8.3

The Equations of Rotational Kinematics

In Chapters 2 and 3 the concepts of displacement, velocity, and acceleration were introduced. As illustrated in the Concepts-at-a-Glance chart in Figure 2-8, we combined these concepts and developed a set of equations called the equations of kinematics for constant acceleration (see Tables 2-1 and 3-1). These equations are a great aid in solving problems involving linear motion in one and two dimensions.
CONCEPTS AT A GLANCE We now take a similar approach for rotational motion. The Concepts-at-a-Glance chart in Figure 8-8 shows that we will bring together the ideas of angular displacement, angular velocity, and angular acceleration to produce a set of equations called the equations of rotational kinematics for constant angular acceleration. These equations, like those developed in Chapters 2 and 3 for linear motion, will prove very useful in solving problems that involve rotational motion.

Figure 8-8 CONCEPTS AT A GLANCE The equations of rotational kinematics for constant angular acceleration are obtained by bringing together the notions of angular displacement, angular velocity, and angular acceleration. The rotational motion of the wheels of these antique bicycles can be described by using these equations, provided that the angular acceleration is constant. Both cyclists have taken their feet off the pedals because they spin too rapidly as the large front wheel accelerates down the hill. (Ron Kuenstler/Journal Times/(c)AP/Wide World Photos)

A complete description of rotational motion requires values for the angular displacement , the angular acceleration , the final angular velocity , the initial angular velocity , and the elapsed time . In Example 4, for instance, only the angular displacement of the fan blades during the 14-s interval is missing. Such missing information can be calculated, however. For convenience in the calculations, we assume that the orientation of the rotating object is given by at time t0 = 0 s. Then, the angular displacement becomes , and the time interval becomes .
In Example 4, the angular velocity of the fan blades changes at a constant rate from an initial value of to a final value of . Therefore, the average angular velocity is midway between the initial and final values:

In other words, when the angular acceleration is constant, the average angular velocity is given by

| (8.5)

With a value for the average angular velocity, Equation 8.2 can be used to obtain the angular displacement of the fan blades:

In general, when the angular acceleration is constant, the angular displacement can be obtained from

| (8.6)

This equation and Equation 8.4 provide a complete description of rotational motion under the condition of constant angular acceleration. Equation 8.4 (with t0 = 0 s) and Equation 8.6 are compared with the analogous results for linear motion in the first two rows of Table 8-1. The purpose of this comparison is to emphasize that the mathematical forms of Equations 8.4 and 2.4 are identical, as are the forms of Equations 8.6 and 2.7. Of course, the symbols used for the rotational variables are different from those used for the linear variables, as Table 8-2 indicates.

Table 8-1 The Equations of Kinematics for Rotational and Linear Motion

Rotational Motion () |  | Linear Motion () |
| (8.4) |  | (2.4)
| (8.6) |  | (2.7)
| (8.7) |  | (2.8)
| (8.8) |  | (2.9)

Table 8-2 Symbols Used in Rotational and Linear Kinematics

Rotational Motion | Quantity Linear | Motion
| Displacement | x
| Initial velocity | v0
| Final velocity | v
| Acceleration | a
t | Time | t

In Chapter 2, Equations 2.4 and 2.7 are used to derive the remaining two equations of kinematics (Equations 2.8 and 2.9). These additional equations convey no new information but are convenient to have when solving problems. Similar derivations can be carried out here. The results are listed as Equations 8.7 and 8.8 below and in Table 8-1; they can be inferred directly from their counterparts in linear motion by making the substitution of symbols indicated in Table 8-2:

| (8.7)

| (8.8)

The four equations in the left column of Table 8-1 are called the equations of rotational kinematics for constant angular acceleration. The following example illustrates that they are used in the same fashion as the equations of linear kinematics.

| Example 5 | | | Blending with a Blender

The blades of an electric blender are whirling with an angular velocity of +375 rad/s while the “puree” button is pushed in, as Figure 8-9 shows. When the “blend” button is pressed, the blades accelerate and reach a greater angular velocity after the blades have rotated through an angular displacement of +44.0 rad. The angular acceleration has a constant value of +1740 rad/s2. Find the final angular velocity of the blades.

Figure 8-9 The angular velocity of the blades in an electric blender changes each time a different push button is chosen.

Reasoning The three known variables are listed in the table below, along with a question mark indicating that a value for the final angular velocity is being sought.

|  |  |  | t
+44.0 rad | +1740 rad/s2 | ? | +375 rad/s |

We can use Equation 8.8, because it relates the angular variables , , , and .

Problem solving insight Each equation of rotational kinematics contains four of the five kinematic variables, , , , , and t. Therefore, it is necessary to have values for three of these variables if one of the equations is to be used to determine a value for an unknown variable.

Solution From Equation 8.8 it follows that

The negative root is disregarded, since the blades do not reverse their direction of rotation.

The equations of rotational kinematics can be used with any self-consistent set of units for , , , , and t. Radians are used in Example 5 only because data are given in terms of radians. Had the data for , , and been provided in rev, rev/s2, and rev/s, respectively, then Equation 8.8 could have been used to determine the answer for directly in rev/s. In any case, the reasoning strategy for applying the kinematics equations is summarized as follows.

Problem solving insight

| Reasoning Strategy

Applying the Equations of Rotational Kinematics
1. | Make a drawing to represent the situation being studied, showing the direction of rotation.
2. | Decide which direction of rotation is to be called positive (+) and which direction is to be called negative (−). In this text we choose the counterclockwise direction to be positive and the clockwise direction to be negative, but this is arbitrary. However, do not change your decision during the course of a calculation.
3. | In an organized way, write down the values (with appropriate + and − signs) that are given for any of the five rotational kinematic variables (, , , , and t). Be on the alert for implied data, such as the phrase “starts from rest”, which means that the value of the initial angular velocity is . The data box in Example 5 is a good way to keep track of this information. In addition, identify the variable(s) that you are being asked to determine.
4. | Before attempting to solve a problem, verify that the given information contains values for at least three of the five kinematic variables. Once the three variables are identified for which values are known, the appropriate relation from Table 8-1 can be selected.
5. | When the rotational motion is divided into segments, the final angular velocity of one segment is the initial angular velocity for the next segment.
6. | Keep in mind that there may be two possible answers to a kinematics problem. Try to visualize the different physical situations to which the answers correspond.

Need more practice?

Interactive LearningWare8.1
After 10.0 s, a spinning roulette wheel has slowed down to an angular velocity of +1.88 rad/s, where the plus sign indicates a counterclockwise rotation. During this time, the wheel rotates through an angular displacement of +44.0 rad. Determine the angular acceleration of the wheel.
Related Homework: Problem 22

Check Your Understanding 2

The blades of a ceiling fan start from rest and, after two revolutions, have an angular speed of 0.50 rev/s. The angular acceleration of the blades is constant. What is the angular speed after eight revolutions? (The answer is given at the end of the book.)
Background: Since the angular acceleration is constant, the equations of kinematics for rotational motion apply. These equations are given in Table 8-1.
For similar questions (including conceptual counterparts), consult Self-Assessment Test 8.1. The test is described next.

Test your understanding of the material in Sections 8.1–8.3:

· | Angular Displacement
· | Angular Velocity
· | Angular Acceleration
· | The Equations of Rotational Kinematics

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