# Gyroscopes

Consider a top or a gyroscope rotating as shown here. As long as the top or gyroscope rotates fairly rapidly about its axis of rotation, whatever other rotation or movement it may have will be minor and the angular momentum L will line up with this axis of rotation. As the top or gyroscope slows down and eventually comes to a stop, of course, this will no longer be true.

What torque acts on this top or gyroscope? Mainly, what direction is the net torque acting on the top or gyroscope? And, then, we will want to ask what effect this has on the gyroscope -- that is, what happens to the gyroscope?

Two forces act on the top or gyroscope -- the normal force n that supports it and the weight M g acting downward. What torques do they produce? Simply from convenience, we say "what is the torque?" but we always mean "what is the torque about some reference point?". The point of contact or the point of support seems like a reasonable point to use. About that point the normal force n produces no torque. The weight, M g, does produce a torque. If the x-, y-, and z-axes are oriented as shown (with the x-axis coming out of the page), then the torque, from

is a vector pointing along the positive y-axis (or to the right). That is the only torque so it is certainly the net torque.

We also know the net torque causes the angular momentum L to change,

or

from which we can immediately find the change in the angular momentum,

To emphasize the vector nature of all of this, we can write this as

As we noted earlier, the torque is a vector that points to the right. This means the change in the angular momentum also points to the right.

How does the angular momentum L move so that the change in the angular momentum is to the right? The change in the angular momentum is perpendicular to the angular momentum so the magnitude does not change. But the direction changes. The angular momentum vector L "precesses" or rotates as shown in the diagram above. The top or gyroscope, too, precesses so that its axis of rotation remains aligned with the angular momentum L.

Here are additional examples or diagrams: