheEnergy?

Description:Aseriesofbasicconceptualandcomputationalquestionsaboutthemotionofablockslidingdownaramp.Motionwithandwithoutthenon-conservativeforcesisconsidered.

LearningGoal:Tounderstandhowtoapplythelawofconservationofenergytosituationswithandwithoutnonconservativeforcesacting.

Thelawofconservationofenergystatesthefollowing:

Inanisolatedsystemthetotalenergyremainsconstant.


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Iftheobjectswithinthesysteminteractthroughgravitationalandelasticforcesonly,thenthetotalmechanicalenergyisconserved.

Themechanicalenergyofasystemisdefinedasthesumofkineticenergy andpotentialenergy .Forsuchsystemswherenoforcesotherthanthegravitationalandelasticforcesdo

work,thelawofconservationofenergycanbewrittenas

,

wherethequantitieswithsubscript"i"refertothe"initial"momentandthosewithsubscript"f"refertothefinalmoment.Awisechoiceofinitialandfinalmoments,whichisnotalwaysobvious,maysignificantlysimplifythesolution.

Thekineticenergyofanobjectthathasmass andvelocity isgivenby

.

Potentialenergy,instead,hasmanyforms.Thetwoformsthatyouwillbedealingwithmostofteninthischapterarethegravitationalandelasticpotentialenergy.Gravitationalpotentialenergyistheenergypossessedbyelevatedobjects.Forsmallheights,itcanbefoundas

,

where isthemassoftheobject, istheaccelerationduetogravity,and istheelevationoftheobjectabovethezerolevel.Thezerolevelistheelevationatwhichthegravitational

potentialenergyisassumedtobe(youguessedit)zero.Thechoiceofthezerolevelisdictatedbyconvenience;typically(butnotnecessarily),itisselectedtocoincidewiththelowestpositionoftheobjectduringthemotionexploredintheproblem.

Elasticpotentialenergyisassociatedwithstretchedorcompressedelasticobjectssuchassprings.Foraspringwithaforceconstant ,stretchedorcompressedadistance ,the

associatedelasticpotentialenergyis

.

Whenallthreetypesofenergychange,thelawofconservationofenergyforanobjectofmass canbewrittenas

.

Thegravitationalforceandtheelasticforcearetwoexamplesofconservativeforces.Whatifnonconservativeforces,suchasfriction,alsoactwithinthesystem?Inthatcase,thetotalmechanicalenergywouldchange.Thelawofconservationofenergyisthenwrittenas

,

where representstheworkdonebythenonconservativeforcesactingontheobjectbetweentheinitialandthefinalmoments.Thework isusuallynegative;thatis,the

nonconservativeforcestendtodecrease,ordissipate,themechanicalenergyofthesystem.

Inthisproblem,wewillconsiderthefollowingsituationasdepictedinthediagram:Ablockofmass slidesataspeed

alongahorizontal,smoothtable.Itnextslidesdownasmoothramp,descendingaheight ,andthenslidesalonga

horizontalroughfloor,stoppingeventually.Assumethattheblockslidesslowlyenoughsothatitdoesnotlosecontactwiththesupportingsurfaces(table,ramp,orfloor).

Youwillanalyzethemotionoftheblockatdifferentmomentsusingthelawofconservationofenergy.

PartA

Whichwordinthestatementofthisproblemallowsyoutoassumethatthetableisfrictionless?

ANSWER: straightsmooth

horizontal

Althoughtherearenotruly"frictionless"surfaces,sometimesfrictionissmallenoughtobeneglected.Theword"smooth"oftendescribessuchlow-frictionsurfaces.Canyoudeducewhattheword"rough"means?

PartB

Supposethepotentialenergyoftheblockatthetableisgivenby .Thisimpliesthatthechosenzerolevelofpotentialenergyis__________.


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HintB.1 Definitionof

Gravitationalpotentialenergyisgivenby

,

where istheheightrelativetothezerolevel.Notethat whentheobjectisabovethechosenzerolevel; whentheobjectisbelowthechosenzerolevel.

ANSWER: adistance abovethefloor

adistance belowthefloor

adistance abovethefloor

adistance belowthefloor

onthefloor

PartC

Ifthezerolevelisadistance abovethefloor,whatisthepotentialenergy oftheblockonthefloor?

Expressyouranswerintermsofsomeorallthevariables , ,and andanyappropriateconstants.

ANSWER: =

PartD

Consideringthatthepotentialenergyoftheblockatthetableis andthatontheflooris ,whatisthechangeinpotentialenergy oftheblockifitismoved

fromthetabletothefloor?

