What's happening to my spinning rugby ball

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A spinning rugby ball rises onto its end due to stability related to its principal axes of rotation. When spun around the maximum or minimum moment of inertia, the motion remains stable, allowing it to precess without deviating from its path. However, spinning around the intermediate principal axis leads to instability, causing any small perturbations to grow. Sufficient initial kinetic energy is required to overcome gravity and raise the ball's center of mass into a stable position. Without adequate spin, the ball may wobble chaotically instead of achieving stable motion.
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Hello everyone, this is my first post here so I hope I don't commit any physicsforums faux pas in this!

I have a curiousity. If you take a rugby ball or any other prolate ellipsoid and give it a good spin, it will rise up onto it's end. I wonder if any of you guys could clear up exactly why this happens!? I have sketchy notions that the ball seeks to take up a position where the concentration of its mass is the lowest possible distance away from the axis of its rotation but as to why this might be I can only think it's because it's more efficient in some way.

Very hazy anyway, so if anyone can enlighten me I would be very happy indeed. TIA.

James
 
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The basic reason is stability.

In general a 3-dimensional body has three unique "principal axes" passing through its center of mass. Two of these are the directions about which the moment of inertia is the maximum and minimum for the body. The third one is at right angles to those two.

In this general case it can be shown that if the body spins about the either the maximium or the minimum inertia directions, the motion is stable. If it is perturbed slightly it will "wobble" (precess) about the original direction, but it won't deviate further from its original motion.

However if it is spinning about the intermediate principal axis, the motion is unstable and any small pertubation will continue to grow.

See http://farside.ph.utexas.edu/teaching/336k/Newton/node71.html for the math.

If two of the principal axes have the same moment of inertia, the only stable rotation is about the thrid axis. For the rugby ball, this is along the length of the ball and it has the minimum possible inertia.

To get the ball the "stand up" as you describe, you have to give it enough initial kinetic energy to do the work required against gravity to raise the center of mass. If you don't spin it fast enough, it may wobble around apparently chaotically, but it doesn't have enough energy to get into a stable motion.

If you spin it slowly, if may appear to be spinning stably, but of course its contact with the ground is stopping it moving as it would in a zero-gravity environment.
 
Fantastic. Thank you so much for your help :approve:
 

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