Angular Momentum 2 • Angular momentum of a particle Text sections 11.2 - 11.4 Practice Problems: Chapter 11, problems 11, 13, 15, 19, 53 Physics 1D03 Review Quiz Two astronauts are held together by a long rope and rotate about their common center of mass. One has twice the mass of other. One astronaut gathers in 1/3 of the rope separating them. Which of the following remains constant? A) Kinetic energy B) Angular velocity C) Angular momentum D) Tension in the rope By what factor do each of the others change? Physics 1D03 Angular momentum of a particle L = r×p = r ×(mv) z L This is the fundamental definition of L. y • L is a vector . O r v • Like torque, it depends on the choice x m φ of origin (or “pivot”). • If the particle motion is all in the x-y plane, L is parallel to the z axis. Physics 1D03 Angular momentum of a particle (2-D): v r ⊥ |L| = mrv ⊥ v = mvr sin φ , etc m φ For a particle travelling in a circle (constant | r|), v⊥ = r ω, and 2 L = mrv ⊥ = mr ω = Iω Physics 1D03 A hockey puck slides in a straight line at constant speed past a physicist at O. How does its angular momentum about O change with time? A) increases, then decreases B) decreases, then increases C) remains constant, but not zero O D) is zero unless the puck is spinning v Physics 1D03 General motion: “orbital” and “spin” angular momentum Angular momentum of a particle: L = r×p = r ×(mv) of a rotating rigid body: L = I ω.ω.ω. In general, for a moving, rotating rigid body, ω L = r×(mvCM ) + ICM The first term is called the “orbital” angular momentum and the second term is the “spin” angular momentum. Example: angular momentum of a planet about the sun. Physics 1D03 Example: Uniform thin hoop (mass M, radius R); axis perpendicular to hoop. Calculate its angular momentum (about P) when it rotates about P at angular speed ω. P CM Physics 1D03 Review Exercise: Compare this with the Parallel-Axis Theorem D CM 2 I = ICM + MD Physics 1D03 Quiz Angular momentum provides an elegant approach to Atwood’s Machine. We will find the acclerations of the masses using “external torque = rate of change of L”. For now: assume equal masses , so that m 1 descends at constant speed (after an initial push), and we calculate the angular momentum about the centre of the pulley (point O). As the mass ω descends, the angular momentum L1 (of m 1 only) O will R A) increase B) remain constant C) decrease v m2 v m1 Physics 1D03 Atwoods Machine, frictionless, massive pulley L For m1 : 1 = | r1 x p 1|= Rp 1 L so 1 = +m 1vR L 2 = +m2vR R O ω Lpulley = Iω = Iv/R R Thus L = ( m + m + I/R 2)vR 1 2 r 2 so dL/dt = ( m1 + m2 + I/R )aR v m2 v m1 Torque, τ = m1gR − m2gR p = (m 1 − m2 )gR 1 p1 Write τ = dL/dt , and complete the calculation to solve for a. Note that we only consider the external torques on the entire system. Physics 1D03 Summary Particle: L = r×p = r ×(mv) ω Rigid body: L = r×(mvCM ) + ICM Physics 1D03.