Built on Facts

Grab a book, or an empty DVD case, or anything else that's a uniform rectangular solid. If it's a book, you might want to secure the book closed with tape or a very lightweight clip, because we'll be throwing it in the air. We want to test a theory.

In classical mechanics we know that each solid object has three special rotational axes, called "principal axes". Intuitively, imagine the object is shaped from styrofoam. The principal axes are the axes along which you can spear the object with a wooden dowel, and the object doesn't try to "wobble" when you rotate the dowel. Aligning a car tire is simply the process of making sure the axle and the principal axis of the tire are collinear. There's a somewhat involved mathematical procedure you can use to calculate the principal axes (there's always three of them*), but frequently we can find them a priori without any math just by looking for axes of symmetry. Pulling a picture from Wikipedia, here's one principal axis of our book:

There are two more principal axes, and we can find them in the same way. One runs through the center, parallel to the spine. One runs through the center, parallel to the lines of text. If we throw the book in the air such that it's rotating exactly about one of those axes, it won't wobble.

But it's not possible to be perfect. We will inevitably be a tiny bit off. In that case, what happens? To answer, we look at Euler's equations which describe this sort of scenario:

The various "I" represent the moment of inertia about each principal axis, and the little omega represent the angular velocity about those axes, which is just the rate of spin. Notice the equations are coupled: the rates of spin about one axis affect the change of the rate of spin in the others. However, we see that if the rotation is initially about just one axis (say, ω1 ) the rotation about the other two will stay zero because their initial rotation rates are zero.

But if the initial rotation isn't perfectly aligned with a principal axis, all three omegas may be nonzero. What happens then? Well, we can solve these equations approximately and we'll see that in fact the rotation stays nearly aligned with the principal axis, with a slight precession with a frequency given by Ω. I'll skip the math. It isn't hard, but it's not very instructive either. In any case, the frequency of precession is:

Here we've numbered the axes such that 3 is the one about which we were mostly initially rotating modulo the very slight rotations about the other two. ω with the 0 subscripted is just the initial rotational speed.

Now here's the tricky part. Ω is square, so it must be a positive number. This will be true if I3 is the largest moment of inertia, as both terms in the denominator will be positive. This will also be true if I3 is the smallest moment of inertia, as both terms will be negative, and negative times negative is positive. But if I3 is the middle-sized moment of inertia, clearly we get a negative, and a squared real number can't be negative. Our assumption that the tiny imperfection in our throw doesn't matter must therefore fail if we're initially rotating about that middle-inertia axis.

So throw that book such that it's spinning about the axis in the picture. It'll work great. Throw it such that it's spinning about the axis parallel to the spine. It'll work great. Throw it so that it's spinning about the axis parallel to the text. It will wobble crazily, no matter how hard you work on the careful precision of your throw. Bet your friend a dollar that he can't do it, and physics will make you a dollar richer.

*With especially symmetrical objects like spheres and car tires the axes are not necessarily uniquely specified. There are still three of them, just with sort of a "phase freedom" in their orientation.