HintD.1 Definitionof

Bydefinition,thechangeinpotentialenergyisgivenby .Ingeneral,changeisalwaysdefinedasthe"final"quantityminusthe"initial"one.


ANSWER: =

Asyoumayhaverealized,thischoiceofthezerolevelwaslegitimatebutnotveryconvenient.Typically,insuchproblems,thezerolevelisassumedtobeonthefloor.Insolvingthisproblem,wewillassumejustthat:thezerolevelofpotentialenergyisonthefloor.

PartE

Whichformofthelawofconservationofenergydescribesthemotionoftheblockwhenitslidesfromthetopofthetabletothebottomoftheramp?

HintE.1 Howtoapproachtheproblem

Thinkaboutthesequestions:

Arethereanynonconservativeforcesactingontheblockduringthispartofthetrip?Arethereanyobjectsinvolvedthatcanstoreelasticpotentialenergy?Istheblockchangingitsheight?Istheblockchangingitsspeed?

ANSWER:

PartF


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Astheblockslidesdowntheramp,whathappenstoitskineticenergy ,potentialenergy ,andtotalmechanicalenergy ?

ANSWER: decreases; increases; staysthesame

decreases; increases; increases

increases; increases; increases

increases; decreases; staysthesame

PartG

Usingconservationofenergy,findthespeed oftheblockatthebottomoftheramp.

HintG.1 Howtoapproachtheproblem

Usetheequationforthelawofconservationofenergythatdescribesthemotionoftheblockasitslidesdowntheramp.Thensubstituteinallknownvaluesandsolvefortheunknown.


ANSWER: =

PartH

Whichformofthelawofconservationofenergydescribesthemotionoftheblockasitslidesonthefloorfromthebottomoftheramptothemomentitstops?

HintH.1 Howtoapproachtheproblem

Thinkaboutthesequestions:

Arethereanynonconservativeforcesactingontheblockduringthispartofthetrip?Arethereanyobjectsinvolvedthatcanstoreelasticpotentialenergy?Istheblockchangingitsheight?Istheblockchangingitsspeed?

ANSWER:

PartI

Astheblockslidesacrossthefloor,whathappenstoitskineticenergy ,potentialenergy ,andtotalmechanicalenergy ?

ANSWER: decreases; increases; decreases

increases; decreases; decreases

decreases; staysthesame; decreases

increases; staysthesame; decreases

decreases; increases; staysthesame

increases; decreases; staysthesame

PartJ

Whatforceisresponsibleforthedecreaseinthemechanicalenergyoftheblock?

ANSWER: tensiongravity

friction

normalforce


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PartK

Findtheamountofenergy dissipatedbyfrictionbythetimetheblockstops.

HintK.1 Howtoapproachtheproblem

Usetheequationforthelawofconservationofenergythatyouselectedasthemostappropriatefortheblockslidingonthefloor.Thensubstituteinallknownvaluesandsolvefortheunknown.Youwillneedtousethevaluefor thatyoufoundearlierinPartG,asyourinitialspeed.


ANSWER: =

VectorDotProduct

Description:Thisproblemreviewstherulesforcomputingdotproductsandprovidespracticeoncalculatingthem.

Letvectors , ,and .

Calculatethefollowing:

PartA

HintA.1 Rememberthedotproductequation

If and ,then

.

ANSWER: =

PartB

Whatistheangle between and ?

HintB.1 Rememberthedefinitionofdotproducts

,where istheanglebetween and .

ANSWER: =

PartC

ANSWER: =

PartD

ANSWER: =

PartE

Whichofthefollowingcanbecomputed?

HintE.1 Dotproductoperator

Thedotproductoperatesonlyontwovectors.Thedotproductofavectorandascalarisnotdefined.


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ANSWER:

and aredifferentvectorswithlengths and respectively.Findthefollowing:

PartF

HintF.1 Whatistheanglebetweenavectoranditself?

Theanglebetweenavectoranditselfis0.

HintF.2 Rememberthedefinitionofdotproducts

,where istheanglebetween and .

Expressyouranswerintermsof

ANSWER:

=

PartG

If and areperpendicular,

HintG.1 Whatistheanglebetweenperpendicularvectors?

Theanglebetweenvectorsthatareperpendicularisequalto radiansor90degrees.

ANSWER:

=

PartH

If and areparallel,

HintH.1 Whatistheanglebetweenparallelvectors?

Theanglebetweenvectorsthatareparallelisequalto0.


ANSWER: =

Problem11.53

Description:Amiceskaterisglidingalongtheice,headingduenorthatv.Theicehasasmallcoefficientofstaticfriction,topreventtheskaterfromslippingsideways,butmu_k=0.Suddenly,awindfromthenortheastexertsaforceofFonthe...

A48.0 iceskaterisglidingalongtheice,headingduenorthat4.30 .Theicehasasmallcoefficientofstaticfriction,topreventtheskaterfromslippingsideways,but

.Suddenly,awindfromthenortheastexertsaforceof4.30 ontheskater.

PartA

Useworkandenergytofindtheskater'sspeedaftergliding100minthiswind.


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ANSWER:

m/s

PartB

Whatistheminimumvalueof thatallowshertocontinuemovingstraightnorth?

ANSWER:

CenterofMassandExternalForces

Description:Findtheposition,velocity,accelerationofthecenterofmassofasystemoftwoblocks;computetheeffectofinternalandexternalforcesontheaccelerationofthecenterofmass.

LearningGoal:Understandthat,formanypurposes,asystemcanbetreatedasapoint-likeparticlewithitsmassconcentratedatthecenterofmass.

Acomplexsystemofobjects,bothpoint-likeandextendedones,canoftenbetreatedasapointparticle,locatedatthesystem'scenterofmass.Suchanapproachcangreatlysimplifyproblemsolving.

Beforeyouusethecenterofmassapproach,youshouldfirstunderstandthefollowingterms:

System:Anycollectionofobjectsthatareofinteresttoyouinaparticularsituation.Inmanyproblems,youhaveacertainfreedominchoosingyoursystem.Makingawisechoiceforthesystemisoftenthefirststepinsolvingtheproblemefficiently.Centerofmass:Thepointthatrepresentsthe"average"positionoftheentiremassofasystem.Thepostionofthecenterofmass canbeexpressedintermsoftheposition

vectors oftheparticlesas

.

Thexcoordinateofthecenterofmass canbeexpressedintermsofthexcoordinates oftheparticlesas

.

Similarly,theycoordinateofthecenterofmasscanbeexpressed.Internalforce:Anyforcethatresultsfromaninteractionbetweentheobjectsinsideyoursystem.Aswewillshow,theinternalforcesdonotaffectthemotionofthesystem'scenterofmass.Externalforce:Anyforceactingonanobjectinsideyoursystemthatresultsfromaninteractionwithanobjectoutsideyoursystem.

Considerasystemoftwoblocksthathavemasses and .Assumethattheblocksarepoint-likeparticlesandarelocatedalongthexaxisatthecoordinates and asshown.

Inthisproblem,theblockscanonlymovealongthexaxis.

PartA

Findthexcoordinate ofthecenterofmassofthesystem.


ANSWER: =


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PartB

If ,thenthecenterofmassislocated:

ANSWER: totheleftof atadistancemuchgreaterthan

totheleftof atadistancemuchlessthan

totherightof atadistancemuchlessthan

totherightof atadistancemuchgreaterthan

totherightof atadistancemuchlessthan

totheleftof atadistancemuchlessthan

PartC

If ,thenthecenterofmassislocated:

ANSWER: atat

half-waybetween and

theanswerdependson and

PartD

Recallthattheblockscanonlymovealongthexaxis.Thexcomponentsoftheirvelocitiesatacertainmomentare and .Findthe componentofthevelocityofthecenterof

mass atthatmoment.Keepinmindthat,ingeneral: .


ANSWER: =

Because and arethexcomponentsofthevelocitiesof and theirvaluescanbepositiveornegativeorequaltozero.

PartE

Supposethat and haveequalmagnitudes.Also, isdirectedtotherightand isdirectedtotheleft.Thevelocityofthecenterofmassisthen:

ANSWER: directedtotheleftdirectedtotheright

zero

theanswerdependsontheratio

PartF

Assumethatthexcomponentsoftheblocks'momentaatacertainmomentare and .Findthexcomponentofthevelocityofthecenterofmass atthatmoment.


ANSWER: =

PartG

Supposethat .Whichofthefollowingmustbetrue?

ANSWER:

noneoftheabove

PartH


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Assumethattheblocksareaccelerating,andthexcomponentsoftheiraccelerationsatacertainmomentare and .Findthexcomponentoftheaccelerationofthecenterofmass

atthatmoment.Keepinmindthat,ingeneral, .


ANSWER: =

Because and arethexcomponentsofthevelocitiesof and theirvaluescanbepositiveornegativeorequaltozero.

