Precession of Euler's Disk


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!


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.