Chapter 11 – Torque and Angular Momentum

1 Chapter 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 = Direction:right hand :FrFrFrFr == = =)sin(sin Torque is calculated with respect to (about) a point. Changing the point can change the Torque s magnitude and TorqueII. Angular Momentum - Vector quantity.)(vrmprl = =Direction:right hand :vmrprprprvmrvmrprl === = = = = )sin(sinsin l positive counterclockwisel negative clockwiseDirection of l is always perpendicular to plane formed by r and :kg m2/sIII. Newton s second law in Angular formdtpdFnet= LinearAngulardtldnet = Single particleThe vector sum of all torques acting on a particle is equal to the time rate of change of the Angular Momentum of that :()()netnetFrFramrdtldarmvvarmvdtrddtvdr mdtldvrml = = = == = + = + = = )()(V.

2 Angular Momentum - System of particles: ==++++= === ==ninetinetniidtLddtlddtLd1,1 Includes 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 net external Torque acting on a system of particles is equal to the time rate of change of the system s total Angular Momentum Rigid body (rotating about a fixed axis with constant Angular speed ):))(()90)(sin)((iiiiiivmrprl== MagnitudeDirection.

3 Li perpendicular to riand piILIrmrmlLziniiiniiniizz == === ===21211 IL=Rotational inertia of a rigid body about a fixed axisiirv = 2)(iiiiiirmrmrl ==extzzdtdLIdtdIdtdL = ==- Conservation of Angular Momentum :dtLdnet = Newton s second lawIf no net external Torque acts on the system (isolated system)cteLdtLd= = 0 Law 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 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.

4 No matter what changes take place within the conservation law holds not only within the frame of Newton s mechanics but also for relativistic particles (speeds close to light) and subatomic =( Ii,f, i,frefer to rotational inertia and Angular speed before and after the redistribution of mass about the rotational axis ).Examples:If< Ii(mass closer to rotation axis) Torque ext =0 Ii i= If f f> iSpinning volunteerSpringboard diver- Center of mass follows parabolic When in air, no net external Torque about COM Diver s Angular Momentum L constant throughout dive (magnitude and direction).

5 - L is perpendicular to the plane of the figure (inward).- Beginning of dive She pulls arms/legs closerIntention:I is reduced increases- End of dive layout positionPurpose:I increases slow rotation rate less water-splash TranslationRotationF Fr = p prl =ForceAngular momentumCOMiivMpP == =iilL Newton s second lawdtPdF =dtLdnet = Conservation lawTorqueLinear momentumAngularmomentumLinear Momentum (system of particles, rigid body)System of particlesRigid body, fixedaxis L=component along that axis. IL=Newton ssecond lawcteP= (Closed isolated system)Conservation lawcteL= (Closed isolated system)IV. Precession of a gyroscopeGyroscope.

6 Wheel fixed to shaft and free tospin about shaft s one end of shaft is placed on a support and released Gyroscope falls by rotating downward about the tip of the = The Torque causing the downward rotation (fall) changes Angular Momentum of caused by gravitational force acting on 90sin Non-spinning gyroscopeIf released with shaft s angle slightly upward first rotates downward, then spins horizontally about vertical axis z precessiondue to non-zero initialangular momentumI = rotational moment of gyroscope about shaft = Angular speed of wheel about shaftSimplification: i) L due to rapid spin >> L due to precessionii) shaft horizontal when precession starts IL=Rapidly spinning gyroscopeVector L along shaft, parallel to rTorque perpendicular to L can only change theDirection of L, not its = IMgrdtLdLd==Rapidly spinning gyroscopeMgrdtdtdLdtLd== = IMgrdtLdLd==Precession rate: IMgrdtd==