1. Conservation of Angular Momentum 8.01 W10D2 Young and Freedman: 10.5-10.6

2. Announcements Math Review Night Next Tuesday from 9-11 pm Pset 10 Due Nov 15 at 9 pm No Class Friday Nov 11 Exam 3 Tuesday Nov 22 7:30-9:30 pm W011D1 Rolling without Slipping, Translation and Rotation Reading Assignment Young and Freedman: 10.3-4

3. Time Derivative of Angular Momentum for a Point Particle Angular momentum of a point particle about S: Time derivative of the angular momentum about S: Product rule Key Fact: Result:

4. Torque and the Time Derivative of Angular Momentum: Point Particle Torque about a point S is equal to the time derivative of the angular momentum about S.

5. Conservation of Angular Momentum Rotational dynamics No torques Change in Angular momentum is zero Angular Momentum is conserved

6. Concept Question: Change in Angular Momentum A person spins a tennis ball on a string in a horizontal circle with velocity (so that the axis of rotation is vertical). At the point indicated below, the ball is given a sharp blow (force ) in the forward direction. This causes a change in angular momentum in the 1. direction 2. direction 3. direction

7. Concept Question: Change in Angular Momentum Answer 3. The torque about the center of the circle points in the positive -direction. The change in the angular momentum about the center of the circle is proportional to the angular impulse about the center of the circle which is in the direction of the torque

8. Table Problem: Cross-section for Meteor Shower A meteor of mass m is approaching earth as shown on the sketch. The distance h on the sketch below is called the impact parameter. The radius of the earth is R e. The mass of the earth is m e. Suppose the meteor has an initial speed of v 0. Assume that the meteor started very far away from the earth. Suppose the meteor just grazes the earth. You may ignore all other gravitational forces except the earth. Find the impact parameter h and the cross section πh 2.

9. Angular Momentum for System of Particles Treat each particle separately Angular momentum for system about S

10. Conservation of Angular Momentum for System of Particles Assumption: all internal torques cancel in pairs Rotational dynamics No external torques Change in Angular momentum is zero Angular Momentum is conserved

11. Angular Impulse and Change in Angular Momentum Angular impulse Change in angular momentum Rotational dynamics

12. Kinetic Energy of Cylindrically Symmetric Body A cylindrically symmetric rigid body with moment of inertia I z rotating about its symmetry axis at a constant angular velocity Angular Momentum Kinetic energy

13. Concept Question: Figure Skater A figure skater stands on one spot on the ice (assumed frictionless) and spins around with her arms extended. When she pulls in her arms, she reduces her rotational moment of inertia and her angular speed increases. Assume that her angular momentum is constant. Compared to her initial rotational kinetic energy, her rotational kinetic energy after she has pulled in her arms must be 1.the same. 2.larger. 3.smaller. 4.not enough information is given to decide.

14. Concept Question: Figure Skater Answer 2. Call the rotation axis the z-axis. When she pulls her arms in there are no external torques in the z-direction so her z-component of her angular momentum about the axis of rotation is constant. The magnitude of her angular momentum about the rotation axis passing through the center of the stool is L= Iω. Her kinetic energy is Since L is constant and her moment of inertia decreases when she pulls her arms, her kinetic energy must increase.

15. Constants of the Motion When are the quantities, angular momentum about a point S, energy, and momentum constant for a system? No external torques about point S : angular momentum about S is constant No external work: mechanical energy constant No external forces: momentum constant

16. Rotational and Translational Comparison QuantityRotationTranslation Force Torque Kinetic Energy Momentum Angular Momentum Kinetic Energy

17. Demo: Rotating on a Chair A person holding dumbbells in his/her arms spins in a rotating stool. When he/she pulls the dumbbells inward, the moment of inertia changes and he/she spins faster.

18. Concept Question: Twirling Person A woman, holding dumbbells in her arms, spins on a rotating stool. When she pulls the dumbbells inward, her moment of inertia about the vertical axis passing through her center of mass changes and she spins faster. The magnitude of the angular momentum about that axis is 1.the same. 2.larger. 3.smaller. 4.not enough information is given to decide.

19. Concept Question: Twirling Person Answer 1. Call the rotation axis the z-axis. Because we assumed that there are no external torques in the z-direction acting on the woman then the z-component of her angular momentum about the axis of rotation is constant. (Note if we approximate the figure skater as a symmetric body about the z-axis than there are no non-z- components of the angular momentum about a point lying along the rotation axis.)

20. Demo: Train B134 (1) At first the train is started without the track moving. The train and the track both move, one opposite the other. (2) Then the track is held fixed and the train is started. When the track is let go it does not revolve until the train is stopped. Then the track moves in the direction the train was moving. (3) Next try holding the train in place until the track comes up to normal speed (Its being driven by the train). When you let go the train remains in its stationary position while the track revolves. You shut the power off to the train and the train goes backwards. Put the power on and the train remains stationary. A small gauge HO train is placed on a circular track that is free to rotate.

21. Table Problem: Experiment 6 A steel washer, is mounted on the shaft of a small motor. The moment of inertia of the motor and washer is I r. Assume that the frictional torque on the axle is independent of angular speed. The washer is set into motion. When it reaches an initial angular velocity ω 0, at t = 0, the power to the motor is shut off, and the washer slows down during the time interval Δt 1 = t a until it reaches an angular velocity of ω a at time t a. At that instant, a second steel washer with a moment of inertia I w is dropped on top of the first washer. The collision takes place over a time Δt col = t b - t a. a)What is the angular deceleration α 1 during the interval Δt 1 = t a ? b)What is the angular impulse due to the frictional torque on the axle during the collision? c) What is the angular velocity of the two washers immediately after the collision is finished?

22. Experiment 6: Conservation of Angular Momentum

23. Appendix: Angular Momentum and Torque for a System of Particles

24. Angular Momentum and Torque for a System of Particles Change in total angular momentum about a point S equals the total torque about the point S

25. Internal and External Torques Total external torque Total internal torque Total torque about S

26. Internal Torques Internal forces cancel in pairs Does the same statement hold about pairs of internal torques? ``By the Third Law this sum becomes The vector points from the j th element to the i th element.

27. Central Forces: Internal Torques Cancel in Pairs Assume all internal forces between a pair of particles are directed along the line joining the two particles then With this assumption, the total torque is just due to the external forces No isolated system has been encountered such that the angular momentum is not constant.