Chapter 9

1. Chapter 9 Rotation of Rigid Bodies

2. Goals for Chapter 9 • To describe rotation in terms of angular coordinate, angular velocity, and angular acceleration • To analyze rotation with constant angular acceleration • To relate rotation to the linear velocity and linear acceleration of a point on a body • To understand moment of inertia and how it relates to rotational kinetic energy • To calculate moment of inertia

3. Introduction • A wind turbine, a CD, a ceiling fan, and a Ferris wheel all involve rotating rigid objects. • Real-world rotations can be very complicated because of stretching and twisting of the rotating body. But for now we’ll assume that the rotating body is perfectly rigid.

4. 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.

5. Units of angles • An angle in radians is  = s/r, as shown in the figure. • One complete revolution is 360° = 2π radians.

6. Angular velocity • The angular displacement of a body is  = 2 – 1. • The average angular velocity of a body is av-z = /t. • The subscript z means that the rotation is about the z-axis. • The instantaneous angular velocity is z = d/dt. • A counterclockwise rotation is positive; a clockwise rotation is negative.

7. Calculating angular velocity • The angular position of a 0.36 m diameter flywheel is given by • (a) Find in radians and in degrees at • and • Find the distance that a particle on the • flywheel rim moves over the interval from • to

8. Example: • Find the average angular velocity in rad/s and • rev/min over that interval. • Find the instantaneous angular velocities at • and

9. Solution: • (a) • (b) • (c)

10. Solution: • (d)

11. 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.

12. 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.

13. Example: • In the previous example, find the • Average angular acceleration between • and • (b) instantaneous acceleration at • and

14. Solution: • (a) • (b)

15. Angular acceleration as a vector • For a fixed rotation axis, the angular acceleration and angular velocity vectors both lie along that axis.

16. Rotation with constant angular acceleration • The rotational formulas have the same form as the straight-line formulas, as shown in Table 9.1 below.

17. Example: • You have finished watching a movie on Blue-ray and the disc is slowing to a stop. The disc’s angular velocity at t = 0 is and its angular acceleration is a constant • A line PQ on the disc’s surface lies along the +x-axis at t = 0. Find the: • (a) disc’s angular velocity at t = 0.300 s. • (b) angle that PQ makes with the +x-axis at this • time.

18. Solution: • (a) • (b)

19. Rotation of a Blu-ray disc • A Blu-ray disc is coming to rest after being played. • Follow Example 9.3 using Figure 9.8 as shown at the right.

20. Relating linear and angular kinematics • For a point a distance r from the axis of rotation: its linear speed is v = r its tangential acceleration is atan = r its centripetal (radial) acceleration is arad = v2/r =

21. Example: • An athlete whirls a discus in a circle of radius 80.0 cm. At a certain instant, the athlete is rotating at 10.0 rad/s and the angular speed is increasing at 50 . At this instant, find the tangential and centripetal components of the acceleration of the discus and the magnitude of the acceleration.

22. Solution:

23. An athlete throwing a discus • Follow Example 9.4 and Figure 9.12.

24. Designing a propeller • Follow Example 9.5 and Figure 9.13.

25. 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. • Follow Example 9.6 using Figure 9.15 below.

26. Moments of inertia of some common bodies • Table 9.2 gives the moments of inertia of various bodies.

27. Example: • We wrap a light, nonstretching cable around a solid cylinder of mass 50 kg and diameter 0.120 m, which rotates in frictionless bearings about a stationary axis. We pull the free end of the cable with a constant 9.0 N force for a distance of 2.0 m; it turns the cylinder as it unwinds without slipping. The cylinder is initially at rest. Find its final angular speed and the final speed of the cable.

28. An unwinding cable • Follow Problem-Solving Strategy 9.1. • Follow Example 9.7.

29. Solution: • The work done on the cylinder is: • The moment of inertia is: • Conservation of energy gives:

30. Solution: • The final tangential speed of the cylinder, and hence the final speed of the cable is:

31. An unwinding cable

32. An unwinding cable

33. An unwinding cable

34. An unwinding cable

35. Gravitational potential energy of an extended body • The gravitational potential energy of an extended body is the same as if all the mass were concentrated at its center of mass: Ugrav = Mgycm.

36. The parallel-axis theorem • The parallel-axis theorem is: IP = Icm + Md2. • Follow Example 9.9 using Figure 9.20 below.

37. Moment of inertia of a hollow or solid cylinder • Follow Example 9.10 using Figure 9.22.

38. Moment of inertia of a uniform solid sphere • Follow Example 9.11 using Figure 9.23.