Bill, For the ball-rolling exhibit at MoMath, I specified custom balls with a metal core and foamy "mantle" to reduce the moment of inertia. But I'm no longer working with MoMath, so I'm not sure about the status of them. There are plenty of sources of error and the timing differences between different tracks are not large enough to make detailed comparisons, but the main lesson which does come through is that the straight line track, though shortest in space, is not shortest in time. Tangentially related: An excellent recent book on Galileo is "Galileo's Muse" by Mark Peterson. It doesn't say much about the cycloid, but explains the importance of mathematical ideas from art that aided his development of science. George http://georgehart.com/ P.S. Check out some of my cool math videos: http://www.youtube.com/channel/UCTl0dASnxto6j2wlVs5Bs2Q On 11/21/2012 12:33 AM, Bill Gosper wrote:
I find unanimity on the web that Galileo thought the semicircle was tautochronous, i.e., that the pendulum period was amplitude-invariant. I find this incredible. He was a great experimenter. One site excuses his error by quoting some small % difference between low and high amplitudes. But I calculate that for a string length of 1.00544 ft = g/32 sec^2, the small amplitude period is 2 π/√32 = 1.107 sec, whereas the theoretical period for a full semicircular swing is ¼!^2 √(8/π) = 1.311 sec.
A related mystery is why Galileo was fascinated by the cycloid. He *named* it. He couldn't compute its area, so he made a metal one and weighed it. Could he have at least suspected it was the tautochrone?
Neil points out that this image http://www.scitechantiques.com/cycloidhtml/images/chrisy_cycloid_SML.jpg from a slightly dodgy website, is probably photoshopped. The page (with a broken video) claims that this beautiful piece of cabinetry admirably demonstrates tautochronicity. How could it, with rolling spheres? Their centers of mass do not follow a cycloid, and they have angular momentum.
An interesting problem would be to manufacture heavy balls with very low moment of inertia. Neil suggests instead a lightweight cylinder with an osmium spindle. To reduce friction, maybe taper the ends slightly and run it on a pair of thin rails.
A challenge to Veits' Mechanics geeks might be to fudge the museum cycloid to make a true tautochrone curve for a sphere of specific mass and diameter. Would the the mass actually matter? Hard to see how, given the Leaning Tower results. This exhibit would convince all the average kids, and confuse the really smart ones. How would George Hart handle this moral dilemma? --rwg _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun