8.2

Angular Velocity and Angular Acceleration

| Angular Velocity

In 2.2 we introduced the idea of linear velocity to describe how fast an object moves and the direction of its motion. The average linear velocity was defined as the linear displacement of the object divided by the time required for the displacement to occur, or (see Equation 2.2). We now introduce the analogous idea of angular velocity to describe the motion of a rigid object rotating about a fixed axis. The average angular velocity (Greek letter omega) is defined as the angular displacement divided by the elapsed time during which the displacement occurs.

| Definition of Average Angular Velocity

| (8.2)

SI Unit of Angular Velocity: radian per second (rad/s)

The SI unit for angular velocity is the radian per second (rad/s), although other units such as revolutions per minute (rev/min or rpm) are also used. In agreement with the sign convention adopted for angular displacement, angular velocity is positive when the rotation is counterclockwise and negative when it is clockwise. Example 3 shows how the concept of average angular velocity is applied to a gymnast.

| Example 3 | | | Gymnast on a High Bar

A gymnast on a high bar swings through two revolutions in a time of 1.90 s, as Figure 8-6 suggests. Find the average angular velocity (in rad/s) of the gymnast.

Figure 8-6 Swinging on a high bar.

Reasoning The average angular velocity of the gymnast in rad/s is the angular displacement in radians divided by the elapsed time. However, the angular displacement is given as two revolutions, so we begin by converting this value into radian measure.
Solution The angular displacement (in radians) of the gymnast is

where the minus sign denotes that the gymnast rotates clockwise (see the drawing). The average angular velocity is

| (8.2)

The instantaneous angular velocity is the angular velocity that exists at any given instant. To measure it, we follow the same procedure used in Chapter 2 for the instantaneous linear velocity. In this procedure, a small angular displacement occurs during a small time interval . The time interval is so small that it approaches zero (), and in this limit, the measured average angular velocity,, becomes the instantaneous angular velocity :

| (8.3)

The magnitude of the instantaneous angular velocity, without reference to whether it is a positive or negative quantity, is called the instantaneous angular speed. If a rotating object has a constant angular velocity, the instantaneous value and the average value are the same.

| Angular Acceleration

In linear motion, a changing velocity means that an acceleration is occurring. Such is also the case in rotational motion; a changing angular velocity means that an angular acceleration is occurring. There are many examples of angular acceleration. For instance, as a compact disc recording is played, the disc turns with an angular velocity that is continually decreasing. And when the push buttons of an electric blender are changed from a lower setting to a higher setting, the angular velocity of the blades increases. We will define the average angular acceleration in a fashion analogous to that used for the average linear acceleration. Recall that the average linear acceleration is equal to the change in the linear velocity of an object divided by the elapsed time (Equation 2.4). When the angular velocity changes from an initial value of at time t0 to a final value of at time t, the average angular acceleration (Greek letter alpha) is defined similarly:

| Definition of Average Angular Acceleration

| (8.4)

SI Unit of Average Angular Acceleration: radian per second squared (rad/s2)
The SI unit for average angular acceleration is the unit for angular velocity divided by the unit for time, or (rad/s)/s = rad/s2. An angular acceleration of +5 rad/s2, for example, means that the angular velocity of the rotating object increases by +5 radians per second during each second of acceleration.
The instantaneous angular acceleration is the angular acceleration at a given instant. In discussing linear motion, we assumed a condition of constant acceleration, so that the average and instantaneous accelerations were identical (). Similarly, we assume that the angular acceleration is constant, so that the instantaneous angular acceleration and the average angular acceleration are the same (). The next example illustrates the concept of angular acceleration.

| Example 4 | | | A Jet Revving Its Engines

A jet awaiting clearance for takeoff is momentarily stopped on the runway. As seen from the front of one engine, the fan blades are rotating with an angular velocity of −110 rad/s, where the negative sign indicates a clockwise rotation (see Figure 8-7). As the plane takes off, the angular velocity of the blades reaches −330 rad/s in a time of 14 s. Find the angular acceleration, assuming it to be constant.

Figure 8-7 The fan blades of a jet engine have an angular velocity in a clockwise direction.

Reasoning Since the angular acceleration is constant, it is equal to the average angular acceleration. The average acceleration is the change in the angular velocity, , divided by the elapsed time, .
Solution Applying the definition of average angular acceleration given in Equation 8.4, we find that

Thus, the magnitude of the angular velocity increases by 16 rad/s during each second that the blades are accelerating. The negative sign in the answer indicates that the direction of the angular acceleration is also in the clockwise direction.

Copyright © 2007 John Wiley & Sons, Inc. All rights reserved.