1. Copyright © 2012 Pearson Education Inc. Rotational Kinematics, Inertia Physics 7C lecture 11 Tuesday November 5, 8:00 AM – 9:20 AM Engineering Hall 1200

2. Copyright © 2012 Pearson Education Inc. Review: Angular coordinate A car’s speedometer needle rotates about a fixed axis, as shown at the right. The angle  that the needle makes with the +x-axis is a coordinate for rotation.

3. Copyright © 2012 Pearson Education Inc. Review: Angular velocity The instantaneous angular velocity is  z = d  /dt. A counterclockwise rotation is positive; a clockwise rotation is negative.

4. Copyright © 2012 Pearson Education Inc. Review: Angular velocity is a vector Angular velocity is defined as a vector whose direction is given by the right-hand rule shown in Figure 9.5 below.

5. Copyright © 2012 Pearson Education Inc. Review: Angular acceleration The average angular acceleration is  av-z =  z /  t. The instantaneous angular acceleration is  z = d  z /dt = d 2  /dt 2. Follow Example 9.2.

6. Copyright © 2012 Pearson Education Inc. A. 0.40 rad B. 0.80 rad C. 1.0 rad D. 2.0 rad Q9.2 A DVD is initially at rest so that the line PQ on the disc ’ s surface is along the +x-axis. The disc begins to turn with a constant  z = 5.0 rad/s 2. At t = 0.40 s, what is the angle between the line PQ and the +x-axis?

7. Copyright © 2012 Pearson Education Inc. A. 0.40 rad B. 0.80 rad C. 1.0 rad D. 2.0 rad A9.2 A DVD is initially at rest so that the line PQ on the disc ’ s surface is along the +x-axis. The disc begins to turn with a constant  z = 5.0 rad/s 2. At t = 0.40 s, what is the angle between the line PQ and the +x-axis?

8. Copyright © 2012 Pearson Education Inc. Review: angular and linear motions combine angular and linear motions

9. Copyright © 2012 Pearson Education Inc. A DVD is rotating with an ever- increasing speed. How do the radial acceleration a rad and tangential acceleration a tan compare at points P and Q? A. P and Q have the same a rad and a tan. B. Q has a greater a rad and a greater a tan than P. C. Q has a smaller a rad and a greater a tan than P. D. P and Q have the same a rad, but Q has a greater a tan than P. Q9.3

10. Copyright © 2012 Pearson Education Inc. A DVD is rotating with an ever- increasing speed. How do the radial acceleration a rad and tangential acceleration a tan compare at points P and Q? A. P and Q have the same a rad and a tan. B. Q has a greater a rad and a greater a tan than P. C. Q has a smaller a rad and a greater a tan than P. D. P and Q have the same a rad, but Q has a greater a tan than P. A9.3

11. Copyright © 2012 Pearson Education Inc. A. a faster linear speed and a faster angular speed. B. the same linear speed and a faster angular speed. C. a slower linear speed and the same angular speed. D. the same linear speed and a slower angular speed. E. none of the above Q9.4 Compared to a gear tooth on the rear sprocket (on the left, of small radius) of a bicycle, a gear tooth on the front sprocket (on the right, of large radius) has

12. Copyright © 2012 Pearson Education Inc. A. a faster linear speed and a faster angular speed. B. the same linear speed and a faster angular speed. C. a slower linear speed and the same angular speed. D. the same linear speed and a slower angular speed. E. none of the above A9.4 Compared to a gear tooth on the rear sprocket (on the left, of small radius) of a bicycle, a gear tooth on the front sprocket (on the right, of large radius) has

13. Copyright © 2012 Pearson Education Inc. Kinetic energy Motionkinetic energy How to calculate kinetic energy for angular motion? K = ½ m v 2 speed:ω mass (inertial): ?

