## The Wobbling of the Earth and Other Curiosities

#### December 17, 1999

Here is a puzzle I posed on the newsgroup sci.physics.research, Jim Heckman's attempt to solve it, and some interesting related stuff. It's all about the motion of the earth, moon and sun.

Here's a nice puzzle about the wobbling of the earth's axis of rotation. I'm sure some real experts on this subject are lurking out there, so I'd appreciate it if they kept quiet until some of the nonexperts have had a crack at it. It's a fun puzzle, because it raises lots of interesting issues.

If you've studied the rigid rotating body in classical mechanics, you'll know that such an body has 3 principal axes that it can rotate around without wobbling. Associated to each axis there's a number called the "moment of inertia", which says how much energy it takes to rotate the body about that axis at a given angular velocity.

Let's call the biggest moment of inertia I1, the middle one I2, and the smallest one I3. It turns out that the body will rotate stably around the first or third axes, but unstably about the second. In other words: if you start it spinning almost but not quite along the first or third axes, it will wobble slightly. In other words, it'll precess. If you start it spinning almost but not quite along the second axis, it will start flopping about in a pretty complicated-looking way.

This is called the "tennis racket theorem", because it's most easily illustrated by tossing a tennis racket into the air while spinning it around each of its principal axes. My physics professor in college did this in class, and it was the most exciting day of the whole class!

Now, the earth is roughly an oblate spheroid, so it's rotating almost about the first axis, the one with the biggest moment of inertia. It's a bit like a spinning frisbee, but not so dramatically flattened. I2 and I3 are almost equal - not quite, but for starters let's pretend they are.

Okay, here comes the puzzle. The earth is not spinning exactly around the first principal axis. It's a bit off, so it wobbles. Estimate the period of this wobble!

I am smugly pleased to report than when Jim Dolan asked me to do this, I was able to guess the answer within a few minutes. Then I looked it up, and it turned out I got it right to within a factor of 3.

The nice thing about this puzzle is that you don't need to look up a bunch of numbers to solve it! You need to understand physics, but there's just one non-obvious number that you need to know - and if you're reasonably lucky, you can guess that number to within an order of magnitude.

Takers, anyone?