Rotational Kinematics, Inertia

1. Physics 7C lecture 11 • Rotational Kinematics, Inertia Tuesday November 5, 8:00 AM – 9:20 AM Engineering Hall 1200

2. 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. Review: Angular velocity • The instantaneous angular velocity is z = d/dt. • A counterclockwise rotation is positive; a clockwise rotation is negative.

4. 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. Review: Angular acceleration • The average angular acceleration is av-z = z/t. • The instantaneous angular acceleration is z = dz/dt = d2/dt2. • Follow Example 9.2.

6. 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 az = 5.0 rad/s2. At t = 0.40 s, what is the angle between the line PQ and the +x-axis? A. 0.40 rad B. 0.80 rad C. 1.0 rad D. 2.0 rad

7. 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 az = 5.0 rad/s2. At t = 0.40 s, what is the angle between the line PQ and the +x-axis? A. 0.40 rad B. 0.80 rad C. 1.0 rad D. 2.0 rad

8. Review: angular and linear motions • combine angular and linear motions

9. Q9.3 A DVD is rotating with an ever-increasing speed. How do the radial acceleration arad and tangential acceleration atan compare at points P and Q? A. P and Q have the same aradand atan. B. Q has a greater arad and a greater atan than P. C. Q has a smaller arad and a greater atan than P. D. P and Q have the same arad, but Q has a greater atan than P.

10. A9.3 A DVD is rotating with an ever-increasing speed. How do the radial acceleration arad and tangential acceleration atan compare at points P and Q? A. P and Q have the same aradand atan. B. Q has a greater arad and a greater atan than P. C. Q has a smaller arad and a greater atan than P. D. P and Q have the same arad, but Q has a greater atan than P.

11. 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 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

12. 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 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

13. Kinetic energy • Motion kinetic energy speed: ω mass (inertial): ? K = ½ m v2 • How to calculate kinetic energy for angular motion?

14. Rotational kinetic energy • The moment of inertia of a set of particles is • I = m1r12 + m2r22 + … = miri2 • The rotational kinetic energy of a rigid body having a moment of inertia I is K = 1/2 I2. easy to spin hard to spin m m

15. Rotational kinetic energy • find the moment of inertiaalong B-C axis, ignore size of the disks.

16. Rotational kinetic energy • find the moment of inertiaalong A axis, ignore size of the disks.

17. Moments of inertia of some common bodies

18. Q9.5 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.

19. A9.5 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.

20. Moments of inertia of a rod

21. Moments of inertia of a cylinder

22. Q9.6 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

23. A9.6 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

24. An unwinding cable • pull the wire with 9N force for 2m, what is the final speed of the wheel?

25. An unwinding cable work: W = F s = 9N 2m = 18 J kinetic energy: K = ½ I ω2 = ½ ½ m r2 ω2 = 0.045 ω2

26. Q9.7 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)mR2. 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

27. A9.7 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)mR2. 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

28. The parallel-axis theorem • What is the moment of inertial along p axis? : IP? We know the answer for the center axis.

29. The parallel-axis theorem • The parallel-axis theorem is: IP = Icm + Md2.

30. The parallel-axis theorem • The parallel-axis theorem is: IP = Icm + Md2. Verify:

31. The parallel-axis theorem • The parallel-axis theorem is: IP = Icm + Md2. Proof: