On Nov 29, 2010, at 8:24 PM, Bill Gosper wrote:
Veit, the center of oscillation seems to be the ratio of moment of inertia to some other moment. The math isn't the problem--it's my physics. --Bill
it's all just natural philosophy ...
Here's the equation of motion,
I phi'' = T
I = moment of inertia about point of support phi = angle about point of support T = torque due to gravity
The torque is the same as if all the mass were located at the center of mass. (That's because the gravitational acceleration is a constant vector field, approximately). Let R be the distance between the point of support and the center of mass. Also, let I_0 be the moment of inertia about the center of mass. By the parallel axis theorem I = I_0 + MR^2, where M is the mass of the pendulum. The equation of motion is therefore
(I_0 + MR^2) phi'' = - MgR sin(phi).
The ratio I_0/M = R_0^2 defines the "radius of gyration". Divide through by MR and you get
(R_0^2/R + R) phi'' = - g sin{phi).
This is the same as the equation for a point-mass pendulum on a massless string of length L = R_0^2/R + R.
Huzzahs to the hereinbefore! As an example, a spherical shell of radius r
supported at its north pole has R_0 = r, R = r, L = 2r (as if all the mass were placed at the south pole).
But hoohah hereat. http://en.wikipedia.org/wiki/List_of_moments_of_inertia and tutor Julian assure me that the hollow sphere moment of inertia is 2 m R^2/3. This puts the effective center 5/6 of the way to the south pole, which I find infinitely more plausible than 6/6. Based on your earlier analysis, I withdraw my claim about hypercubes. The effective center varies (slightly) depending on what plane the centroid oscillates in. --Bill
Veit