Chapter 7.5 - Angular Momentum, Stability, Precession
The underlying lens of this chapter is that the change in angular momentum is:
If you want to change a body’s momentum, you apply a force; either linearly or rotationally:
Linear | Rotational |
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If we push something, it gains momentum, in the direction we pushed it. If it was originally stationary, now it’s moving in the direction we pushed it. However, if it was already moving, then we find the new direction by adding using vector addition. Correct vector addition (whether graphically or by components) is particularly important when the applied force (and impulse ) is perpendicular to the original momentum.
As illustrated above, there is no force in the direction, so this component of momentum doesn’t change. The ball exhibits a parabolic trajectory while there is an applied vertical force. After the force is released, the ball again continues in a straight line in a new direction determined from
Imagine now, pushing on the rim of a spinning bicycle wheel. A bicycle wheel can be thought of as an infinite number of masses that are the same distance from the axle. Again, the applied force will also change of direction of the momentum of the piece of mass on the rim.
For most of us, this is an unexpected and very surprising result: We push the front end of the rim upward, and rather than rotating the top of the axle back away from us, the top of the axle tips to the left. And after the force is removed, the orientation of the wheel stays fixed! Additionally, please convince yourself that if the wheel were rotating in the opposite direction, the same push would tip the top of the axle to the right.
A wheel is more than one mass, so let’s consider all the masses along the rim, and let’s apply our forces to the axle. If we were to push away at the top of the axle and pull the bottom of the axle (applying a torque to the left), it would have the same effect. All the masses closer to us would be pushed upward (gaining upward momentum), and all the masses on the far side of the wheel would be pushed downward (gaining downward momentum). We might expect this torque on the axle to the left would cause the axle to to the left, but instead, the axle rotates out at us (or counter clockwise as we see it above) only while the torque is applied and remains fixed thereafter.
We can make sense of this if we consider that we are used to dealing with objects that are not spinning – that have no angular momentum. Non-spinning objects rotate in the direction of applied torque. We see from the first diagram, if then when pushing upward on the ball, the ball moves upward. Accordingly, the non spinning wheel will then rotate in the direction of the applied torque.
Additionally, remember the first four equations in this section. Torque produces a change in angular momentum: If then and the object is rotating in the direction of applied torque. However, if it’s already spinning, then , added correctly like vectors.
As we apply the torque to the left, the axle precesses to a new orientation as gets bigger with increased time. When again, the precession stops and the wheel remains fixed at the new orientation.
Exercise 7.5.1 (please draw pictures)
With a wheel or gyroscope, please repeat the process described above:
- Can you predict the direction the axle will turn?
- Predict what happens when the wheel is spinning in the opposite direction? Why?
- How does the rate of precession change if the wheel is spinning faster? Can you explain why?
- How does the rate of precession change when you push harder on the axle? Why?
Spinning wheels (called gyroscopes) are very stable. While they don't resist translational movement any more than a non-spinning body, they resist angular change and are used to maintain the orientation of satellites, for instance. To see this, consider the first diagram in this section. When you push on a moving object, you can change its trajectory. However, the more initial momentum the object has, the less your impulse will change the object’s trajectory. Thus, if an object is traveling faster, your push will change its direction less. Similarly, if an object is spinning faster, the masses have greater momentum, and the spinning object has greater angular momentum. The greater is, the less the orientation will change with the same angular impulse,
Exercise 7.5.2
Please answer the following questions. A diagram of may help.
- Why do we spin a Frisbee (or football) when we throw it?
- Please notice sometime that the rim of the Frisbee is very thick. Why do you think that they are made this way?
Gravitational precession of a gyroscope or top
If the axle of a spinning wheel, top, or gyroscope is not vertical and is supported only on one side, then gravity provides a constant torque perpendicular to the axis of orientation. This torque produces a constant supply of changes of angular momentum: Again, we might think that this would cause the axle to rotate downward. However, is added to the present angular momentum of the spinning wheel, and the axle instead precesses in a circle in the horizontal plane. Please verify this from the diagrams below.
Exercise 7.5.3
Spin a wheel or a gyroscope, support it from one side of the axle, and verify the behavior of the precession.
- Can you predict what will happen?
- What changes if you spin the wheel the other way? Why?
- What changes if you spin the wheel faster? Why?
- What changes if you switch sides and support the axle on the other side? Why?
- What happens if you support the axle closer to the center of the wheel? Why?