Wewillnowconsidertheeffectofexternalandinternalforcesontheaccelerationofthecenterofmass.

PartI

Considerthesamesystemoftwoblocks.Anexternalforce isnowactingonblock .Noforcesareappliedtoblock asshown.Findthexcomponentoftheaccelerationofthe

centerofmass ofthesystem.

HintI.1 UsingNewton'slaws

FindtheaccelerationofeachblockfromNewton'ssecondlawandthenapplytheformulafor foundearlier.

Expressyouranswerintermsofthexcomponent oftheforce, ,and .

ANSWER: =

PartJ

Considerthesamesystemoftwoblocks.Now,therearetwoforcesinvolved.Anexternalforce isactingonblock andanotherexternalforce isactingonblock .Findthe

xcomponentoftheaccelerationofthecenterofmass ofthesystem.

Expressyouranswerintermsofthexcomponents and oftheforces, and .

ANSWER: =

Notethat,inbothcases,theaccelerationofmasscanbefoundas

where isthenetexternalforceappliedtothesystem,and isthetotalmassofthesystem.Eventhougheachforceisonlyappliedtooneobject,itaffectsthe

accelerationofthecenterofmassoftheentiresystem.


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Thisresultisespeciallyusefulsinceitcanbeappliedtoageneralcase,involvinganynumberofobjectsmovinginalldirectionsandbeingacteduponbyanynumberofexternalforces.

PartK

Considertheprevioussituation.Underwhatconditionwouldtheaccelerationofthecenterofmassbezero?Keepinmindthat and representthecomponents,ofthe

correspondingforces.

ANSWER:

PartL

Considerthesamesystemoftwoblocks.Now,therearetwointernalforcesinvolved.Aninternalforce isappliedtoblock byblock andanotherinternalforce is

appliedtoblock byblock .Findthexcomponentoftheaccelerationofthecenterofmass ofthesystem.

Expressyouranswerintermsofthexcomponents and oftheforces, and .

ANSWER:

=

Newton's3rdlawtellsyouthat .Fromyouranswersabove,youcanconcludethat .Theinternalforcesdonotchangethevelocityofthecenterof

massofthesystem.Intheabsenceofanyexternalforces, and isconstant.

Youjustdemonstratedthistobethecaseforthetwo-bodysituationmovingalongthexaxis;however,itistrueinmoregeneralcasesaswell.

IntroductiontoMomentsofInertia

Description:Conceptualquestionsaboutmomentofinertia;severalbasiccomputationalquestionsforbothdiscreteandcontinuousmassdistribution.

LearningGoal:Tounderstandthedefinitionandthemeaningofmomentofinertia;tobeabletocalculatethemomentsofinertiaforagroupofparticlesandforacontinuousmassdistributionwithahighdegreeofsymmetry.

Bynow,youmaybefamiliarwithasetofequationsdescribingrotationalkinematics.Onethingthatyoumayhavenoticedwasthesimilaritybetweentranslationalandrotationalformulas.Suchsimilarityalsoexistsindynamicsandinthework-energydomain.

Foraparticleofmass movingataconstantspeed ,thekineticenergyisgivenbytheformula .Ifweconsiderinsteadarigidobjectofmass rotatingataconstant

angularspeed ,thekineticenergyofsuchanobjectcannotbefoundbyusingtheformula directly:differentpartsoftheobjecthavedifferentlinearspeeds.However,they

allhavethesameangularspeed.Itwouldbedesirabletoobtainaformulaforkineticenergyofrotationalmotionthatissimilartotheonefortranslationalmotion;suchaformulawouldincludetheterm insteadof .

Suchaformulacan,indeed,bewritten:forrotationalmotionofasystemofsmallparticlesorforarigidobjectwithcontinuousmassdistribution,thekineticenergycanbewrittenas

.

Here, iscalledthemomentofinertiaoftheobject(orofthesystemofparticles).Itisthequantityrepresentingtheinertiawithrespecttorotationalmotion.


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Itcanbeshownthatforadiscretesystem,sayof particles,themomentofinertia(alsoknownasrotationalinertia)isgivenby

.

Inthisformula, isthemassoftheithparticleand isthedistanceofthatparticlefromtheaxisofrotation.

Forarigidobject,consistingofinfinitelymanyparticles,theanalogueofsuchsummationisintegrationovertheentireobject:

.

Inthisproblem,youwillanswerseveralquestionsthatwillhelpyoubetterunderstandthemomentofinertia,itsproperties,anditsapplicability.Itisrecommendedthatyoureadthecorrespondingsectionsinyourtextbookbeforeattemptingthesequestions.

