# Q & A: Stable and Unstable Rotations: The Tennis Racket Theorem

Q:
Why is when you flip a candy bar in the air end to end (like a Hershey's milk chocolate with almonds bar) will it not only flip end to end but also rotate on its axis?
- Dave Bacon (age 54)
Appleton, WI USA
A:

Hi Dave,

I'm curious, did you hear about this somewhere, or are you a good observer and experimentalist? If you haven't tried this, you really should. (But it requires a steady arm and practice.) Find a textbook whose length, width, and height are all different, and tape it shut. If you spin it in the air about different symmetry axes, you will find that it spins nicely and stably about the axes that have the smallest and largest moment of inertia.

However, if you try spinning it about the axes with the intermediate moment of inertia, you won't possibly be able to make it rotate without tumbling unstably. This effect is more obvious in space, where the book doesn't hit the floor after a rotation or two. You can find a pile of cool videos of the effect all over the web:

The effect is known as the "Tennis racket theorem," or the Dzhannibekov effect. When the object is tumbling unstably, it basically rotates a few times about the unstable axis, and then suddenly the axis of rotation flips end-to-end. Depending on its shape, the object may rotate many times before this flip occurs.

I'm not entirely sure how to understand this effect physically (solid-body rotations can be hard to visualize). However, the effect falls quite simply out of the mathematics, if you understand differential equations: . These differential equations come from the torque-free Euler equations, which describe how free bodies rotate.

So is the Earth in danger of flipping over all of a sudden? Probably not, for various reasons. Most obviously, it is rotating about its largest moment of inertia, so the rotation should be stable. In addition, the earth isn't really a solid body, and there may be effects from the moon. That said, however, it has been proposed that a large volcano or other tectonic activity might create extra mass somewhere on earth. If this bulge were massive enough, it might slowly drift down towards the equator, changing the orientation of the earth with respect to its rotation axis! Supposedly there is some evidence for this effect in magnetic layers under the sea*, but I don't think the idea is at all accepted by the general scientific community... so far it's mostly just a neat idea.

*This effect is completely different from the magnetic pole reversals which are well known to happen every few hundred thousand years. Here, we are talking about the physical earth rotating with respect to its spin axis.

Cheers!

David Schmid

(published on 10/08/2013)