Precession of a Top

Introduction

A precessing mechanical top is a good prelude to understanding the effects of a magnetic field on electrons and nucleons in atoms. The animation that this document describes should give the learner some hands-on experience with that precession,

Figures

Figure 1: The top precesses about the z axis at angle q. The diagram shows the instantaneous angular momentum vector, L, as well as the center of mass. The force, F=mg, is that due to gravity. This force provides toque at radius r=h sin(q) where h is the height of the center of mass above the x-y plane when the top is aligned with the z axis.

Calculations

In steady precession the angular momentum can be described as:

(1)

where the bold x and y are unit vectors in the x and y directions, respectively, q is the tilt angle with respect to the z axis and w is the angular rate of precession.

The torque, t, that causes the precession has to be synchronous with L and is:

(2)

And the pertinent equation relating the precession to the torque is:

(3)

From equation 3 we easily obtain the angular rate of precession as:

(4)

Summary

Note that the precession frequency is independent of the angle of tilt because both the torque and dL/dt are proportional to the sine of the tilt angle. As the top slows down, however, L₀ decreases, increasing the rate of precession and the tilt will become larger until finally it will lie on the floor.