Tennis racket theorem

 

If you try to spin a tennis racket (or a book, etc.) around any of its three principal axes, you will find that different things happen with the different axes. Assuming that the principal moments (relative to the CM) are labeled according to I₁ > I₂ > I₃ (see Fig. 8.40), you will find that the racket will spin nicely around the ₁ and ₃ axes, but it will wobble in a rather messy manner if you try to spin it around the ₂ axis. Verify this claim experimentally with a book (preferably lightweight, and wrapped with a rubber band), or a tennis racket (if you happen to study with one on hand). Verify this claim mathematically. The main point here is that you can’t start the motion off with ω pointing exactly along a principal axis. Therefore, what you want to show is that the motion around the ₁ and ₃ axes is stable (that is, small errors in the initial conditions remain small); whereas the motion around the ₂ axis is unstable (that is, small errors in the initial conditions get larger and larger, until the motion eventually doesn’t resemble rotation around the ₂ axis).²³ Your task is to use Euler’s equations to prove these statements about stability. (Exercise 3 gives another derivation of this result.)