9.1

The Action of Forces and Torques on Rigid Objects

The mass of most rigid objects, such as a propeller or a wheel, is spread out and not concentrated at a single point. These objects can move in a number of ways. Figure 9-1a illustrates one possibility called translational motion, in which all points on the body travel on parallel paths (not necessarily straight lines). In pure translation there is no rotation of any line in the body. Because translational motion can occur along a curved line, it is often called curvilinear motion or linear motion. Another possibility is rotational motion, which may occur in combination with translational motion, as is the case for the somersaulting gymnast in Figure 9-1b.

Figure 9-1 Examples of (a) translational motion and (b) combined translational and rotational motions.

We have seen many examples of how a net force affects linear motion by causing an object to accelerate. We now need to take into account the possibility that a rigid object can also have an angular acceleration. A net external force causes linear motion to change, but what causes rotational motion to change? For example, something causes the rotational velocity of a speedboat’s propeller to change when the boat accelerates. Is it simply the net force? As it turns out, it is not the net external force, but rather the net external torque that causes the rotational velocity to change. Just as greater net forces cause greater linear accelerations, greater net torques cause greater rotational or angular accelerations.
Figure 9-2 helps to explain the idea of torque. When you push on a door with a force , as in part a, the door opens more quickly when the force is larger. Other things being equal, a larger force generates a larger torque. However, the door does not open as quickly if you apply the same force at a point closer to the hinge, as in part b, because the force now produces less torque. Furthermore, if your push is directed nearly at the hinge, as in part c, you will have a hard time opening the door at all, because the torque is nearly zero. In summary, the torque depends on the magnitude of the force, on the point where the force is applied relative to the axis of rotation (the hinge in Figure 9-2), and on the direction of the force.

Figure 9-2 With a force of a given magnitude, a door is easier to open by (a) pushing at the outer edge than by (b) pushing closer to the axis of rotation (the hinge). (c) Pushing nearly into the hinge makes it very difficult to open the door.

For simplicity, we deal with situations in which the force lies in a plane that is perpendicular to the axis of rotation. In Figure 9-3, for instance, the axis is perpendicular to the page and the force lies in the plane of the paper. The drawing shows the line of action and the lever arm of the force, two concepts that are important in the definition of torque. The line of action is an extended line drawn colinear with the force. The lever arm is the distance l between the line of action and the axis of rotation, measured on a line that is perpendicular to both. The torque is represented by the symbol (Greek letter tau), and its magnitude is defined as the magnitude of the force times the lever arm:

| Definition of Torque

| (9.1)

Figure 9-3 In this top view, the hinges of a door appear as a black dot (•) and define the axis of rotation. The line of action and lever arm are illustrated for a force applied to the door (a) perpendicularly and (b) at an angle. (c) The lever arm is zero because the line of action passes through the axis of rotation.

Direction: The torque is positive when the force tends to produce a counterclockwise rotation about the axis, and negative when the force tends to produce a clockwise rotation.
SI Unit of Torque: newton · meter (N · m)

Equation 9.1 indicates that forces of the same magnitude can produce different torques, depending on the value of the lever arm, and Example 1 illustrates this important feature.

| Example 1 | | | Different Lever Arms, Different Torques

In Figure 9-3 a force (magnitude = 55 N) is applied to a door. However, the lever arms are different in the three parts of the drawing: (a) , (b) , and (c) . Find the torque in each case. Reasoning In each case the lever arm is the perpendicular distance between the axis of rotation and the line of action of the force. In part a this perpendicular distance is equal to the width of the door. In parts b and c, however, the lever arm is less than the width. Because the lever arm is different in each case, the torque is different, even though the magnitude of the applied force is the same.
Solution Using Equation 9.1, we find the following values for the torques:

a. |
b. |
c. |

In parts a and b, the torques are positive, since the forces tend to produce a counterclockwise rotation of the door. In part c the line of action of F passes through the axis of rotation (the hinge). Hence, the lever arm is zero, and the torque is zero.

In our bodies, muscles and tendons produce torques about various joints. Example 2 illustrates how the Achilles tendon produces a torque about the ankle joint.

| Example 2 | | | The Achilles Tendon

Figure 9-4a shows the ankle joint and the Achilles tendon attached to the heel at point P. The tendon exerts a force of magnitude F = 720 N, as Figure 9-4b indicates. Determine the torque (magnitude and direction) of this force about the ankle joint, which is located 3.6 × 10−2 m away from point P.

Figure 9-4 The force generated by the Achilles tendon produces a clockwise (negative) torque about the ankle joint.

The physics of the Achilles tendon.

Reasoning To calculate the magnitude of the torque, it is necessary to have a value for the lever arm . However, the lever arm is not the given distance of 3.6 × 10−2 m. Instead, the lever arm is the perpendicular distance between the axis of rotation at the ankle joint and the line of action of the force . In Figure 9-4b this distance is indicated by the dashed red line.
Solution From the drawing, it can be seen that the lever arm is . The magnitude of the torque is

| (9.1)

The force tends to produce a clockwise rotation about the ankle joint, so the torque is negative: .

Check Your Understanding 1

The drawing shows an overhead view of a horizontal bar that is free to rotate about an axis perpendicular to the page. Two forces act on the bar, and they have the same magnitude. However, one force is perpendicular to the bar, and the other makes an angle with respect to it. The angle can be 90°, 45°, or 0°. Rank the values of according to the magnitude of the net torque (the sum of the torques) that the two forces produce, largest net torque first. (The answer is given at the end of the book.)

Background: This question deals with the concept of torque and how it depends on the lever arm, the magnitude of the force, and the direction of the rotation that the force causes.
For similar questions (including calculational counterparts), consult Self-Assessment Test 9.1. This test is described at the end of Section 9.3.

Need more practice?

Interactive LearningWare9.1
A thin rod is mounted on an axle that passes perpendicularly through the center of the rod and allows the rod to rotate in the vertical plane. The weight of the rod is negligible. The rod is held at an angle of 33° above the horizontal by the action of two forces. These two forces produce a total torque of zero with respect to the axle. One force has a magnitude F and is applied perpendicularly to one end of the rod. The other force is due to a 49-N weight that hangs vertically from the other end of the rod. Find F.
Related Homework: Problem 8

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