PartA

Onwhichofthefollowingdoesthemomentofinertiaofanobjectdepend?

Checkallthatapply.

ANSWER: linearspeed

linearacceleration

angularspeed

angularacceleration

totalmass

shapeanddensityoftheobject

locationoftheaxisofrotation

Unlikemass,themomentofinertiadependsnotonlyontheamountofmatterinanobjectbutalsoonthedistributionofmassinspace.Themomentofinertiaisalsodependentontheaxisofrotation.Thesameobject,rotatingwiththesameangularspeed,mayhavedifferentkineticenergydependingontheaxisofrotation.

Considerthesystemoftwoparticles,aandb,showninthefigure.Particleahasmass ,andparticlebhasmass .

PartB

Whatisthemomentofinertia ofparticlea?

ANSWER:

undefined:anaxisofrotationhasnotbeenspecified.

PartC

Findthemomentofinertia ofparticleawithrespecttothexaxis(thatis,ifthexaxisistheaxisofrotation),themomentofinertia ofparticleawithrespecttotheyaxis,and

themomentofinertia ofparticleawithrespecttothezaxis(theaxisthatpassesthroughtheoriginperpendiculartoboththexandyaxes).

Expressyouranswersintermsof and separatedbycommas.

ANSWER:

, , =


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PartD

Findthetotalmomentofinertia ofthesystemoftwoparticlesshowninthediagramwithrespecttotheyaxis.


ANSWER: =

ForpartsEthroughG,supposethatbothparticlesrotatewiththesameangularspeed abouttheyaxiswhilemaintainingtheirdistancesfromtheyaxis.

PartE

Usingthetotalmomentofinertia ofthesystemfoundinPartD,findthetotalkineticenergy ofthesystem.


ANSWER:

=

Itisusefultoseehowtheformulaforrotationalkineticenergyagreeswiththeformula forthekineticenergyofanobjectthatisnotrotating.Toseetheconnection,let

usfindthekineticenergyofeachparticle.

PartF

Usingtheformulaforkineticenergyofamovingparticle ,findthekineticenergy ofparticleaandthekineticenergy ofparticleb.

HintF.1 Findthelinearspeed

Usingtheformula ,findthelinearspeed ofparticlea.


ANSWER: =

Expressyouranswersintermsof , ,and separatedbyacomma.

ANSWER:

, =

PartG

Usingtheresultsforthekineticenergyofeachparticle,findthetotalkineticenergy ofthesystemofparticles.


ANSWER:

=

Notsurprisingly,theformulas and givethesameresult.Theyshould,ofcourse,sincetherotationalkineticenergyofasystemofparticlesissimplythe

sumofthekineticenergiesoftheindividualparticlesmakingupthesystem.

ConstrainedRotationandTranslation

Description:Determinetheconstraintequationsthatrelatelineartorotationalmotionforthecasesofanunrollingtapemeasureandarollingtire.


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LearningGoal:Tounderstandthatcontactbetweenrollingobjectsandwhattheyrollagainstimposesconstraintsonthechangeinposition(velocity)andangle(angularvelocity).

Thewayinwhichabodymakescontactwiththeworldoftenimposesaconstraintrelationshipbetweenitspossiblerotationandtranslationalmotion.Aballrollingonaroad,ayo-younwindingasitfalls,andabaseballleavingthepitcher'shandareallexamplesofconstrainedrotationandtranslation.Inasimilarmanner,therotationofonebodyandthetranslationofanothermaybeconstrained,ashappenswhenafiremanunrollsahosefromitsstoragedrum.

Situationslikethesecanbemodeledbyconstraintequations,relatingthecoupledangularandlinearmotions.Althoughtheseequationsfundamentallyinvolveposition(theangleofthewheelataparticulardistancedowntheroad),itisusuallytherelationshipofvelocitiesandaccelerationsthatarerelevantinsolvingaprobleminvolvingsuchconstraints.Thevelocitiesareneededintheconservationequationsformomentumandangularmomentum,andtheaccelerationsareneededforthedynamicalequations.

Itisimportanttousethestandardsignconventions:positiveforcounterclockwiserotationandpositiveformotiontowardtheright.Otherwise,yourdynamicalequationswillhavetobemodified.Unfortunately,afrequentresultwillbetheappearanceofnegativesignsintheconstraintequations.

