# Rotational Motion and Angular Momentum

## Gyroscopic Effects: Vector Aspects of Angular Momentum


### Learning Objectives

By the end of this section, you will be able to:
1. Describe the right-hand rule to find the direction of angular velocity, momentum, and torque.
2. Explain the gyroscopic effect.
3. Study how Earth acts like a gigantic gyroscope.
Angular momentum is a vector and, therefore, has direction as well as magnitude. Torque affects both the direction and the magnitude of angular momentum. What is the direction of the angular momentum of a rotating object like the disk in ? The figure shows the right-hand rule used to find the direction of both angular momentum and angular velocity. Both  and  are vectors—each has direction and magnitude. Both can be represented by arrows. The right-hand rule defines both to be perpendicular to the plane of rotation in the direction shown. Because angular momentum is related to angular velocity by , the direction of  is the same as the direction of . Notice in the figure that both point along the axis of rotation.

Now, recall that torque changes angular momentum as expressed by
This equation means that the direction of  is the same as the direction of the torque  that creates it. This result is illustrated in , which shows the direction of torque and the angular momentum it creates.
Let us now consider a bicycle wheel with a couple of handles attached to it, as shown in . (This device is popular in demonstrations among physicists, because it does unexpected things.) With the wheel rotating as shown, its angular momentum is to the woman's left. Suppose the person holding the wheel tries to rotate it as in the figure. Her natural expectation is that the wheel will rotate in the direction she pushes it—but what happens is quite different. The forces exerted create a torque that is horizontal toward the person, as shown in (a). This torque creates a change in angular momentum  in the same direction, perpendicular to the original angular momentum , thus changing the direction of  but not the magnitude of .  shows how  and  add, giving a new angular momentum with direction that is inclined more toward the person than before. The axis of the wheel has thus moved perpendicular to the forces exerted on it, instead of in the expected direction.

            

This same logic explains the behavior of gyroscopes.  shows the two forces acting on a spinning gyroscope. The torque produced is perpendicular to the angular momentum, thus the direction of the torque is changed, but not its magnitude. The gyroscope precesses around a vertical axis, since the torque is always horizontal and perpendicular to . If the gyroscope is not spinning, it acquires angular momentum in the direction of the torque (), and it rotates around a horizontal axis, falling over just as we would expect.
Earth itself acts like a gigantic gyroscope. Its angular momentum is along its axis and points at Polaris, the North Star. But Earth is slowly precessing (once in about 26,000 years) due to the torque of the Sun and the Moon on its nonspherical shape.

            

### Test Prep for AP Courses


### Section Summary

1. Torque is perpendicular to the plane formed by  and  and is the direction your right thumb would point if you curled the fingers of your right hand in the direction of . The direction of the torque is thus the same as that of the angular momentum it produces.
2. The gyroscope precesses around a vertical axis, since the torque is always horizontal and perpendicular to . If the gyroscope is not spinning, it acquires angular momentum in the direction of the torque (), and it rotates about a horizontal axis, falling over just as we would expect.
3. Earth itself acts like a gigantic gyroscope. Its angular momentum is along its axis and points at Polaris, the North Star.

### Conceptual Questions


### Problem Exercises
