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The Tennis Racket Theorem

Sometimes it is easy to spin an object; sometimes not. A case in point is described by the Tennis racket theorem.

In Brief

Below we show a rectangular solid whose sides are in the ratio 6:3:2. (This results in moments of inertia in the ratio of 1:2:3, see below.)

Spinning this solid around the short or long axis (in midair, or in space) is stable: it will continue to do so until it hits the ground, or some other object. But spinning it around the medium axis is unstable: no matter how perfectly you try to do it, the object will start spinning around its other two axes (i.e. tumbling).

To explain this, we will use the terminology of rolling, pitching, and yawing.

Try It Yourself

To see the object roll about the X-axis (stable), drag this slider.

To see the object pitch about the Y-axis (unstable), drag this slider.

To see the object yaw about the Z-axis (stable), drag this slider.


What does this all have to do with tennis rackets? It is easy to spin (roll) a racket about its longitudinal axis. People waiting for a serve do it all the time, as a sort of nervous tic. You can also — with some dexerity — give it an underhanded boost so that it rotates once and you catch it by the handle. That is, if you see the strings edge-on. But if you try the same maneuver with the strings facing you, the racket will tumble. Why is that?

Every 3D object has two distinguished axes: one about which its moment of inertia is maximum, and one about which the MOI is minimum. Rotation about both of these axes is stable: if you get it going it will stay going. Rotation about a third axis, whose MOI is intermediate, is unstable: the slightest deviation will set the object tumbling.

This is well known to skateboarders too: no matter how you do a kickflip, the board will not pitch perfectly end-over-end. It will always start to roll.

Try It Another Way

To rotate the sphere, drag this slider.

More Remarks

The sphere in the figure represents angular momentum space of a spinning object with moments of inertia 3, 2, and 1 about the X (stable), Y (unstable), and Z (stable) axes respectively. The red lines are unstable attractors.

You can see this behavior with almost any book — as long as it is not square. It will be easy to spin it about two of its axes, but rotation around the third will inevitably decay into a chaotic combination of spins about the other two axes.

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