Concept Summary

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Topic | Discussion | Learning Aids
Angular displacement | 8.1 ROTATIONAL MOTION AND ANGULAR DISPLACEMENT When a rigid body rotates about a fixed axis, the angular displacement is the angle swept out by a line passing through any point on the body and intersecting the axis of rotation perpendicularly. By convention, the angular displacement is positive if it is counterclockwise and negative if it is clockwise. |
Radian | The radian (rad) is the SI unit of angular displacement. In radians, the angle is defined as the circular arc of length s traveled by a point on the rotating body divided by the radial distance r of the point from the axis:

| (8.1)

Average angular velocity | 8.2 ANGULAR VELOCITY AND ANGULAR ACCELERATION The average angular velocity is the angular displacement divided by the elapsed time :

| (8.2)

Instantaneous angular velocity | As approaches zero, the average angular velocity becomes equal to the instantaneous angular velocity . The magnitude of the instantaneous angular velocity is called the instantaneous angular speed. |
Average angular acceleration | The average angular acceleration is the change in the angular velocity divided by the elapsed time :

| (8.4)

Instantaneous angular acceleration | As approaches zero, the average angular acceleration becomes equal to the instantaneous angular acceleration . |
Equations of rotational kinematics | 8.3 THE EQUATIONS OF ROTATIONAL KINEMATICS The equations of rotational kinematics apply when a rigid body rotates with a constant angular acceleration about a fixed axis. These equations relate the angular displacement , the angular acceleration , the final angular velocity , the initial angular velocity , and the elapsed time . Assuming that at , the equations of rotational kinematics are

| (8.4)

| (8.6)

| (8.7)

| (8.8)

These equations may be used with any self-consistent set of units and are not restricted to radian measure.

Use Self-Assessment Test 8.1 to evaluate your understanding of Sections 8.1–cutnell3158c08-sec1-0004.

Relations between angular and tangential variables | 8.4 ANGULAR VARIABLES AND TANGENTIAL VARIABLES When a rigid body rotates through an angle about a fixed axis, any point on the body moves on a circular arc of length s and radius r. Such a point has a tangential velocity (magnitude = vT ) and, possibly, a tangential acceleration (magnitude = aT). The angular and tangential variables are related by the following equations:

| (8.1)

| (8.9)

| (8.10)

These equations refer to the magnitudes of the variables involved, without reference to positive or negative signs, and only radian measure can be used when applying them.
Centripetal acceleration | 8.5 CENTRIPETAL ACCELERATION AND TANGENTIAL ACCELERATION The magnitude ac of the centripetal acceleration of a point on an object rotating with uniform or nonuniform circular motion can be expressed in terms of the radial distance r of the point from the axis and the angular speed :

| (8.11)

This point experiences a total acceleration that is the vector sum of two perpendicular acceleration components, the centripetal acceleration and the tangential acceleration ; .
Total acceleration | 8.6 ROLLING MOTION The essence of rolling motion is that there is no slipping at the point where the object touches the surface upon which it is rolling. As a result, the tangential speed vT of a point on the outer edge of a rolling object, measured relative to the axis through the center of the object, is equal to the linear speed v with which the object moves parallel to the surface. In other words, we have

| (8.12)

The magnitudes of the tangential acceleration aT and the linear acceleration a of a rolling object are similarly related:

| (8.13)

Use Self-Assessment Test 8.2 to evaluate your understanding of Sections 8.4–8.6.

Right-hand rule | 8.7 THE VECTOR NATURE OF ANGULAR VARIABLES The direction of the angular velocity vector is given by a right-hand rule. Grasp the axis of rotation with your right hand, so that your fingers circle the axis in the same sense as the rotation. Your extended thumb points along the axis in the direction of the angular velocity vector.
The angular acceleration vector has the same direction as the change in the angular velocity.

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