14. Copyright © 2012 Pearson Education Inc. Rotational kinetic energy The moment of inertia of a set of particles is I = m 1 r 1 2 + m 2 r 2 2 + … =  m i r i 2 The rotational kinetic energy of a rigid body having a moment of inertia I is K = 1/2 I  2. m m easy to spin hard to spin

15. Copyright © 2012 Pearson Education Inc. Rotational kinetic energy find the moment of inertia along B-C axis, ignore size of the disks.

16. Copyright © 2012 Pearson Education Inc. Rotational kinetic energy find the moment of inertia along A axis, ignore size of the disks.

17. Copyright © 2012 Pearson Education Inc. Moments of inertia of some common bodies

18. Copyright © 2012 Pearson Education Inc. You want to double the radius of a rotating solid sphere while keeping its kinetic energy constant. (The mass does not change.) To do this, the final angular velocity of the sphere must be A. 4 times its initial value. B. twice its initial value. C. the same as its initial value. D. 1/2 of its initial value. E. 1/4 of its initial value. Q9.5

19. Copyright © 2012 Pearson Education Inc. You want to double the radius of a rotating solid sphere while keeping its kinetic energy constant. (The mass does not change.) To do this, the final angular velocity of the sphere must be A. 4 times its initial value. B. twice its initial value. C. the same as its initial value. D. 1/2 of its initial value. E. 1/4 of its initial value. A9.5

20. Copyright © 2012 Pearson Education Inc. Moments of inertia of a rod

21. Copyright © 2012 Pearson Education Inc. Moments of inertia of a cylinder

22. Copyright © 2012 Pearson Education Inc. The three objects shown here all have the same mass M and radius R. Each object is rotating about its axis of symmetry (shown in blue). All three objects have the same rotational kinetic energy. Which one is rotating fastest? A. thin-walled hollow cylinder B. solid sphere C. thin-walled hollow sphere D. two or more of these are tied for fastest Q9.6

23. Copyright © 2012 Pearson Education Inc. The three objects shown here all have the same mass M and radius R. Each object is rotating about its axis of symmetry (shown in blue). All three objects have the same rotational kinetic energy. Which one is rotating fastest? A. thin-walled hollow cylinder B. solid sphere C. thin-walled hollow sphere D. two or more of these are tied for fastest A9.6

24. Copyright © 2012 Pearson Education Inc. An unwinding cable pull the wire with 9N force for 2m, what is the final speed of the wheel?

25. Copyright © 2012 Pearson Education Inc. An unwinding cable work: W = F s = 9N 2m = 18 J kinetic energy: K = ½ I ω 2 = ½ ½ m r 2 ω 2 = 0.045 ω 2

26. Copyright © 2012 Pearson Education Inc. A thin, very light wire is wrapped around a drum that is free to rotate. The free end of the wire is attached to a ball of mass m. The drum has the same mass m. Its radius is R and its moment of inertia is I = (1/2)mR 2. As the ball falls, the drum spins. At an instant that the ball has translational kinetic energy K, the drum has rotational kinetic energy A. K.B. 2K.C. K/2.D. none of these Q9.7

27. Copyright © 2012 Pearson Education Inc. A thin, very light wire is wrapped around a drum that is free to rotate. The free end of the wire is attached to a ball of mass m. The drum has the same mass m. Its radius is R and its moment of inertia is I = (1/2)mR 2. As the ball falls, the drum spins. At an instant that the ball has translational kinetic energy K, the drum has rotational kinetic energy A. K.B. 2K.C. K/2.D. none of these A9.7

28. Copyright © 2012 Pearson Education Inc. The parallel-axis theorem What is the moment of inertial along p axis? : I P ? We know the answer for the center axis.

29. Copyright © 2012 Pearson Education Inc. The parallel-axis theorem The parallel-axis theorem is: I P = I cm + Md 2.

30. Copyright © 2012 Pearson Education Inc. The parallel-axis theorem The parallel-axis theorem is: I P = I cm + Md 2. Verify:

31. Copyright © 2012 Pearson Education Inc. The parallel-axis theorem The parallel-axis theorem is: I P = I cm + Md 2. Proof: