October 25 2013, 05:31:29 PM
It's a property of rotating rigid bodies.
Rotation around the principal axis with the highest or the lowest moment of inertia will be stable, but rotation around the third axis (which has neither the largest or the smallest moment of inertia) will be unstable. You can try it with any object, preferably something where you can easily identify the axis, like a deck of cards. Let's call the smallest face of the box with the cards in it 'a', the intermediate one 'b' and the largest one 'c'. If you try throwing the box so that it rotates around the axis going through 'a' or 'c' you'll see that it rotates normally, but if you try to rotate it around the 'b' axis it'll spin around like crazy. Why? Math. Maybe someone can provide you with a good explanation, but it's not a very intuitive result IMO. In principle I think you can make something spin around the intermediate axis, but if it's even slightly off it will start to wobble.
If you want to get a bit more technical, the magnitude of the angular momentum vector & the total kinetic energy has to be preserved in the absence of external forces. For rotation close to the 'a' and 'c' axis this will result in precession around those axis, but for the 'b' axis there's no similarly stable way to conserve the momentum:
The ellipsoid corresponds to the three moments of inertia (which corresponds to the energy conservation constraint), so the 'a' axis is the longest and 'c' the shortest. The curves in the picture are the intersections between an ellipsoid and a sphere. The sphere obviously comes from the condition for conservation of angular momentum. So the curves are just the points which satisfy both conditions.
edit: made a correction
edit2: to be a bit more clear, the ellipsoid itself is a representation of how fast the object has to spin around a given axis to have a given rotational energy. So because 'a' is the smallest moment of inertia it needs to spin the fastest around 'a' of all the other axis to have a given energy, that's why the ellipse is longest there. The curves represent points which have the same energy AND the same angular momentum.
Last edited by kyrieee; October 25 2013 at 05:59:18 PM.
October 25 2013, 05:56:53 PM
A mathematical treatment of the problem
I think it is pretty clear.