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Chapter11-AngularMomentum MichaelChen
Definitions:
CrossProduct: = ()
AngularMomentum:momentumofarotatingobjectrelativetoanaxis;the
rotationalequivalentoflinearmomentum
Equations:
Torque: =
= AngularMomentum: = = forarigidobjectofknownI
KineticEnergy: =
Solving:
Themaintechniqueusedforangularmomentumisconservationofmomentumby
equatingtheinitialandfinalmomentumstogether.Becarefulnottodosowhenthe
problemdictatesthatthesystemisnotisolated.
Solveforlinearmomentumandangularmomentumseparately,andcombineusinga
systemofequationsifnecessary.!!! + !!! = !!! + !!!! = !
Right-handRule:
CrossProductDirections:
Applied
Direction
(Middle)
Initial
Direction(Index)
Resulting
Force/Direction
(Thumb)
TheRight-handRuleallowsus
todeterminethefinaldirectionofacrossproduct.
Becarefulnottouseyourlefthandwhendeterminingthe
right-handrule.
k
(+)
(-)
DirectionofMotion(Curledfingers)
Momentum:Momentumisconservedinisolated
systems.Thesame
appliesforangular
momentum.
! !"!#$ = !"# + !"!
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Chapter11-AngularMomentum MichaelChen
Problems:1.
Aballtiedtoafishingpoleisswungaroundonahorizontalplane.The
ballofmassmswingswithalengthofLlineuntilithasavelocityv.a.
Whatistheinitialangularspeed?b.
Ifthelineisshortenedbylengthl,whatwilltheangularspeedbecomeprovidedthatnoexternalworkisdonetothesystem?
2.
Agiantmetalplankwithavolume.25x3x10m3andaweightof75kgisstooduponitssmallend(sothatthe10mgoesup)andplacedatthe
edgeofachasm.Inordertopushitovertobridgethechasm,thereisa
smallcannonthatcanshoota10kgleadballamaximumof60abovethe
horizon.Byshootingtheballatthetopoftheplank,theleadwillmerge
withtheplankandpushtheplankover.
a.
Howfarawaydoesthecannonhavetobeinordertohitthetopoftheplank?Treattheballandthecannonaspoints.
b.
Whatisthemomentofinertiaabouttheaxisofrotationaftertheballhits?
c.
Theballhitstheplankwithavelocityof15m/s.Whatwastheangularmomentumaftercollision?Treatthisasanisolated
system.3.
AcylindricalbarrelofmassMandradiusrisallowedtospinfreelyalong
itscenteraxiswithoutfriction.Astudentattemptstoshootanarrowof
massmintothebarreladistancedawayfromtheaxisofrotation.
a.
Ifthecorktravelsatvelocityv,whatistheangularspeedofthebarrel-with-arrowwhenthearrowhits?
b.
Ifdweretoequalzero,whatwouldbetheangularspeed?Explainqualitativelyandquantitatively.
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Chapter11-AngularMomentum MichaelChen
1.
a.
! =!
!
! =
!
!
b.
! = !!! = !!!! =
!!
! =!!!!
!!! !
2.
a.
tan 60 =!"
!
= 5.77
b.
=
85 = ! = 85 10
! = 8500!
m
v
L
m
vf
L-l
=
Themscancel. =
! = ! =
10m
!"!#$=!"#$% +!"##
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Chapter11-AngularMomentum MichaelChen
c.
= =
= 8500
1560
10
= 11041!/
3.
a.
! = ! = =
!+
!
=
+ !
b.
Ifd=0,then=0.
Shootingthecenterofaxiswouldnotapplyanytorqueonthebarrel,soitwouldnotrotate.
Thesin60takesonlythe
horizontalvelocitysinceverticalvelocitydoesnot
affecttheplank.
m
MR
d
!"#$%&'( = !
!""#$=
!risthedistanceofthe
arrowfromtheaxis
