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Physics116 Lecture:Rota3onalDynamics 2016 Announcements • WorkshopsmeetinARC336WedsandThurs thisweek. • HWisveryimportantthisweek More Motion Analogues Recap • Add Force / Torque to Correspondences: x t vx t K ax K TRANSLATION vx vx 0 a xt x x0 vx 0t F t 0 1 a t2 2 x vx20 2ax x K 1 mv 2 2 ma I ROTATION vx2 F m 0t 0 2 K 2 0 1 2 I I 2 2 1 2 t2 TorqueisaVECTORPRODUCT RHR: • Counterclockwiserota3onisconsideredOUTOFTHEPAGE¤ • Clockwiserota3onisconsideredINTOTHEPAGE. nd Newton’s2 Law Recallthatlinearaccelera3oniscausedby unbalancedexternalforces,andisinversely propor3onaltotheobject’smass: a=Fnet/m Analogously,angularaccelera3oniscausedby unbalancedexternaltorquesandisinversely propor3onaltotheobject’smomentofiner3a: α=τnet/I nd Newton’s2 Law Angularaccelera3oniscausedby unbalancedexternaltorquesandis inverselypropor3onaltotheobject’s momentofiner3a: α=τnet/I Adiskisfreetorotateaboutanaxis.Aforce appliedatadistancedfromtheaxiscausesan angularaccelera3onα.Whatangular accelera3onisproducedifthesameforceis appliedadistance2dfromtheaxis? 1. α 2. 2α 3. α/2 4. 4α 5. α/4 AwheelofradiusR1hasanaxleofradius R2=¼R1.IfaforceF1isappliedtangent tothewheel,aforceF2,appliedtangent totheaxlethatwillkeepthewheelfrom turning,isequalto 1. F1/4 2. F1 3. 4F1 4. 16F1 5. F1/16 Tostarttheplaygroundmerry-go-roundrota3ng (radius=2m),aropeiswrappedarounditand pulled.Aforceof200Nisexertedontherope for10seconds.Duringthis3methemerry-goroundmakesonecompleterevolu3on. 1. Findthetorqueexertedbytheropeonthe merry-go-round. 2. Findtheangularspeedofthemerry-gorounda_er10seconds. 3. Forthemomentofthenurseofthemerrygo-round. Ineachfigurebelow,thejetengineisslowingdownduetotheapplica3onof aconstanttorque.Alloftheenginesareiden3cal,buttheystartwith differentangularspeedsandhavetorquesofdifferentmagnitudesappliedto therota3ngsha_swithintheengines.Magnitudesoftheini3alangular speedsandtorquesaregiveninthefigures. Rankthesesitua-onsonthebasisofthemagnitudeoftheangular accelera-onoftheenginesastheyslowdown,fromGREATESTtoLEAST. 1. B>D>E>F>A>C 2. C>A>F>E>D>B 3. C=D=F>A=B=E 4. B=D>E=F>A=C 5. A=B=C=D=E=F Aweightis3edtoaropethatiswrappedaroundapulley.Thepulleyis ini3allyrota3ngcounterclockwiseandispullingtheweightup.Thetensionin theropecreatesatorqueonthepulleythatopposesthisrota3on.Consider thefollowingstatementswhichMIGHTbetrueattheinstantthepulleystops rota3ngcounterclockwisebeforestar3ngtorotateclockwise: A. Thetorqueonthepulleyaboutitsaxisisequaltozero. ω B. Theangularaccelera3onofthepulleyisequaltozero. C. Theangularmomentumofthepulleyabouttheaxisisequaltozero. Whichofthefollowingis(are)TRUE? 1. Aonly 2. Bonly 3. Conly 4. AandB 5. A,B,andC Axis of rotation Angularmomentum Recallthatlinearmomentumisthefundamental quan3tyoflinearmo3on.Itistheproductofthe massandthevelocity: Linearmomentum=mv Analogously,angularmomentumisthe fundamentalquan3tyofrota3onalmo3on.Itisthe productofthemomentofiner3aandtheangular velocity: Angularmomentum L=Iω Direc3onofangularmomentum Foranobjectspinningaroundanaxisofsymmetry,theangular momentumvectorandtheangularvelocityvectorareparallel. L L Fouriden3calsmallcylindersrestonacircular horizontalturntableatthevariousposi3onsshown inthetop-viewdiagrambelow.Theturntableis rota3ngclockwiseataconstantangularspeed. Ranktheangularmomentumofthecylinders abouttheaxisofrota-onoftheturntable. 1. 2. 3. 4. 5. R>P=S=T P=S=T>R P=R=S=T P=R=S=T=0 Cannotbedeterminedwithoutunknownangularvelocity Abicyclewheelhasadiameterof64.0cmanda momentofiner3aof.5kgm2.Thebicycleis placedspinningwithaposi3veangularvelocity onasta3onarystand,andaresis3vebraking forceof60Nisappliedtotherimofthe3re. Whatistheangularaccelera3onofthewheel? 1. 38rad/s2 2. 19rad/s2 3. -38rad/s2 4. -19rad/s2 5. noneoftheabove Conserva3onofangularmomentum Recallthatlinearmomentumisconservedifthereareno unbalancedexternalforcesac3ngonthesystem. IfΣFnet=0,thenthetotalmomentumofthesystemis constant. Σmvisconstant Analogously,angularmomentumisalsoconservedifthere arenounbalancedexternaltorquesac3ngonthesystem. IfΣτnet=0,thenthetotalmomentumofthesystemis constant. L=Σiωisconstant Rota3onalcollisions Recallthatlinearmomentumisconservedinall collisions,whilekine3cenergyisconservedonlyin perfectlyelas3ccollisions. ΣMvbefore=ΣmvaCer ButKi≠Kfunlessthecollisionisperfectlyelas3c Analogously,angularmomentumisalsoconservedinall collisions,whilekine3cenergyisconservedonlyin perfectlyelas3cones.For2-dobjectsthatrotateabout anaxisofsymmetry: ΣIωbefore=ΣIωaCer Abulletofmassmisshotatahingedrodandhits therodadistancedfromthehinge.Therodwas Initial bullet ini3allyatrestandhasamomentofiner3aofIo velocity v aboutanaxisthroughitshinge.Thebulletisfiredat d anangleθtotherod,asshowninthetopview,with θ Hinge anini3alvelocityofvatadistancedfromthehinge. Rod Theangularspeedoftherodwiththebullet embeddedrighta_erthecollisionisωf.Assumethere isnofric3oninthehingeandthattherodisfreeto rotateaboutthehingeaxis.Considerthefollowing twostatements: A. Theangularmomentumaboutthehingeoftherodandbullettogethera:erthe bulletisembeddedisthesameastheini3alangularmomentumofthebulletabout thehinge. B. Thetotalkine3cenergyoftherodandbulleta:erthebulletisembeddedisthe sameastheini3alkine3cenergyofthebullet. 1. Aonly 2. Bonly 3. bothAandB 4. Thereisnotenoughinforma3ontodetermine. Adisk,forwhichthemomentofiner3aisI1,rotatesabouta ver3cal,fric3onlessaxlewithangularvelocityω0.Asecond disk,whichhasmomentofiner3aI2andisnotrota3ng,drops ontothefirstdisk,asillustratedbelow.Thediskss3ck together,whichresultsintheirbothspinningwithanangular velocityωf. 1. Whatobjectsgointoyoursystem? 2. Isthelinearmomentumofthesystemconserved?Why orwhynot? 3. Istheangularmomentumofthesystemconserved?Why orwhynot? 4. Calculateωf 5. Isthechangeinthekine3cenergyofthesystemgreater than,lessthanorequaltozero? 6. Ifthechangeinkine3cenergyisnotzero,wheredidthe energycomefrom(orgoto)?