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

To analyze rotation with constant angular acceleration 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

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 period T is the time for one revolution, and arad = 4π2R/T2.

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.

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.

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

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.

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

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.

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.

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

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

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2

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

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.

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

An unwinding cable Follow Example 9.7.

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

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.

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

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

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