# Ask a Physicist Answers

## I have a question about a tennis racquet: when I hold the racquet like a frying pan (plane of the head horizontal) and give it one flip, it changes from smooth up to rough up (or vice versa). When I flip it with the plane of the head vertical, nothing peculiar happens. In both cases, I try not to impart any rotation about the long axis. - GF, Shawnee, Kansas

You've happened upon a favorite example of the intermediate axis theorem, a famous concept of nineteenth century classical physics. The tennis racquet has three important or "principal" axes about which it can rotate and the frying pan axis is the intermediate one. To understand why it's intermediate, we need to look at the three principal axes and their differences in rotational inertia.

The easiest axis about which to spin a tennis racquet is the axis that runs right down through its handle. When you twist the racquet about this axis, it turns easily because most of its mass is located close to this axis and its rotational inertia about this axis is therefore low.

The hardest axis about which to spin the racquet is the one that is perpendicular to both the racquet's face and its handle. When you hold the racquet so that the plane of its face is vertical and flip the racquet vertically, you are spinning it about this axis. The racquet is difficult to spin about this axis because most of the racquet's mass is located far from this axis and its rotational inertia about this axis is therefore high.

The intermediate axis is the one you describe: the axis that lies in the plane of the racquet's face and is perpendicular to its handle. Spinning or flipping the racquet about this axis has a difficulty that is in between that of the other two axes. The racquet's rotational inertia about this axis is intermediate between those of the other two axes.

If you could flip or spin the racquet so that it turned exclusively about one of its three principal axes, it would continue to spin about that axis indefinitely. That's why they're called principal axes. But in a real flip there is always some mixture of motions about all three axes. Here is where the intermediate axis theorem enters the picture: while a racquet spinning mostly about either the low-rotational-inertia axis or high-rotational-inertia axis will be relatively unaffected by extraneous motion about the other two axes, a racquet that is spinning mostly about the intermediate-rotational-inertia axis is exquisitely sensitive to any accidental motion about those other two axes. Even a tiny amount of unintended motion about those axes will cause the racquet to wobble significantly. In your case, the racquet's face undergoes an unexpected half twist in the time it takes the racket to flip once in the "frying pan" mode. That half twist is interesting in a tennis racquet, but don't try it with a real frying pan full of pancakes!

Answered by Lou A. Bloomfield of the University of Virginia.