2024/9, Hiizu Nakanishi

Story from Feynman's autobiography "Surely You're Joking, Mr. Feynman!":

Within a week, while I was in the cafeteria, a guy playfully threw a plate into the air. As I watched it spin, I noticed it was wobbling, and the red medallion printed on the plate was rotating faster than the wobble.

I had nothing better to do, so I began calculating the motion of the spinning plate. When the tilt angle is small, the medallion spins about twice as fast as the wobble rate. A 2-to-1 ratio. But that result came from some rather complicated equations. Then I started thinking, "Is there a more fundamental explanation in terms of forces or dynamics?"

I don't remember exactly how I derived it, but I eventually worked out the motion of a mass point and how the balance of all accelerations led to the 2-to-1 ratio. (Ref. 1. End of quote)

This web simulator visualizes the rotational motion of a spinning axisymmetric object like a plate, thrown into the air while spinning around its symmetry axis.

The angle $\theta$ is the deviation between the spin axis and the plate’s symmetry axis ($Z$ axis). If this angle is nonzero, the spin axis wobbles—this is the wobbling motion.

You can use the sliders to change the tilt angle $\theta$ and the plate thickness $d$.

With the display options, you can show:

• Angular momentum vector ($\vec L$, red arrow) and angular velocity vector ($\vec\omega$, yellow arrow),
• The body-fixed $XYZ$ axes,
• The trajectories of the $X$ and $Y$ axes (green and yellow curves respectively)

Watching the trajectories of the $X$ and $Y$ axes, you'll see that in small-angle motion of a thin plate, the axis tips trace out tilted circles intersecting at two points and rotating in opposite directions (in fact, the circles themselves rotate slowly too). Each particle making up the plate undergoes similar motion. Every small piece of the plate moves along a fixed circle in space, and circles of motion at points separated by 90 degrees intersect at two points.

The motion of the plate in the air follows Euler’s equations without external force. Solving them reveals that when $\theta$ is small, the wobble angular velocity $\Omega$ is twice the spin angular velocity $\omega$. That is, the wobble is twice as fast, contrary to Feynman's quote.

This factor of 2 depends on the ratio of the moment of inertia about the symmetry axis ($I_Z$) to that about a perpendicular axis ($I_X$). In general, for a nearly symmetric spinning object with small deviation, \[ {\Omega\over\omega} \approx {I_Z\over I_X} \] For a thin disk, this ratio is 2, but it decreases as the disk becomes thicker.

[References]

1. Richard P. Feynman, "Surely, You Are Joking, Mr. Feynman!". Norton, New York, 1997, pp.173-174.
   
2. B. F. Chao, Physics Today, 1989, Feb. p.15.
   A. Ruina, Physics Today, 1989, Nov. p.127.
3. Tuleja, S., Gazovic, B., Tomori, A., & Hanc, J. "Feynman's Wobbling Plate". American Journal of Physics, (2007), 75(3), 240-244.