Download angular momentum

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)?