Solving the Wobble Rate for a Plate Thrown Off Axis

[abercrombiems02]

Hey everyone, I need some help here. The question is to show that the wobble rate of a plate when thrown slightly off axis, is twice that of the spin rate. I don't even know how to start this problem. All I really know is that the inertia matrix can be written simply about body fixed axes. Thus the axial moment of intertia is twice that of both the transverse axes. I know that the angular momentum is most likely a constant of the motion because gravity will not produce a moment about the center of mass. Also, H = Iw, but I do not know how to write an expression for w. I know i want to use body fixed coordinates, but do not know how to express the precession (or wobble) rate in terms of these body fixed coordinates? Can someone please show me how to start this problem?

 

[elhinnaw]

First off, its an inertia tensor not an inertia matrix.

Second, without actually solving the problem right now, I would suggest trying to work things out in terms of your angular momentum.

If you have a plate spinning, its angular momentum will be about the Z axis. if the plate is wobbling, the angular momentum vector is now rotating (sweeping out a cone). When they say wobble rate, I would assume this to be the the time derivatave of PHI(L), where PHI(L) is the angle in the XY plane of the angular momentum vector. Your task would then be to find a relationship between dPHI(L)/dt and w. You might have to create a ficticious angle for the plate to wobble thru in order to do the calculations.

As with all the advice I give, I could be wrong, but that's how I would start the problem

 

[Ralph]

I would have imagined this to be a difficult problem, since supposedly Feynman spent a fair amount of time on it at... I think it was Cornell, after he left Los Alamos and was feeling depressed. (I read this in one of his autobiographical books, maybe "Sure You're Joking" or "Why Do You Care What Anyone Thinks".)

If Feynman had to spend a while getting it, I guess you don't have to feel too bad if you are stumped. I think he said the general equations for a rotating body are quite complicated. I never got past the mysterious "principle axes" and "moment of inertia" calculations, which I didn't like, because nobody could explain exactly what a "principle axis" really was. Some sort of mystical entity, it seems. There must be some kind of line integrals plus group theory to figure out the principle axes, but it must be really bad since apparently nobody wants to talk about it.

 

[abercrombiems02]

Thanks, I got it! Turns out that the observation in text is incorrect. The wobble rate is actually twice the spin rate. The solution comes from Eulers EOMS with the assumption that there are no external torques and that the oblate body is axially symmetric. Also, for the case where the wobbling angle is small (theta is close to 90) it turns out that the relative spin rate approaches the observed spin rate. Quite the simplification from a very complicated problem!