John Baez said something about "staring at the Euler equations", until it was clear that the ratio of wobble-rate to spin-rate was (I1 - I2) / I1. No need for that, it's really just a simple change of basis vectors (in plain old Euclidean 3-space, nothing fancy!) It's practically the same situation as the famous Feynman plate experiment; I append my old post on that.
One datum you need is the ratio I1 / I2. That in turn depends (somehow!) on the oblateness of the Earth --- the ratio of the polar diameter to the equatorial diameter. I say somehow because it's not obvious to me that this diameter ratio must equal the ratio of the moments of inertia. Even if you assume that the Earth is an oblate spheroid of uniform density, don't you have to do an integral or something? The wobble thread ignored this fine point, unless I missed it. I don't feel like doing integrals on New Year's eve, so I'll ignore it too.
How to get the oblateness? John just guessed, Jim Hecke seemed to know it off the top of his head. But in keeping with the spirit of jim dolan's original puzzle, I think you should derive the oblateness --- at least within a factor of two. Use Newton's argument:
If our Earth was not higher about the equator than at the poles, the seas would subside about the poles, and, rising towards the equator, would lay all things there under water.So the only data you need are the gravitational acceleration, the rotation rate, and the diameter (either one, since they are nearly the same). Have fun!--Principia, Book III, Proposition 18
A bit of historical trivia. In Newton's time, controversy reigned as to whether the Earth was a prolate spheroid or an oblate spheroid -- i.e., stretched like an plum tomato or flattened like a tangerine. The issue was finally settled by an expedition to Lapland, headed by Maupertuis (the least-action guy). Newton was vindicated. Voltaire wrote to Maupertuis:
Vous avez confirme dans les lieux pleins d'ennui Ce que Newton connut sans sortir de chez luior in a free translation: "To a far-off clime you had to roam, To confirm what Newton knew without leaving home." (At least that's how Chandrasekhar tells the story in his Newton's Principia for the Common Reader. Andrade's address at the Newton tercentenary (reproduced in Newman's World of Mathematics) has a slightly different couplet, and says Voltaire wrote this to La Condamine, who went to the equator instead of Lapland.)
Getting back to the wobble, Grossman's Sheer Joy of Celestial Mechanics, gives a figure of 303 days for the so-called Euler free period of the Earth (i.e., what we've been talking about). Then he says this:
It is found that a single periodic term does not suffice for representing the empirical variations in latitude. Two periodic terms suffice to a high degree of satisfaction. The first has amplitude 0".09 and period about one year. The second-- the Chandler wobble -- has amplitude 0".18 and period about 14 months. There are, in fact, torques acting on the Earth. Among the causes of external forces and their torques are changes in the atmosphere from seasonal and other activities; tidal forces from water on the surface of the Earth; and forces within the Earth from shifting of crustal plates and viscous sloshing of a molten core.Also, of course, the Earth's shape is not precisely an oblate spheroid.
The spinning plate is a classic bit of Feynman lore. In the cafeteria at Cornell, Feynman saw a student toss a plate in the air, and noticed that the rate of wobble bore a simple relation to the rate of spin. In fact, the wobble rate is about twice the spin rate, for small wobbles.
Feynman told the story in one of his collections; working out the details helped bring him out of an emotional slump, a sort of scientific "writer's block". Eventually he resumed his work on QED and path integrals, and so he went from understanding the spinning plate to understanding spinning electrons.
When Feynman retold the incident, he misreported the relationship: he said the spin was twice as fast as the wobble. An old letter to Physics Today speculated that this "mistake" was deliberate, another Feynman game with the reader. (The letter writer noticed something curious about the page numbers in the book; I forget the details.)
In Jagdish Mehra's Feynman bio, The Beat of a Different Drum, he quotes Feynman as saying that the exercise was a nice example of how a simple relationship emerged from a complicated calculation. When I looked at the problem recently, I was pleasantly surprised to find that the result actually falls out from some very simple vector algebra. All you need are two facts:
(1) The angular momentum vector L is fixed.Then express W in terms of L and W_sym, and stare at it for a bit. In other words, just a change of basis!(2) If the angular velocity vector W is expressed as a sum of two other angular velocities:
W = W_sym + W_perp
where W_sym is along the symmetry axis, and W_perp is perpendicular to W_sym, then
L = k (2 W_sym + W_perp)
or in other words, the moment of inertia about the symmetry axis is twice the moment of inertia about a perpendicular axis.
The "spinning plate to spinning electrons" is a line I've run across somewhere before; I thought it was in Mehra's book, but now I can't find it. It seems unlikely that there could be any technical relation between the two physical problems--- even though the factor 2 makes a famous appearance in the theory of electron spin (Dirac's value for the gyromagnetic ratio), and (according to Mehra), Feynman was working hard to fit spin-1/2 into his formulation of QM just around the time of the spinning plate incident.
Is there some mathematical connection?
Here are some more details.
Let W be the angular velocity vector, and L the angular momentum vector. Also let W_sym be the component of W along the symmetry axis, and W_perp the perpendicular component. L = k(2W_sym + W_perp), but the factor of k plays no further role except to clutter up things, and I'll ignore it. (Consider that a redefinition of what L stands for.) So:
W = W_sym + W_perp
L = 2W_sym + W_perp
Now we rearrange:
W = L - W_sym
OK, how to interpret a sum of angular velocity vectors? Let's take a page from the ancient Greek astronomers (Eudoxus, to be precise), and think of a sphere rotating on an axis with angular velocity L; mounted inside this sphere is a spindle, aligned along the symmetry axis, and the plate spins on this spindle with velocity W_sym.
L is fixed, by conservation of angular momentum. With this picture in mind, it's pretty clear that the rate of the wobble is just |L|: i.e., if you wait for a period of time 2pi/|L|, then the symmetry axis will have returned to the same orientation.
|W_sym| is the rate of spin.
In the limit of small wobbles, L .=. 2W_sym, so the wobble rate is twice the spin rate.
It's worth thinking about the meaning of that minus sign in
W = L - W_sym
If you imagine setting W_sym=0 for a moment without changing L, then you'll see that the plate would be carried through a full rotation every 2pi/|L| time-units. The minus sign means that the plate doesn't spin fast enough to keep up with this rotation; in other words, the -W_sym refers to the body frame, not the lab frame (or should that be the cafeteria frame?)
If you prefer to measure the spin rate in the lab frame, then (again for small wobbles), you get approximately 2W_sym - W_sym = W_sym, so either way the spin rate is half the wobble rate.