Mastering Physics Mechanics 2 - assessed PDF

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09/04/2017

MasteringPhysics: Mechanics 2 - assessed

Question7 LearningGoal: Tounderstandthedefinitionandthemeaningofmomentofinertia;tobeabletocalculatethemomentsofinertiaforagroup ofparticlesandforacontinuousmassdistributionwithahighdegreeofsymmetry. Bynow,youmaybefamiliarwithasetofequationsdescribingrotationalkinematics.Onethingthatyoumayhavenoticed wasthesimilaritybetweentranslationalandrotationalformulas.Suchsimilarityalsoexistsindynamicsandinthework energydomain. Foraparticleofmass

movingataconstantspeed ,thekineticenergyisgivenbytheformula

considerinsteadarigidobjectofmass befoundbyusingtheformula

.Ifwe

rotatingataconstantangularspeed ,thekineticenergyofsuchanobjectcanno directly:differentpartsoftheobjecthavedifferentlinearspeeds.However,they

allhavethesameangularspeed.Itwouldbedesirabletoobtainaformulaforkineticenergyofrotationalmotionthatis similartotheonefortranslationalmotion;suchaformulawouldincludetheterm insteadof . Suchaformulacan,indeed,bewritten:forrotationalmotionofasystemofsmallparticlesorforarigidobjectwithcontinuou massdistribution,thekineticenergycanbewrittenas . Here, iscalledthemomentofinertiaoftheobject(orofthesystemofparticles).Itisthequantityrepresentingtheinertia withrespecttorotationalmotion. Itcanbeshownthatforadiscretesystem,sayof particles,themomentofinertia(alsoknownasrotationalinertia)isgiven by . Inthisformula, isthemassoftheithparticleand isthedistanceofthatparticlefromtheaxisofrotation. Forarigidobject,consistingofinfinitelymanyparticles,theanalogueofsuchsummationisintegrationovertheentireobject

. Inthisproblem,youwillanswerseveralquestionsthatwillhelpyoubetterunderstandthemomentofinertia,itsproperties, anditsapplicability.Itisrecommendedthatyoureadthecorrespondingsectionsinyourtextbookbeforeattemptingthese questions.

PartA Onwhichofthefollowingdoesthemomentofinertiaofanobjectdepend? Checkallthatapply. ANSWER:

linearspeed linearacceleration angularspeed angularacceleration totalmass shapeanddensityoftheobject locationoftheaxisofrotation

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Correct Unlikemass,themomentofinertiadependsnotonlyontheamountofmatterinanobjectbutalsoonthe distributionofmassinspace.Themomentofinertiaisalsodependentontheaxisofrotation.Thesameobject, rotatingwiththesameangularspeed,mayhavedifferentkineticenergydependingontheaxisofrotation.

Considerthesystemoftwoparticles,aandb,showninthefigure. Particleahasmass ,andparticlebhasmass .

PartB Whatisthemomentofinertia ofparticlea? ANSWER:

   undefined:anaxisofrotationhasnotbeenspecified.

Correct

PartC Findthemomentofinertia ofparticleawithrespecttothexaxis(thatis,ifthexaxisistheaxisofrotation),the momentofinertia ofparticleawithrespecttotheyaxis,andthemomentofinertia ofparticleawithrespecttothe zaxis(theaxisthatpassesthroughtheoriginperpendiculartoboththexandyaxes). Expressyouranswersintermsof

and separatedbycommas.

ANSWER: ,

,

=

Correct

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PartD Findthetotalmomentofinertia ofthesystemoftwoparticlesshowninthediagramwithrespecttotheyaxis. Expressyouranswerintermsof

and .

ANSWER: =

Correct

ForpartsEthroughG,supposethatbothparticlesrotatewiththesameangularspeed abouttheyaxiswhilemaintaining theirdistancesfromtheyaxis.

PartE Usingthetotalmomentofinertia ofthesystemfoundinPartD,findthetotalkineticenergy Rememberthatbothparticlesrotateabouttheyaxis. Expressyouranswerintermsof

ofthesystem.

, ,and .

ANSWER: =

Correct

Nextwewillproveexplicitlythattherotationalkineticenergyformula( formula(

)agreeswiththelinearkineticenergy

).Toseetheconnection,letusfindthekineticenergyofparticleaandb.

PartF Usingtheformulaforkineticenergyofamovingparticle kineticenergy

,findthekineticenergy

ofparticleaandthe

ofparticleb.Rememberthatbothparticlesrotateabouttheyaxis.

Expressyouranswersintermsof

, ,and separatedbyacomma.

Hint1.Findthelinearspeed Usingtheformula

,findthelinearspeed

ofparticlea.

Expressyouranswerintermsof and . ANSWER: =

ANSWER:

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,

=

Correct

PartG Now,usingtheresultsofPartF,findthetotalkineticenergy theyaxis. Expressyouranswerintermsof

ofthesystem.Rememberthatbothparticlesrotateabou

, ,and .

ANSWER:

=

Correct Notsurprisingly,theformulas

and

givethesameresult.Theyshould,ofcourse,since

therotationalkineticenergyofasystemofparticlesissimplythesumofthekineticenergiesoftheindividual particlesmakingupthesystem.

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