Instability of a thrown tennis racquet

Question

Someone once mentioned to me that it's impossible to throw a tennis racquet (or similarly shaped object) into the air, perpendicularly to the string plane, in such a way that it won't turn.

What is this effect, or was he talking rubbish?

Answer 1

The "effect" is quite real, but I don't believe there's a name for it. It has to do with rotational stability and the intermediate axis theorem.

I'm sure you've also observed this effect when throwing a spatula, frying pan, or remote control.

In terms of rotational inertia, the "easiest" axis of rotation for the racquet is straight through the handle. The "hardest" axis of rotation would be straight down (perpendicular to the strings).

But when there's rotation along the "intermediate" axis (the one you describe), the other axes (especially the "easiest") become extremely sensitive to random perturbations.

If you could flip or spin the racquet so that it turned exclusively about one of its three principal axes, it would continue to spin about that axis indefinitely. That's why they're called principal axes. But in a real flip there is always some mixture of motions about all three axes. Here is where the intermediate axis theorem enters the picture: while a racquet spinning mostly about either the low-rotational-inertia axis or high-rotational-inertia axis will be relatively unaffected by extraneous motion about the other two axes, a racquet that is spinning mostly about the intermediate-rotational-inertia axis is exquisitely sensitive to any accidental motion about those other two axes. Even a tiny amount of unintended motion about those axes will cause the racquet to wobble significantly.

Answer 2

See this thread on math overflow. (And the nice videos in the question and the answers.)