PARTIIIProblemsProblem1Acomputerdiskstartsrotatingfromrestatconstantangularacceleration.Ifittakes0.750stocompleteitssecondrevolution:a)Howlongdoesittaketocompletethefirstcompleterevolution;b)Whatistheangularacceleration?ANSWER:a)1.81s;b) α =3.84rad i s−2 Problem 2 Amachinist turns the power on to a grindingwheel, at rest, at time t = 0 s. Thewheelacceleratesuniformlyfor10sandreachestheoperatingangularvelocityof42rad/s.Thewheelisrunatthatangularvelocityfor39sandthenpowerisshutoff.Thewheeldeceleratesuniformlyat2.6rad/s2untiltheitstops.Inthissituation,theangularaccelerationofthewheelbetweent=0sandt=10sisclosestto:a)5.9rad/s2b)7.6rad/s2c)4.2rad/s2d)5.0rad/s2e)6.7rad/s2
PartialSolution:Use
�
ωz = ω0z + α zt→α z = ωz −ω0z
t with
�
ω0z = 0and
�
ωz = 42 rads
Answer:cProblem3AlighttriangularplateOABisinahorizontalplane.Threeforces,F1=3N,F2=1N,andF3=9N,actontheplate,whichispivotedaboutaverticalaxesthroughpointO.InFigurebelow,themagnitudeofthetorqueduetoforceF1abouttheaxisthroughpointOisclosestto:
PartialSolution:Hereuse
!τ = !r ×
!F thatgivesτ1 = rF1 sin60
0 ,whereyoushouldbeabletodeducethattheanglebetween
!r and !F1 is 60! answere)
Problem4***Inthediagramonthenextpage,themassisreleasedfromrest.Asitdescendstheroperotatesthepulleyandthehollowspherewithoutslipping.T1isthetensioninthebottompartoftherope,andT2isthetensioninthetoppartoftherope,with
T1 ≠T2 .i) Drawafree-bodydiagramonm,anduseNewton’ssecondlaw(equation
5-1)toobtain,theequation,T1 −mg= −ma ,whereaistheacceleration.ii) Drawafree-bodydiagramonthepulleythatincludesthefourforces
actingonit,anduseNewton’ssecondlaw(equation10-45)forrotationtoobtaintheequationT1r −T2r = Iα p = I /r2( )α pr .
a)1.1N•mb)1.4N•mc)0.90N•md)1.8N•me)1.6N•m
iii) Drawafree-bodydiagramonthehollowthatincludesthethreeforcesactingonit,anduseNewton’ssecondlaw(equation10-45)forrotationtoobtaintheequationT2R = Isα s = 2/3( )MR2α s .
iv) Usetheno-slipconditions,a= rα p anda= Rα s ,tofindtheacceleration
ofm,andtheangularaccelerationofthesphereandpulley,aswellasthetensionT1andT2.Compareyouranswerswiththeonesyouobtainbytheenergymethod.ANSWER: a=1.2mi s−2 ;T1=36.9NandT2=8.8N.
Problem5Ablockofmass5.00kgisreleasedfromrest,slidesdownasurfaceinclinedat !9.36tothehorizontal.Thecoefficientofkineticfrictionis 1.0=kµ .AstringattachedtotheblockiswrappedaroundasolidcylindricalflywheelofM=25.0kgandR=0.15m.Thereisfrictionbetweentheropeandtheflywheelsothattheropedoesnotslipagainstthewheel.
A)Drawafreeenergydiagramontheblockshowingallforcesactingonit,andhencedeterminethekineticforceoffrictionontheblock.B)Usingtheconservationoftotalenergyorthework-energytheorem,determinethelinearspeedoftheblockafterithasslid1.1mdowntheincline(asshownindiagram).Don’tforgetthattheflywheelwillbespinning.C)Calculatetheangularvelocityoftheflywheelaftertheblockhasslid1.1m?ANSWER:A)fk =3.92N ;B)UseconservationofenergyWext = − fkd = ΔEmech = ΔU +ΔK = −mgh+ 1/2( )mv2 + 1/2( )Iω 2 ,
d=1.1m, h= dsin36.9! ,m=5kg,I = 1/2( )MR2 ,M=25kg,R=.15m,withno-slipconditionωR = v , v =1.1mi s−1 ;C)ω =7.441s−1 .
