Precession of Euler's Disk

## Question:

I am trying to figure out why the precession of Euler's disk
increases, specifically why the center of mass of the disk will
become closer and closer to the vertical Y axis of which it is
spinning around. (Why does the angle created by the axis which
runs straight through the center of the disk and the Y axis
decrease? How does this affect the increasing precession?) Any
information would be GREATLY appreciated. thanx!
## Answer:

Imagine a disk with an axle through the center of mass so that it
can be spun in the horizontal plane, resting on the tip of the
vertical axle. In this situation the center of mass stays on the
y axis and the moment of inertia is the normal 1/2mr^2 for a
disk. This is normal spinning top motion.
Now consider the disk with the axle removed spinning on its
edge. The motion of the disk may be broken down into rotation
about the axis through the center of mass, just the top-like
motion we saw above, and the precession of that axis of symmetry
about the vertical.

In this situation the center of mass of the disk will not be
directly over the point of contact with the surface on which the
disk is spinning. The torque produced by gravity acting
vertically on this center of mass will cause the axis about which
the disk is spinning to precess. Conservation of angular momentum
is responsible for this effect. See
gyro theory
for more details on
this part.

Energy lost due to friction must come from the potential
energy of the elevated center of mass so the center of mass of
the disk becomes lower. This increases the moment arm between the
point of contact with the surface and the center of mass,
increasing the gravitational torque. The increase of torque
causes faster precession about an axis that is increasingly
vertical. Finally loss of energy brings the whole thing to a
halt.

Just before it stops the center of mass is at the maximum
distance from the point of contact and the rotation about the
axis of symmetry has just about stopped. The precession is
whipping the rotational axis rapidly around a tight little
circle. The disk finally comes to rest with the axis of symmetry
returned to the vertical.