The problem with having odd-shaped balls…

…by which, of course, I mean rugby balls.

To be precise, a rugby ball is a prolate ellipsoid – that is, something that is like a 3d version of an ellipse, but having a cross-section along its long axis of a circle.  (A flying-saucer would be an oblate ellipsoid) 

Rugby balls behave awkwardly. In one sense that’s obvious – their awkward shape will lead to awkward movement, on the ground at least. But what about when in the air?  Why is it hard to get a rugby ball to spin nicely and stay in one orientation (like the professionals can do?) Or, put another way, why is it so easy to make a rugby ball tumble all over the place in flight, making it more difficult for the opposing full-back to catch?

Once in the air, the rugby ball undergoes torque-free movement. (We’ll neglect things like the Magnus effect). That means there’s no external forces trying to create extra spin on the ball.  The behaviour of an object in this situation can be described by Euler’s equations (after the Swiss mathematician Leonhard Euler). I won’t write them in a blog, but they are not that difficult to interpret if you know about rotational inertia and angular velocities. 

These equations can be solved fairly simply in some cases. It turns out, that if you have a rugby ball that is spinning on its long axis (like you try to do when you pass to one of your teammates) the rotation is stable. That means, once you have established it rotating in this manner, it will remain doing so. If the ball is perturbed slightly off this spin (perhaps by a piece of flying mud) it will return to this direction.

However, the same is not true if you spin the ball about a short axis (so it tumbles end on end). In this case, the motion is  critically stable – meaning that any perturbation is not corrected. The ball can easily lose this manner of motion, and start moving in another way. If you don’t start it exactly in this way of movement, it’s unlikely to stay there. Find yourself a rugby ball and try it.

Perhaps more interesting is the case when you have an object with three axes of different lengths (e.g. a paperback book). You’ll find that if you try to spin the book about its long axis, or its short axis, these are stable, but about the middle axis the rotation is unstable.  Try it. Take a book (a cuboid with three different length axes), hold it with its cover the right way up and facing you, and grip its bottom two corners. Now flip it in the air – try to get it to spin 360 degrees and catch it again, by its bottom two corners.  See what happens.  It’s very difficult to do – the book will tumble around, because rotation about this middle axis is unstable. Any small perturbation from it will be magnified as the book spins.  However, you try the same experiment spinning about one of the other axes of the book, and you’ll find the motion stable.

Incidentally, if you take an object with three identical axes (e.g. a soccer ball), its rotations are critically stable in all directions, which is one reason why a soccer ball in flight can spin (and bend) in all kinds of whacky ways.