8.8

Concepts & Calculations

In this chapter we have studied the concepts of angular displacement, angular velocity, and angular acceleration. We conclude with some examples that review important features of these ideas. Example 9 illustrates that the angular acceleration and the angular velocity can have the same or the opposite direction, depending on whether the angular speed is increasing or decreasing.

| Concepts & Calculations Example 9 | | | Riding a Mountain Bike

A rider on a mountain bike is traveling to the left in Figure 8-18. Each wheel has an angular velocity of +21.7 rad/s, where, as usual, the plus sign indicates that the wheel is rotating in the counterclockwise direction. (a) To pass another cyclist, the rider pumps harder, and the angular velocity of the wheels increases from +21.7 to +28.5 rad/s in a time of 3.50 s. (b) After passing the cyclist, the rider begins to coast, and the angular velocity of the wheels decreases from +28.5 to +15.3 rad/s in a time of 10.7 s. In both instances, determine the magnitude and direction of the angular acceleration (assumed constant) of the wheels.

Figure 8-18 (a) When the angular speed of the wheel is increasing, the angular velocity and the angular acceleration point in the same direction (counterclockwise in this drawing). (b) When the angular speed is decreasing, the angular velocity and the angular acceleration point in opposite directions.

Concept Questions and Answers Is the angular acceleration positive or negative when the rider is passing the cyclist and the angular speed of the wheels is increasing?

Answer Since the angular speed is increasing, the angular acceleration has the same direction as the angular velocity, which is the counterclockwise, or positive, direction (see Figure 8-18a).

Is the angular acceleration positive or negative when the rider is coasting and the angular speed of the wheels is decreasing?

Answer Since the angular speed is decreasing during the coasting phase, the direction of the angular acceleration is opposite to that of the angular velocity. The angular velocity is in the counterclockwise (positive) direction, so the angular acceleration must be in the clockwise (negative) direction (see Figure 8-18b).

Solution

a. | The angular acceleration is the change in the angular velocity, , divided by the elapsed time t:

| (8.4)

As expected, the angular acceleration is positive (counterclockwise).
b. | Now the wheels are slowing down, but still rotating in the positive (counterclockwise) direction. The angular acceleration for this part of the motion is

| (8.4)

Now, as anticipated, the angular acceleration is negative (clockwise).

Example 10 reviews the two different types of acceleration, centripetal and tangential, that a car can have when it travels on a circular road.

| Concepts & Calculations Example 10 | | | A Circular Roadway and the Acceleration of Your Car

Suppose you are driving a car in a counterclockwise direction on a circular road whose radius is r = 390 m (see Figure 8-19). You look at the speedometer and it reads a steady 32 m/s (about 72 mi/h). (a) What is the angular speed of the car? (b) Determine the acceleration (magnitude and direction) of the car. (c) To avoid a rear-end collision with a vehicle ahead, you apply the brakes and reduce your angular speed to in a time of 4.0 s. What is the tangential acceleration (magnitude and direction) of the car?

Figure 8-19 (a) A car is traveling in a counterclockwise direction at a constant angular speed. The tangential velocity of the car is and its centripetal acceleration is . (b) The car is slowing down, so the tangential acceleration points opposite to the tangential velocity .

Concept Questions and Answers Does an object traveling at a constant tangential speed (for example, ) along a circular path have an acceleration?

Answer Yes. Recall that an object has an acceleration if its velocity is changing in time. The velocity has two attributes, a magnitude (or speed) and a direction. In this instance the speed is not changing, since it is steady at 32 m/s. However, the direction of the velocity is changing continually as the car moves on the circular road. As Sections 5.2 and 8.5 discuss, this change in direction gives rise to an acceleration, called the centripetal acceleration . The centripetal acceleration is directed toward the center of the circle (see Figure 8-19a).

Is there a tangential acceleration when the angular speed of an object changes (e.g., when the car’s angular speed decreases to )?

Answer Yes. When the car’s angular speed decreases, for example, its tangential speed vT also decreases. This is because they are related by (Equation 8.9), where r is the radius of the circular road. A decreasing tangential speed vT, in turn, means that the car has a tangential acceleration . The direction of the tangential acceleration must be opposite to that of the tangential velocity , because the tangential speed is decreasing. (If the two vectors were in the same direction, the tangential speed would be increasing.) Figure 8-19b shows these two vectors.

Solution

a. | The angular speed of the car is equal to its tangential speed vT (the speed indicated by the speedometer) divided by the radius of the circular road:

| (8.9)

b. | The acceleration is the centripetal acceleration and arises because the tangential velocity is changing direction as the car travels around the circular path. The magnitude of the centripetal acceleration is

| (8.11)

As always, the centripetal acceleration is directed toward the center of the circle.
c. | The tangential acceleration arises because the tangential speed of the car is changing in time. The magnitude of the tangential acceleration is (Equation 8.10), where is the magnitude of the angular acceleration. The angular acceleration is given by Equation 8.4 as

The magnitude of the angular acceleration is . Therefore, the magnitude of the tangential acceleration is

| (8.10)

Since the car is slowing down, the tangential acceleration is directed opposite to the direction of the tangential velocity, as shown in Figure 8-19b.

At the end of the problem set for this chapter, you will find homework problems that contain both conceptual and quantitative parts. These problems are grouped under the heading Concepts & Calculations.

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