1. Class 28 - Rolling, Torque and Angular Momentum Chapter 11 - Friday October 29th Reading: pages 275 thru 281 (chapter 11) in HRW Read and understand the sample problems Assigned problems from chapter 11 (due at 11pm on Sunday November 7th): 2, 6, 8, 12, 22, 24, 32, 38, 40, 50, 54, 64 Review of rolling motion Torque and angular momentum Newton's second law in angular form Conservation of angular momentum Demos and sample problems

2. Rolling motion as rotation and translation The wheel moves with speed ds/dt

3. Rolling motion as rotation and translation The wheel moves with speed ds/dt The kinetic energy of rolling

4. Rolling down a ramp However, we do not really have to compute f s (see section 12-3).However, we do not really have to compute f s (see section 12-3). We can, instead, analyze the motion about P, in which case, F g sin  is the only force component with a moment arm about P.We can, instead, analyze the motion about P, in which case, F g sin  is the only force component with a moment arm about P. Use:torque = I  Thus:

5. Torque and angular momentum Torque was discussed in the previous chapter; cross products are discussed in chapter 3 (section 3-7) and at the end of this presentation; torque also discussed in this chapter (section 7).Torque was discussed in the previous chapter; cross products are discussed in chapter 3 (section 3-7) and at the end of this presentation; torque also discussed in this chapter (section 7). Here, p is the linear momentum mv of the object.Here, p is the linear momentum mv of the object. SI unit is Kg.m 2 /s.SI unit is Kg.m 2 /s.

6. Torque and angular momentum Here, p is the linear momentum mv of the object.Here, p is the linear momentum mv of the object. SI unit is Kg.m 2 /s.SI unit is Kg.m 2 /s. Torque was discussed in the previous chapter; cross products are discussed in chapter 3 (section 3-7) and at the end of this presentation; torque also discussed in this chapter (section 7).Torque was discussed in the previous chapter; cross products are discussed in chapter 3 (section 3-7) and at the end of this presentation; torque also discussed in this chapter (section 7).

7. Newton's second law in angular form Linear form The vector sum of all the torques acting on a particle is equal to the time rate of change of the angular momentum. For a system of many particles, the total angular momentum is: The net external torque acting on a system of particles is equal to the time rate of change of the system's total angular momentum. angular form No surprise:

8. Angular momentum of a rigid body about a fixed axis We are interested in the component of angular momentum parallel to the axis of rotation: In fact:

9. Conservation of angular momentum 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 system. It follows from Newton's second law that:

10. Conservation of angular momentum 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 system. It follows from Newton's second law that: What happens to kinetic energy? Thus, if you increase  by reducing I, you end up increasing K.Thus, if you increase  by reducing I, you end up increasing K. Therefore, you must be doing some work.Therefore, you must be doing some work. This is a very unusual form of work that you do when you move mass radially in a rotating frame.This is a very unusual form of work that you do when you move mass radially in a rotating frame. The frame is accelerating, so Newton's laws do not hold in this frameThe frame is accelerating, so Newton's laws do not hold in this frame

11. More on conservation of angular momentum

12. The vector product, or cross product