Considerameasuringtapeunwindingfromadrumofradius .Thecenterofthedrumisnotmoving;thetapeunwindsasits

freeendispulledawayfromthedrum.Neglectthethicknessofthetape,sothattheradiusofthedrumcanbeassumednottochangeasthetapeunwinds.Inthiscase,thestandardconventionsfortheangularvelocity andforthe(translational)velocity

oftheendofthetaperesultinaconstraintequationwithapositivesign(e.g.,if ,thatis,thetapeisunwinding,then

also).

PartA

Assumethatthefunction representsthelengthoftapethathasunwoundasafunctionoftime.Find ,theanglethroughwhichthedrumwillhaverotated,asafunctionof

time.

HintA.1 Findtheamountoftapethatunrollsinonecompleterevolutionofthedrum

Ifthemeasuringtapeunwindsonecompleterevolution( ),howmuchtape, ,willhaveunwound?

ANSWER: =

Expressyouranswer(inradians)intermsof andanyothergivenquantities.

ANSWER: =

radians

PartB

Thetapeisnowwoundbackintothedrumatangularrate .Withwhatspeedwilltheendofthetapemove?(Notethatourdrawingspecifiesthatapositivederivativeof

impliesmotionawayfromthedrum.Becarefulwithyoursigns!Thefactthatthetapeisbeingwoundbackintothedrumimpliesthat ,andfortheendofthetapetomove

closertothedrum,itmustbethecasethat .

HintB.1 Howtoapproachtheprobelm

Thefunction isgivenbythederivativeof withrespecttotime.Computethisderivativeusingtheexpressionfor foundinPartAandthefactthat .


ANSWER: =

Answerintermsof andothergivenquantitiesfromtheproblemintroduction.

ANSWER: =

PartC

Since isapositivequanitity,theansweryoujustobtainedimpliesthat willalwayshavethesamesignas .Ifthetapeisunwinding,bothquanititeswillbepositive.Ifthe


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tapeisbeingwoundbackup,bothquantitieswillbenegative.Nowfind ,thelinearaccelerationoftheendofthetape.

Expressyouranswerintermsof ,theangularaccelerationofthedrum: .

ANSWER: =

PartD

Perhapsthetrickiestaspectofworkingwithconstraintequationsforrotationalmotionisdeterminingthecorrectsignforthekinematicquantities.Consideratireofradius rollingto

theright,withoutslipping,withconstantxvelocity .Find ,the(constant)angularvelocityofthetire.Becarefulofthe

signsinyouranswer;recallthatpositiveangularvelocitycorrespondstorotationinthecounterclockwisedirection.


ANSWER: =

Thisisanexampleoftheappearanceofnegativesignsinconstraintequations--atirerollinginthepositivedirectiontranslationallyexhibitsnegativeangularvelocity,sincerotationisclockwise.

PartE

Assumenowthattheangularvelocityofthetire,whichcontinuestorollwithoutslipping,isnotconstant,butratherthatthetireaccelerateswithconstantangularacceleration .Find

,thelinearaccelerationofthetire.


ANSWER: =

SpinningtheWheels:AnIntroductiontoAngularMomentum

Description:Amixofqualitativeandcomputationalquestionsintroducingangularmomentumanditsbasicapplications.

LearningGoal:Tolearnthedefinitionandapplicationsofangularmomentumincludingitsrelationshiptotorque.

Bynow,youshouldbefamiliarwiththeconceptofmomentum,definedastheproductofanobject'smassanditsvelocity:

.

Youmayhavenoticedthatnearlyeverytranslationalconceptorequationseemstohaveananalogousrotationalone.So,whatmightbetherotationalanalogueofmomentum?

Justastherotationalanalogueofforce ,calledthetorque ,isdefinedbytheformula

,

therotationalanalogueofmomentum ,calledtheangularmomentum ,isgivenbytheformula

,

forasingleparticle.Foranextendedbodyyoumustadduptheangularmomentaofallofthepieces.

Thereisanotherformulaforangularmomentumthatmakestheanalogytomomentumparticularlyclear.Forarigidbodyrotatingaboutanaxisofsymmetry,whichwillbetrueforallpartsinthisproblem,themeasureofinertiaisgivennotbythemass butbytherotationalinertia(i.e.,themomentofinertia) .Similarly,therateofrotationisgivenbythebody's

angularspeed, .Theproduct givestheangularmomentum ofarigidbodyrotatingaboutanaxisofsymmetry.(Notethatifthebodyisnotrotatingaboutanaxisofsymmetry,

thentheangularmomentumandtheangularvelocitymaynotbeparallel.)


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PartA

WhichofthefollowingistheSIunitofangularmomentum?

