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What drives the motion of a Spinning Top?

There are many forms and shapes of spinning tops, and they are put into motion in an interesting variety of ways. Some are spun by snap-twisting a center stem with your fingers and releasing, while the top remains on the ground. Others are held by a support at the top while a cord wound around the top is pulled to spin it. The spinning top many of us know is launched from about waist level to the floor by snapping your wrist as you release it, while maintaining a grip on the cord wound around its body. However they are spun, each type behaves in a similar fashion.

The physics of rotation.

The body of a top has at least one axis about which it will spin steadily and smoothly. This rotation axis is a symmetry axis of the top, known as a principal axis. For example, the red hoop in the figure below has two unique symmetry axes indicated, for rotations of the type specified by the blue arrows.

The two symmetric axes for the hoop discussed in the text.

For each unique symmetry axis, the object has a moment of inertia value that determines how it will spin when a torque is applied. The way this all works through is described by Newton's Laws of Rotation . While this can get pretty complicated in detail, there are some circumstances where the object will spin in a very simple manner. The object's spin about the rotation axis gives it an angular momentum, which will remain constant until some outside torque works on it.

The ideal top.

Suppose a top is so perfectly fashioned that its principal rotation axis (spin axis) goes through its center of mass. (The center of mass, also known as the center of gravity, is the balance point of the object.) If we spin this top carefully, so that it remains perfectly upright while spinning (and gravity can't exert a torque on it about its point), it will spin at a steady angular velocity almost indefinitely. Sliding friction between its tip and the floor does slow it gradually. But if the point is very sharp, sliding friction there exerts very little torque on the top about its rotational axis. Because it's unable to exert a torque on the ground, the top can't exchange angular momentum with the earth. It spins on until it slowly gets rid of its angular momentum through sliding friction and air resistance.

A more realistic top.

In general, the world is not this accomodating. A slight mismatch between the spin axis and the center of mass will guarantee that gravity exerts a torque on the top about its tip. The rapidly spinning top will precess in a direction determined by the torque exerted by its weight. The precession angular velocity is inversely proportional to the spin angular velocity, so that the precession is faster and more pronounced as the top slows down.

Viewed another way, the torque applied by the top's weight does not change much for small changes in tip angle, so the increment of angular momentum change also stays the same. But, it increases as a fraction of the total angular momentum when the top slows down, producing a larger fractional change in the spin direction for no change in the applied torque -- effectively giving a bigger bang for the buck.

Either way, we get the commonly observed behavior of a spinning top. When it is first launched and spinning its fastest, the top is most nearly vertical and stable in its spin. As it begins to slow down, its precession becomes more pronounced and its tilt angle off of vertical increases.

Diagram with veotor quantities, directions as marked.

With a relatively light top, this precession is most of the behavior we tend to notice. As an example, here is a film clip showing the smooth, stable precession of a gyroscope . (Size: 2.35 MB)

However, it is not the only thing going on. Precession was caused by the gravitational torque acting on the slightly tipped, or just slightly misshapen, top - producing an "orbiting" of the top's spin angular momentum around the vertical direction. In reality, the precession angular velocity corresponds to another angular momentum - a precession angular momentum (which is typically much smaller than its spin angular momentum). Now, if this precession angular momentum is exactly vertical and the top is ideally balanced, there is no effect of the torque from the top's weight on it. But, that kind of perfection is hard to come by. Anything that causes the precession angular momentum to be a bit off vertical will lead to a kind of "precession of the precession."

This secondary effect, called nutation, is usually not significant unless something happens to disturb the motion of the top. To see what it looks like, view this film clip of a gyroscope that has been deliberately nudged . (Size: 2.56 MB)

Again, the story need not end there. Each new type of precession carries along with it a new, related angular momentum, and these angular momenta can in turn be made to precess by an applied torque. It is just that the effect becomes observably less noticeable and significant with every new stage of this process. So, we usually end the story with nutation. Thus, The End!

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