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2. Chapter 9 Rotations of Rigid Bodies

3. Main Points of Chapter 9 Rotation of a rigid body – angular velocity, angular acceleration Rotational kinetic energy Rotational inertia Torque Angular momentum Conservation of angular momentum Rolling

4. 9-1 Simple Rotations of a Rigid Body When a rigid body rotates around a single axis, all points on it move through the same angle in the same time. Distance of the point from the axis doesn’t change

5. 9-1 Simple Rotations of a Rigid Body Angular Velocity (9-2) (9-3) Simple rotations are one- dimensional – the only variable needed is the angle θ If the angular velocity ω is constant, the angle increases linearly with time

6. 9-1 Simple Rotations of a Rigid Body Right-hand rule for finding direction of angular velocity:

7. 9-1 Simple Rotations of a Rigid Body (9-7) (9-8) (9-9) (9-10) For simple rotations, the equations of motion are analogous to those for one- dimensional linear motion:

8. 9-1 Simple Rotations of a Rigid Body Acceleration of a Point in a Rotating Rigid Body If the angular velocity is increasing, there is a tangential component to the acceleration as well as the radial component. (9-11) (9-12)

9. 9-2 Rotational Kinetic Energy Kinetic energy of rotation: (9-14) Rotational inertia: (9-15) The kinetic energy of rotation can be found by adding up the kinetic energy of each piece of the object: This is made simpler by defining the rotational inertia:

10. 9-3 Evaluation of Rotational Inertia Rotational inertia of a continuous object is found by integrating: (9-19) These integrals have been done for many standard geometries.

11. 9-3 Evaluation of Rotational Inertia

12. Parallel-Axis Theorem (9-27) The parallel-axis theorem lets us calculate the rotational inertia around any axis once we know it around a parallel axis passing through the center of mass. I pa is the rotational inertia around the parallel axis, I cm the rotational inertia around the center of mass, M the total mass, and d the distance from the parallel axis to the center of mass.

13. 9-3 Evaluation of Rotational Inertia This shows an example of axes used in the parallel-axis theorem:

14. 9-4 Torque What causes rotation? Force Lever arm – how far is force from axis of rotation? Angle – does force have components toward or away from the axis of rotation?

15. (9-28) (9-29) 9-4 Torque The torque depends on the force, the lever arm, and the angle: The torque causes angular acceleration; just how much depends on the rotational inertia:

16. 9-4 Torque A right-hand rule gives the direction of the torque.

17. 9-4 Torque Gravity and Extended Objects Gravitational torque acts at the center of mass, as if all mass were concentrated there:

18. 9-5 Angular Momentum and Its Conservation (9-37) (9-38) (9-39) Angular momentum is analogous to linear momentum: Torque is given by the change in angular momentum: Rotational kinetic energy can also be written in terms of angular momentum:

19. 9-6 Rolling (9-40) Connection between speed and angular velocity: This assumes rolling without slipping.

20. (9-41) (9-42) 9-6 Rolling Motion is rotation around contact point (with respect to that point), giving the kinetic energy: We can use the parallel-axis theorem to find I contact :

21. 9-6 Rolling Equivalently, we can write the kinetic energy of a rolling object as the sum of: the kinetic energy of rotation about its center of mass, and the kinetic energy of the linear motion of the object as if all the mass were at the center of mass. (9-43)

22. 9-6 Rolling Therefore, the rate at which a rolling object accelerates depends on its rotational inertia – this determines how much of its kinetic energy will be rotational and how much translational.

23. Summary of Chapter 9 Rotations of a rigid body are one-dimensional Equations parallel those of linear motion If angular velocity is increasing, acceleration has a transverse component Kinetic energy of rotation depends on rotational inertia Parallel-axis theorem – once rotational inertia is known about any axis, the rotational inertia around any parallel axis can be found easily

24. Summary of Chapter 9, cont. Torque depends on force, lever arm, and angle Angular acceleration depends on torque and on rotational inertia Torque is change in angular momentum Rolling without slipping: combination of rotation and translation Rotational inertia determines how much potential energy becomes translational kinetic energy and how much rotational