ANSWER:

PartB

Anobjecthasrotationalinertia .Theobject,initiallyatrest,beginstorotatewithaconstantangularaccelerationofmagnitude .Whatisthemagnitudeoftheangularmomentum

oftheobjectaftertime ?


Findtheangularvelocityfirst;thenusethe definitionofangularmomentum.


ANSWER: =

PartC

Arigid,uniformbarwithmass andlength rotatesabouttheaxispassingthroughthemidpointofthebarperpendiculartothebar.Thelinearspeedoftheendpointsofthebaris

.Whatisthemagnitudeoftheangularmomentum ofthebar?


Findseparatelytherotationalinertiaandtheangularvelocity;thenusethe definitionofangularmomentum.

HintC.2 Rotationalinertiaofthebar

Forthisaxis,therotationalinertiaofthebarisgivenby

,

where isthelengthofthebar.

HintC.3 Findingthemagnitudeoftheangularvelocity

Themagnitudeoftheangularvelocity canbeobtainedfromtherelation

,

where istheradiusofrotation.

Expressyouranswerintermsof , , ,andappropriateconstants.

ANSWER: =


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Youmayrecallthat,accordingtoNewton's2ndlaw,therateofchangeofmomentumofanobjectequalsthenetforceactingontheobject:

.

Similarly,therateofchangeofangularmomentumofanobjectequalsthenettorqueactingontheobject:

.

Therefore,ifthenettorqueactingonanobject(orasystemofobjects)iszero(i.e.,thesystemis"closed"),thentherateofchangeofangularmomentumisalsozero.Inotherwords,thenetangularmomentumofaclosedsystemisconstant(conserved).

Thisstatementisknownasthelawofconservationofangularmomentum.Justlikethelawsofconservationofenergyandmomentum,thelawofconservationofangularmomentumplaysamajorroleinmechanics.

PartD

Theuniformbarshowninthediagramhasalengthof0.80m.Thebarbeginstorotatefromrestinthehorizontalplaneabouttheaxispassingthroughitsleftend.Whatwillbethemagnitudeoftheangularmomentum ofthebar6.0safterthemotionhasbegun?Theforcesactingonthebarareshown.

HintD.1 Howtoapproachtheproblem

First,findthenettorque,andthenuse

.

Expressyouranswerin totwosignificantfigures.

ANSWER: =

PartE

Eachofthefourbarsshowncanrotatefreelyinthehorizontalplaneaboutitsleftend.Forwhichdiagramsisthenettorqueequaltozero?

Typeinalphabeticalordertheletterscorrespondingtothecorrectdiagrams.Forinstance,ifyouthinkthatonlydiagramsA,B,andCanswerthequestion,typeABC.

ANSWER: BC

Ifthesumoftheforcesonabodyiszero,thenthenettorqueisindependentofthepointaboutwhichthetorqueiscalculated.Ifthenetforceonthebodyisnotzero,asistruefor


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mostofthebeamsinthispart,thenthetorquewilldependonthepointaboutwhichyoucalculatethetorque.

PartF

ConsiderthefiguresforPartE.Forwhichdiagramsistheangularmomentumconstant?

HintF.1 Determiningwhenangularmomentumisconstant

Recallthat

.

Thismeansthatifthenettorqueiszero,thentheangularmomentumisaconstant.Thatis,iftherateofchangefor iszero,then can'tbechanging.

Typealphabeticallytheletterscorrespondingtothecorrectdiagrams.Forinstance,ifyouthinkthatonlydiagramsA,B,andCanswerthequestion,typeABC.

ANSWER: BC

Angularmomentumisconservedwhenthenettorqueiszero.Thisisanalogoustothestatementfromlineardynamicsthatmomentumisconservedwhenthenetforceiszero.

PartG

Eachofthedisksinthefigurehasradius .Eachdiskcanrotatefreelyabouttheaxispassingthroughthecenterofthediskperpendiculartotheplaneofthefigure,asshown.For

whichdiagramsistheangularmomentumconstant?Inyourcalculations,usetheinformationprovidedinthediagrams.

Typealphabeticallytheletterscorrespondingtothecorrectdiagrams.Forinstance,ifyouthinkthatonlydiagramsA,B,andCanswerthequestion,typeABC.

ANSWER: AD

PartH

Threedisksarespinningindependentlyonthesameaxlewithoutfriction.Theirrespectiverotationalinertiasandangularspeedsare (clockwise); (counterclockwise);and

(clockwise).Thedisksthenslidetogetherandsticktogether,formingonepiecewithasingleangularvelocity.Whatwillbethedirectionandtherateofrotation ofthe

singlepiece?

