8.12 Euler’s disk*
If you spin a coin or solid uniform disk about a vertical axis on a
hard surface, it will eventually lose energy and begin to wobble and
perhaps make a buzzing sound as it rolls. With a massive disk—a
Ω 0 toy called Euler’s disk—the wobbling motion can be astonishing,
with the frequency of buzzing increasing as the rotational speed of the disk decreases. The motion seems at first sight to be like gyro- scope motion, but it can be argued that Euler’s disk is an antigy- roscope. The goal of this problem is to understand the underlying physics of the disk’s motion.
Consider a thin disk of radius R and mass M that is executing the rapidly buzzing motion. The plane of the disk is at angle α with respect to the table. The axis of the disk precesses around the vertical at rate Ω p . The disk also spins about its own axis with spin angular velocity Ω 0 . The result of the two motions is for the contact point on the table to be momentarily at rest.
Find Ω p for a given small angle α .
Find the total angular velocity of the disk and the angular momentum. From this, find the precession rate Ω p .
Find the rate at which the disk appears to rotate as viewed
from above, when α is very small.
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