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
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
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.
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
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,
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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