HintH.1 Howtoapproachtheproblem

Theangularmomentumforthesystemofthethreedisksisconserved.Therefore,ifyoufindtheangularmomentumofthesystembeforethediskssticktogetherandthemomentofinertiaaftertheyhavestuck,youcansolvefortheangularspeed aftertheyhavestucktogether.

HintH.2 Findtherotationalinertia

Whatistherotationalinertia ofthesinglepiece?

Expressyouranswerintermsof .

ANSWER: =

Expressyouranswerintermsofoneorbothofthevariables and andappropriateconstants.Useaminussignforclockwiserotation.

ANSWER: =


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TorqueaboutthezAxis

Description:Guidesstudentthroughboththe"tangentialforcecomponent"and"momentarm"methodsforfindingtorque.

LearningGoal:Tounderstandtwodifferenttechniquesforcomputingthetorqueonanobjectduetoanappliedforce.

Imagineanobjectwithapivotpointpattheoriginofthecoordinatesystemshown.Theforcevector liesinthexyplane,

andthisforceofmagnitude actsontheobjectatapointinthexyplane.Thevector isthepositionvectorrelativetothe

pivotpointptothepointwhere isapplied.

Thetorqueontheobjectduetotheforce isequaltothecrossproduct .When,asinthisproblem,theforce

vectorandleverarmbothlieinthexyplaneofthepaperorcomputerscreen,onlythezcomponentoftorqueisnonzero.

Whenthetorquevectorisparalleltothezaxis( ),itiseasiesttofindthemagnitudeandsignofthetorque, ,interms

oftheangle betweenthepositionandforcevectorsusingoneoftwosimplemethods:theTangentialComponentofthe

ForcemethodortheMomentArmoftheForcemethod.

Notethatinthisproblem,thepositivezdirectionisperpendiculartothecomputerscreenandpointstowardyou(givenbytheright-handrule ),soapositivetorquewouldcausecounterclockwiserotationaboutthezaxis.

Tangentialcomponentoftheforce

PartA

Decomposetheforcevector intoradial(i.e.,parallelto )andtangential(perpendicularto )componentsasshown.

Findthemagnitudeoftheradialandtangentialcomponents, and .Youmayassumethat isbetweenzeroand90

degrees.

HintA.1 Magnitudeof

Usethegivenanglebetweentheforcevector anditsradialcomponent tocomputethemagnitude .

Enteryouranswerasanorderedpair.Express and intermsof and .

ANSWER:

=

PartB

Isthefollowingstatementtrueorfalse?

Thetorqueaboutpointpisproportionaltothelength ofthepositionvector .

ANSWER: truefalse

PartC


Boththeradialandtangentialcomponentsof generatetorqueaboutpointp.

ANSWER: truefalse


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PartD


Inthisproblem,thetangentialforcevectorwouldtendtoturnanobjectclockwisearoundpivotpointp.

ANSWER: truefalse

PartE

Findthetorque aboutthepivotpointpduetoforce .Youranswershouldcorrectlyexpressboththemagnitudeandsignof .

Expressyouranswerintermsof and orintermsof , ,and .

ANSWER:

=

Momentarmoftheforce

Inthefigure,thedashedlineextendingfromtheforcevectoriscalledthelineofactionof .Theperpendiculardistance fromthepivotpointptothelineofactioniscalledthe

momentarmoftheforce.

PartF

Whatisthelength, ,ofthemomentarmoftheforce aboutpointp?


ANSWER: =

PartG

Findthetorque aboutpdueto .Youranswershouldcorrectlyexpressboththemagnitudeandsignof .

Expressyouranswerintermsof and orintermsof , ,and .

ANSWER:

=

Threeequivalentexpressionsforexpressingtorqueaboutthezaxishavebeendiscussedinthisproblem:

1.Torqueisdefinedasthecrossproductbetweenthepositionandforcevectors.Whenboth and lieinthexyplane,onlythezcomponentoftorqueisnonzero,andthecross

productsimplifiesto:

.

Notethatapositivevaluefor indicatesacounterclockwisedirectionaboutthezaxis.

2.Torqueisgeneratedbythecomponentof thatistangentialtothepositionvector (thetangentialcomponentofforce):


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.3.Themagnitudeoftorqueistheproductoftheforceandtheperpendiculardistancebetweenthezaxisandthelineofactionofaforce, ,calledthemomentarmoftheforce:

.

Score Sum m ary:Your score on this assignment is 0%.You received 0 out of a possible total of 105 points, plus 0 points of extra c redit.


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