## The Wobbling of the Earth and Other Curiosities

#### John Baez

#### 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?

To continue click here.

© 1999 John Baez

baez@math.removethis.ucr.andthis.edu