《大學物理簡明教程》 課件 6-2 Rotational dynamics

DisplacementVelocityAccelerationForceMassLinearKineticEnergyMomentumAnglechangeAngularVelocityAngularAccelerationMomentofForce(Torque)MomentofinertiaRotationalKineticEnergyAngularMomentumFormotioninacircle,thereareangularequivalentsforthequantitiesforlinearmotion.LinearCircularCircularMotionvsLinearMotiondF

F

massaccelerationangularaccelerationMassmr

angularacceleration

momentofinertia(I)

forapointmassdoingcircularmotion

forrotatingobjects

o

MomentofInertia

TheSIunitofmomentofinertiaiskgm2orNewton’s2ndlawinrotationThemomentofinertiaofarotatingobjectdependsbothonthemass

andthegeometry(size,shape,rotationalaxis,…)Itdependsonhowmassisdistributedinspace,soitalsodependsonwhichaxistheobjectrotatesaround.I=mr2rI=?mr2rExample:MomentofInertiaofauniformrigidrod

I=?dHowaboutiftheaxisisatoneendoftherod?

ringrodroddisksolidcylindersolidspheresphericalshellMomentofInertiab)

=I

(soliddisc,I=?mr2)

=0.5×100×0.52×/30

=5/12

=1.31NmAmotorisattachedtoa100kg,1mdiametersoliddiscflywheel.Themotorspinstheflywheelfromrestto600rpmin10minutes.Find:Theangularacceleration.Thetorqueexertedbythemotor.Ifthetorqueisfromthebeltattherimoftheflywheel,whatisthefrictionalforcebetweenthebeltandtheflywheel?Example

c)

=fr

f=/r

f=1.31/0.5

=2.62Nii.W=θ

=1.31×6000=2.47×104JAmotorisattachedtoa100kg,1mdiametersoliddiscflywheel.Themotorspinstheflywheelfromrestto600rpmin10minutes.Find:d)Theworkdonebythemotorinspinninguptheflywheel.e)Thekineticenergyoftheflywheel.e)Work=gaininrotationalenergy=2.47×104J

RotationalkineticEnergyW=θTheworkdonebyatorqueispositiveif

and

θ

areinthesamedirectionnegativeif

and

θ

areinoppositedirectionsRotationalkineticenergyhasasimilarexpressionasitslinearequivalent–thelinearkineticenergy:

ExampleCalculatethekineticenergyofacylinderofthemass12kgandtheradius0.20m,rollingalongahorizontalplanewithatranslationalvelocityof0.30ms-1.Themomentofinertiaofthecylinderis0.24kgm2.Thereisnosliding,sopointPmustbestationaryattheinstantofcontact.Theforwardsvelocityviscancelledoutbythe“backwards”speed

r,so

=v/r.

LinearMomentum:P=mvAngularmomentumL=I

AngularMomentumCompareitwiththelinearmomentum?Massmr

L=I=rmvForthemotionofapointmassTherateofchangeofangularmomentum:

vOmdθL=mvd·sinθrLdependsonwheretherotationalaxisis.ConservationofAngularMomentumUsingtheSunasthereferencepoint,thegravitationalforceontheplanethasnotorque,sotheangularmomentumoftheplanetisconserved.mr

Ifthenetexternaltorqueonasystemiszero,thetotalangularmomentumisconserved.-LawofConservationofAngularMomentum

Howaboutthelinearmomentum?Istheangularmomentumoftheplanetconserved?ExampleAbullethitsawoodenblockhungonaropeandembeddedinit.Whatisthespeedoftheblockrightafterthecollision?

mMv0Conservationoflinearmomentuminhorizontaldirectiongives:

Conservationofangularmomentumgives:

ExampleAbullethitsawoodenblockhungonaropeandembeddedinit.Whatisthespeedoftheblockrightafterthecollision?Conservationoflinearmomentuminhorizontaldirectiongives:

Conservationofangularmomentumgives:

θmMTv0θmMlv0vl·cosθvExampleAbullethitsawoodenrodhungatahingeandembeddedinit.Whatistheangularspeedoftherodrightafterthecollision?Theforceatthehingecanbeinanydirec