Rotation of Rigid Bodies: Angular Coordinates, Velocity, and Acceleration

1. Chapter 9 Rotation of Rigid Bodies Modifications by Mike Brotherton

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. Review: Acceleration for uniform circular motion • For uniform circular motion, the instantaneous acceleration always points toward the center of the circle and is called the centripetal acceleration. • The magnitude of the acceleration is arad = v2/R. • The periodT is the time for one revolution, and arad = 4π2R/T2.

4. Introduction – Rigid Rotating Bodies • 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.

5. Angular coordinates • 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.

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

7. 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.

8. Calculating angular velocity • We first investigate a flywheel. • Follow Example 9.1.

9. 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.

10. 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.

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

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

13. 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 = r

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

15. 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.

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

17. An unwinding cable • Follow Example 9.7.

18. More on an unwinding cable • Follow Example 9.8 using Figure 9.17 below.

19. 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.

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

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

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