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. 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). Veit