1Chapter 11 – Torque and Angular MomentumI. TorqueII. Angular momentum - DefinitionIII. Newton’s second law in angular formIV. Angular momentum- System of particles- Rigid body- Conservation- Vector quantity.Fr×=τDirection: right hand rule.Magnitude:FrFrFrFr⊥⊥==⋅=⋅=)sin(sinϕϕτTorque is calculated with respect to (about) a point. Changing the point can change the torque’s magnitude and direction.I. Torque2II. Angular momentum- Vector quantity.)( vrmprl×=×=Direction: right hand rule.Magnitude:vmrprprprvmrvmrprl ⋅===⋅=⋅⋅=⋅⋅=⋅=⊥⊥⊥⊥)sin(sinsinϕϕϕl positive counterclockwisel negative clockwiseDirection of l is always perpendicular to plane formed by r and p.Units: kg m2/sIII. Newton’s second law in angular formdtpdFnet=Linear Angulardtldnet=τSingle particleThe vector sum of all torques acting on a particle is equal to the time rate of change of the angular momentum of that particle.Proof:( )( )netnetFrFramrdtldarmvvarmvdtrddtvdrmdtldvrmlτ=×=×=×==×=×+×=×+×=→×=∑)()(V. Angular momentum- System of particles:∑==++++=niinlllllL1321...3∑∑===→==ninetinetniidtLddtlddtLd1,1ττIncludes internal torques (due to forces between particles within system) and external torques (due to forces on the particles from bodies outside system).Forces inside system third law force pairs torqueintsum =0 The only torques that can change the angular momentum of a system are the external torques acting on a system.The net external torque acting on a system of particles is equal to the time rate of change of the system’s total angular momentum L.- Rigid body (rotating about a fixed axis with constant angular speed ω):))(()90)(sin)((iiiiiivmrprl == MagnitudeDirection: li perpendicular to riand piILIrmrmlLziniiiniiniizzωωωω==⋅===∑∑∑===21211ωIL=Rotational inertia of a rigid body about a fixed axisiirv ⋅=ω2)(iiiiiirmrmrlωω==extzzdtdLIdtdIdtdLταω=→==4- Conservation of angular momentum:dtLdnet=τNewton’s second lawIf no net external torque acts on the system (isolated system)cteLdtLd=→=0Law of conservation of angular momentum:)( systemisolatedLLfi=If the net external torque acting on a system is zero, the angular momentum of the system remains constant, no matter what changes take place within the system.Net angular momentum at time ti= Net angular momentum at later time tfIf the component of the net external torque on a system along a certain axis is zero, the component of the angular momentum of the system along that axis cannot change, no matter what changes take place within the system.This conservation law holds not only within the frame of Newton’s mechanics but also for relativistic particles (speeds close to light) and subatomic particles.ffiiIIωω=( Ii,f, ωi,frefer to rotational inertia and angular speed before and after the redistribution of mass about the rotational axis ).5Examples:If< Ii(mass closer to rotation axis)Torque ext =0 Iiωi= Ifωfωf> ωiSpinning volunteerSpringboard diver- Center of mass follows parabolic path.- When in air, no net external torque about COM Diver’s angular momentum L constant throughout dive (magnitude and direction).- L is perpendicular to the plane of the figure (inward).- Beginning of dive She pulls arms/legs closerIntention: I is reduced ω increases- End of dive layout positionPurpose: I increases slow rotation rate less “water-splash”6TranslationRotationFFr×=τpprl×=ForceAngular momentumCOMiivMpP==∑∑=iilLNewton’s second lawdtPdF=dtLdnet=τConservation lawTorqueLinear momentumAngularmomentumLinear momentum(system of particles, rigid body)System of particlesRigid body, fixedaxis L=component along that axis.ωIL=Newton’ssecond lawcteP =(Closed isolated system)Conservation lawcteL =(Closed isolated system)IV. Precession of a gyroscopeGyroscope: wheel fixed to shaft and free tospin about shaft’s axis.If one end of shaft is placed on a support and released Gyroscope falls by rotating downward about the tip of the support.dtLd=τThe torque causing the downward rotation (fall) changes angular momentum of gyroscope.Torque caused by gravitational force acting on COM.MgrMgr == 90sinτNon-spinning gyroscope7If released with shaft’s angle slightly upward first rotates downward, then spins horizontally about vertical axis z precession due to non-zero initialangular momentumI = rotational moment of gyroscope about shaftω = angular speed of wheel about shaftSimplification: i) L due to rapid spin >> L due to precessionii) shaft horizontal when precession starts ωIL=Rapidly spinning gyroscopeVector L along shaft, parallel to rTorque perpendicular to L can only change theDirection of L, not its magnitude.MgrdtdtdLdtLd ==→=ττωϕIMgrdtLdLd ==Rapidly spinning gyroscopeMgrdtdtdLdtLd ==→=ττωϕIMgrdtLdLd ==Precession
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