Angular MomentumMoment of MomentumMomentum Cross ProductLaw of Angular InertiaApplying TorqueStarting the RideSpinning MassInitial SpinSingle Axis RotationLimitationsAngular Momentum VectorAngular MomentumAngular MomentumMoment of Momentum Moment of Momentum To continue the analysis of To continue the analysis of rotational motion, we must also rotational motion, we must also extend the idea of momentum.extend the idea of momentum.Define an angular momentum Define an angular momentum LL..•Based on the momentum Based on the momentum p = mvp = mv•Includes a lever armIncludes a lever arm•Follows the rules for torqueFollows the rules for torquerpsinrpL Momentum Cross Product Momentum Cross Product Angular momentum is a Angular momentum is a vector.vector.•Vector cross product of the Vector cross product of the lever arm and momentum.lever arm and momentum.•Direction follows the right-Direction follows the right-hand rulehand rule•Magnitude from sine ruleMagnitude from sine rulerpsinrpLprLLLaw of Angular InertiaLaw of Angular InertiaThe time derivative of The time derivative of angular momentum vector is angular momentum vector is the net torque vector.the net torque vector.By the law of reaction, all By the law of reaction, all internal torques come in internal torques come in canceling pairs.canceling pairs.•Add external torquesAdd external torquesextiiiiiiiiiiiiiidtLdFrdtLdvmvFrdtLdpdtrddtpdrdtLdprLL)()()()(Applying TorqueApplying TorqueAn external torque changes angular momentum.An external torque changes angular momentum.•dLdL//dtdt = = LL+rpsinpStarting the RideStarting the RideA child of 180 N runs at 4.5 A child of 180 N runs at 4.5 m/s and hops on the edge of m/s and hops on the edge of a merry-go-round with radius a merry-go-round with radius 2.0 m.2.0 m.If the motion is perpendicular If the motion is perpendicular to the radius, what is angular to the radius, what is angular momentum?momentum?The child starts with linear The child starts with linear momentum.momentum.•WW = = mgmg, , mm = = WW//gg = 18 kg. = 18 kg.•pp = = mvmv = 81 kg m/s = 81 kg m/sAt right angles, all the linear At right angles, all the linear momentum contributes.momentum contributes.•L = rp L = rp = 160 kg m= 160 kg m22/s/sThe torque is due to the The torque is due to the friction at contact.friction at contact.m rSpinning MassSpinning MassThe moment of inertia is the analog of mass for The moment of inertia is the analog of mass for rotational motion.rotational motion.The analog for angular momentum would be:The analog for angular momentum would be:IL Initial SpinInitial SpinA child of 180 N runs at 3.0 A child of 180 N runs at 3.0 m/s and hops on the edge of m/s and hops on the edge of a merry-go-round with radius a merry-go-round with radius 2.0 m and mass of 160 kg.2.0 m and mass of 160 kg.What is the period of What is the period of rotation?rotation?The moment of inertia and The moment of inertia and the angular momentum for the angular momentum for the child on the ride was the child on the ride was found before.found before.•II = = IIdd + + IIcc = = 390 kg m390 kg m22•L = rp L = rp = 160 kg m= 160 kg m22/s/sThe period is related to the The period is related to the angular velocity.angular velocity.•LL = = II = = II(2(2//TT))•TT = 2= 2II / / LL = 15 s = 15 smMrSingle Axis RotationSingle Axis RotationAn axis of rotation that is fixed in direction gives a An axis of rotation that is fixed in direction gives a single axis rotationsingle axis rotation..•Simplest case has the axis through the center of massSimplest case has the axis through the center of mass•Angular momentum vector is parallel to the angular velocityAngular momentum vector is parallel to the angular velocityIL LLimitationsLimitationsThere are limitations to the There are limitations to the relationship between angular relationship between angular momentum and angular momentum and angular velocity.velocity.•Moving axis of rotationMoving axis of rotation•Asymmetric axis of rotationAsymmetric axis of rotationAngular momentum and Angular momentum and angular velocity can have angular velocity can have different directions.different directions.LrpAngular Momentum VectorAngular Momentum VectorThe vector form of the law of rotational action is The vector form of the law of rotational action is generalized to use angular momentum vectors.generalized to use angular momentum vectors.•Correct for all axesCorrect for all axes•Correct for changes in direction as well as angular velocityCorrect for changes in direction as well as angular