1. Chapter 11 Angular Momentum; General Rotation 10-9 Rotational Kinetic Energy 11-2 Vector Cross Product; Torque as a Vector 11-3Angular Momentum of a Particle

2.  orque  orque  orque  orque  orque  orque  orque  orque  orque 10-7 Determining Moments of Inertia If a physical object is available, the moment of inertia can be measured experimentally. Otherwise, if the object can be considered to be a continuous distribution of mass, the moment of inertia may be calculated:

3.  orque  orque  orque  orque  orque  orque  orque  orque  orque 10-7 Determining Moments of Inertia Example 10-13: Parallel axis. Determine the moment of inertia of a solid cylinder of radius R 0 and mass M about an axis tangent to its edge and parallel to its symmetry axis.

4.  orque  orque  orque  orque  orque  orque  orque  orque  orque 10-7 Determining Moments of Inertia The parallel-axis theorem gives the moment of inertia about any axis parallel to an axis that goes through the center of mass of an object:

5.  orque  orque  orque  orque  orque  orque  orque  orque  orque 10-7 Determining Moments of Inertia The perpendicular-axis theorem is valid only for flat objects.

6.  orque  orque  orque  orque  orque  orque  orque  orque  orque 10-8 Rotational Kinetic Energy Translational Kinetic Energy r Recall v=r   m

7.  orque  orque  orque  orque  orque  orque  orque  orque  orque 10-8 Rotational Kinetic Energy A object that both translational and rotational motion also has both translational and rotational kinetic energy:

8.  orque  orque  orque  orque  orque  orque  orque  orque  orque 10-8 Rotational Kinetic Energy: Rotating Rod Example 10-15: A rod of mass M is pivoted on a frictionless hinge at one end. The rod is held at rest horizontally and then released, Determine the angular velocity of the rod when it reaches the vertical position, and the speed of the rod’s tip at this moment.

9.  orque  orque  orque  orque  orque  orque  orque  orque  orque 10-8 Rotational Kinetic Energy The torque does work as it moves the wheel through an angle θ:

10.  orque  orque  orque  orque  orque  orque  orque  orque  orque 10-8 Rotational Kinetic Energy

11.  orque  orque  orque  orque  orque  orque  orque  orque  orque 10-8 Rotational Kinetic Energy When using conservation of energy, both rotational and translational kinetic energy must be taken into account. All these objects have the same potential energy at the top, but the time it takes them to get down the incline depends on how much rotational inertia they have.

12.  orque  orque  orque  orque  orque  orque  orque  orque  orque 10-9 Rotational Plus Translational Motion; Rolling In (a), a wheel is rolling without slipping. The point P, touching the ground, is instantaneously at rest, and the center moves with velocity (rotation and translation) In (b) the same wheel is seen from a reference frame where C is at rest. Now point P is moving with velocity – (Pure rotation). The linear speed of the wheel is related to its angular speed:

13.  orque  orque  orque  orque  orque  orque  orque  orque  orque Chap10: 6, 10, 19, 22, 24, 27, 30, 32, 57, 64, 73 Exam 3 Study Guide Problems

14.  orque  orque  orque  orque  orque  orque  orque  orque  orque 11-1 Angular Momentum—Objects Rotating About a Fixed Axis: Ice Skater Movie http://www.youtube.com/watch?v=AQLtcEAG9v0

15.  orque  orque  orque  orque  orque  orque  orque  orque  orque Question You are skating and you spin with your arms outstretched. When you bring your arms in close to your body, your moment of inertia A) increases B) decreases C) stays the same

16.  orque  orque  orque  orque  orque  orque  orque  orque  orque Question You are skating and you spin with your arms outstretched. When you bring your arms in close to your body, your angular velocity A) increases B) decreases C) stays the same

17.  orque  orque  orque  orque  orque  orque  orque  orque  orque Problem 4 4. (II) A figure skater can increase her spin rotation rate from an initial rate of 1.0 rev every 1.5 s to a final rate of 2.5 rev/s. If her initial moment of inertia was 4.6 kg.m 2 what is her final moment of inertia? How does she physically accomplish this change?