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toroidal perpetual motion machine



At 06:13 AM 6/26/00 -0400, David Bowman asked for more puzzles that
illustrate physics principles. Well, here's a juicy one:

Version 1: Take an inner tube, an ordinary toroidal rubber inner
tube. Grip it its normal resting configuration, whatever that is, and keep
a grip on it while placing it in a pool of water. In general, meniscus
will form at the point where the water-surface meets the rubber. Assume
for concreteness that we have a positive meniscus in this case.

According to basic physics, to form the meniscus requires a force to lift
the water above the general surface level. The rubber pulls up on the
water, and by the same token the water in the meniscus pulls down on the
rubber.

Obviously the length of the meniscus on the outside of the toroid is longer
than the length of the meniscus on the inside. But the height of the
meniscus is the same everywhere, to a more-than-sufficient
approximation. So there's more total force on the outside.

Now release your grip. According to the foregoing analysis, the tube
should rotate a little bit like a vortex; the outer edge rotates down
while the inner edge rotates up.

After the tube rotates a little bit, the meniscus will climb back up to its
original uniform height.

It appears that some energy (force times distance) was imparted to the tube
when it rotated, but this energy did not come from the water in any obvious
way, because the water is in the same state before and after the rotation.

That's impossible. The foregoing analysis must be wrong (or at best
incomplete). So what is really going on here?

Version 2: This is the same as the previous version, except imagine a
not-quite ordinary inner tube, which is made to have rather low (not
necessarily zero) internal friction, and rather low (not necessarily zero)
asymmetry, so that it offers only a tiny bit resistance to the vortex-like
rolling we're interested in. (Such an inner tube would have the property
that if you cut it, the rubber would spring into a uniform cylinder;
conversely you might make such a device by starting with a uniform rubber
cylinder and joining the ends.)

In this version, the paradox is even more blatant: Apparently we have a
perpetual motion machine of the first kind! The difference in
meniscus-weight overcomes the small resistance to motion. The tube moves,
the meniscus re-establishes itself, and then the tube moves some more. If
the tube is sufficiently symmetric, the tube will rotate forever at a
steady rate. (Friction will limit the rate, but even a slow perpetual
motion machine is quite a conversation piece.)

Question: What really happens? Why? How?

Hint #1: This is not a word game. You will be able to answer the question
when and only when you understand the physics of the situation.

You won't learn anything by nit-picking the foregoing analysis. Minor nits
won't suffice; we all know there is some _major_ reason why the foregoing
analysis is wrong or incomplete. But what is it? Imagine really doing
the experiment, and do your own analysis.

Hint #2: The correct analysis is simple and beautiful. Anybody who did
well in high-school physics should immediately recognize it as the right
answer.

Warning: That doesn't mean the answer will be easy to find.

Acknowledgement: I got this from Richard Feynman, and I believe it was
original with him.