Physics 201: Lecture 20, Pg 1 Lecture 20 Goals: Revisit vector cross product Introduce angular momentum Discuss conservation of angular momentum

Physics 201: Lecture 20, Pg 2 Linear Momentum and Angular Momentum comparison l Newton’s 2 nd Law (inertia times acceleration): l Units of angular momentum are kg m 2 /s l Momentum: (inertia time velocity) l And the relationship to linear momentum?

Physics 201: Lecture 20, Pg 3 Question: How does the angular momentum vary? l A 10 kg cart is moving with a speed of 5 m/s following the path shown How do the magnitudes of the angular momenta at positions shown compare (  is changing)? Axis r1r1 r2r2 r3r3

Physics 201: Lecture 20, Pg 4 Question: How does the angular momentum vary? l A 10 kg cart is moving with a speed of 5 m/s. l How do the magnitudes of the angular momenta at positions shown compare? Axis r1r1 r2r2 r3r3

Physics 201: Lecture 20, Pg 5 Question: How does the angular momentum vary? l A 10 kg cart is moving with a speed of 5 m/s l How do the magnitudes of the angular momenta at positions shown compare? Axis r1r1 r2r2 r3r3

Physics 201: Lecture 20, Pg 6 Angular momentum question Does a hockey puck sliding straight along the ice have angular momentum? A. Yes B. No C. Not Sure D. It depends on the mass E. It depends on whether the puck is moving in a straight line or around a curved path “axis of rotation” v

Physics 201: Lecture 20, Pg 7 Angular momentum comparison l Three pucks of equal mass are sliding smoothly along on a horizontal surface. There trajectories are given by the respective dashed lines. l How do the magnitudes of their angular momenta compare? L = I  = I v / r = m v r I = m r 2 l L= m r v A. L A < L B < L C B. L A = L B = 0 < L C C. 0 < L A = L B < L C D. L A = L B = L C r v/2 Axis of rotation r v v A B C

Physics 201: Lecture 20, Pg 8 Putting it all together

Physics 201: Lecture 20, Pg 9 Conservation of momentum

Physics 201: Lecture 20, Pg 10 Example: A Two Disk Collision A disk of mass M and radius R rotates around the z axis with angular velocity  0. A second identical disk, initially not rotating, is dropped on top of the first. There is friction between the disks, and eventually they rotate together with angular velocity  F. 00 z FF z

Physics 201: Lecture 20, Pg 11 Example: Two Disks A disk of mass M and radius R rotates around the z axis with initial angular velocity  0. A second identical disk, at rest, is dropped on top of the first. There is friction between the disks, and eventually they rotate together with angular velocity  F. 00 z FF z No External Torque so L z is constant L i = L f  I  i = I  f  ½ mR 2  = ½ 2mR 2  f

Physics 201: Lecture 20, Pg 12 Another Demonstration: Conservation of Angular Momentum l Figure Skating : AA z BB z Arm I A I B  A  B L A = L B No External Torque so L z is constant even if internal work done.

Physics 201: Lecture 20, Pg 13 Demonstration: Conservation of Angular Momentum l Figure Skating : AA z BB z Arm I A < I B  A >  B ½ I A  A 2 > ½ I B  B 2 (work needs to be done) I A  A = L A = L B = I B  B No External Torque so L z is constant even if internal work done.

Physics 201: Lecture 20, Pg 14 Example: Bullet hitting stick l A uniform stick of mass M and length D is pivoted at the center. A bullet of mass m is shot through the stick at a point halfway between the pivot and the end. The initial speed of the bullet is v 1, and the final speed is v 2.  What is the angular speed  F of the stick after the collision? (Ignore gravity) v1v1 v2v2 M FF beforeafter m D D/4

Physics 201: Lecture 20, Pg 15 Example: Bullet hitting stick What is the angular speed  F of the stick after the collision? (Ignore gravity). l Process: (1) Define system (2) Identify Conditions (1) System: bullet and stick (No Ext. torque, L is constant) (2) Momentum is conserved (I stick = I = MD 2 /12 ) L init = L bullet + L stick = m v 1 D/4 + 0 = L final = m v 2 D/4 + I  f v1v1 v2v2 M FF beforeafter m D D/4

Physics 201: Lecture 20, Pg 16 For Tuesday l Statics: Read Chapter 12.1 to 12.3