36.9°
L
Solidcylinderwithmomentofinertia
L=1.1m
HangingBoxm=4.3kgPulley, ,r=0.099m
HollowSphere, ,
M=11kg,R=0.820
Problem6***In the diagram below, a uniform spherical boulder starts from rest and rolls down a 50.0 m high hill. The top half of the hill is rough so that the boulder rolls without slipping, but the bottom half is covered with ice, and so is frictionless. What is the translational speed of the boulder when it reaches the bottom of the hill? ANSWER: 29 m/s
Problem7Indiagrambelowa2kgrockisatpointPtravelinghorizontallywithaspeedof12m/s.Atthisinstantwhatisthemagnitudeanddirectionoftheangularmomentum?Iftheonlyforceactingontherockisitsweight,whatistherateofchange(magnitudeanddirection)oftheangularmomentum?AngularMomentum
!L = !r × !p = m!r × !v validforpointparticlew.r.t.pointO
!r ,r=8m
36.9°143.1°
!v Forthemagnitude
L =
!L = mvr sin143.1" = 2kg ×12m / s × 8m × .6
L=115.2kg-m2/s.
25.0m
25.0m
Solidsphere(table9.2)
+ Clockwiseispositive
Point1
Point2
Point3
RoughRollwithoutslipping
SmoothSphereslips
M–massofsphereR–radiusofsphere
Directionperpendiculartox-yplaneindicates+zoutofthepageindicates–zintopage
Also+xright+yup
Usingtherighthandruleonitiseasytoseethatthe
directionof is–zorintothepage
Torqueduetogravityonparticlew.r.t.pointO.
!τ = !r ×
!Fg ,Fg = mg = 2kg × 9.8m / s
2 = 19.6N !r ,r=8m
53.1°36.9°
!Fg
Sincetherateofchangeofangularmomentum !τ = d
!L / dt hasoppositedirection
(+z)comparedtothedirectionofthecurrentangularmomentum !L = !r × !p = m!r × !v
(-z),theangularmomentumisdecreasing.Canyouseethesimilaritywithourmuchearlierdiscussiononlinearkinetics?FINALCOMMENTANDADVICE:Angularmomentum
!L andTorque
!τ dependon
theorigin(O).Forexample, !L = 0 and
!τ = 0maybezerofororiginO,butnonzero
!L ≠ 0 and
!τ ≠ 0 inanotheroriginO/.Studyangularmomentumproblemsand
staticequilibriumproblems.Problem8A carousel has a radius of 3.0m and amoment of inertia of IC = 8000kg •m2 , forrotation about axis perpendicular to the its center. The carousel is rotatingunpoweredandwithoutfrictionwithanangularvelocityof1.2rad/s.An80-kgmanruns with a velocity of 5.0 m/s, on a line tangent to the rim of the carousel,overtakingit.Themanrunsontothecarouselandgrabsholdofapoleontherim.
a)Beforethecollision,whatisthemagnitudeoftheangularmomentumoftherotatingcarousel,
!LC ,withrespecttothecenterofthecarousel?Whatisthe
directionof !LC ?ANSWER:
LC = 9600 kg im2
s,direction ⊙ outofpage,+z,or
!LC = 9600kg im2
s⎛
⎝⎜⎞
⎠⎟k̂ b)Beforethecollision,whatisthemagnitudeoftheangular
momentoftherunning80-kgman, !LM ,withrespecttothecenterofthecarousel?
Usingtherighthandruleon itiseasytoseethatthedirectionof is+zoroutthepage.
Thiscanbeexpressedinadifferentunit
Usingsecondlawforrotationintermsofangularmomentum
.Hencethenettorqueistherateofchangeofangular
momentum
+y
+x
Directionperpendiculartox-yplaneindicates+zoutofthepageindicates–zintopage
USETHISCONVENTIONTOINDICATETHEDIRECTIONOFANGULARMOMENTUM
Whatisthedirectionof !LM ?ANSWER:
LM = 1200 kg im2
s,direction ⊙ outofpage
(+z),or
!LC = 1200kg im2
s⎛
⎝⎜⎞
⎠⎟k̂ .c)Afterthecollisionwhenthemanisonthe
carousel,whatisthemagnitudeofthefinalangularvelocityofthecarousel(withthemanonit),ω fC ?Whatisthedirectionofthefinalangularvelocity
!ω fC ?Note:
Itotal = IC +mR2 .ANSWER:ω fc = 1.24
radsoutofpage ⊙ or+z,
!ω fc = 1.24
radsk̂