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812 Euler’s disk If you spin a coin or solid uniform disk about a vertical axis on a α hard surface it will eventually lose energy and begin to wobble and perhaps make a buzzing sound as it

8.12 Euler’s disk*

If you spin a coin or solid uniform disk about a vertical axis on a

α

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 toy called Euler’s disk—the wobbling motion can be astonishing,

α

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 and mass 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 . The disk also spins about its own axis with spin angular velocity Ω . The result of the two motions is for the contact point on the table to be momentarily at rest.

Find Ω for a given small angle α .

  1. Find the total angular velocity of the disk and the angular momentum. From this, find the precession rate Ω .

  2. Find the rate at which the disk appears to rotate as viewed

from above, when α is very small.

 

Jun 19 2020 View more View Less

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