First of all, a spherical bob (on a massless string) centered L below the pivot is a compound pendulum. Its period, for a given amplitude, depends on its radius R! It acts like a point mass centered at L+(3/5)*R^2/L rather than L. Unless it's hollow, when it's just L+R^2/L. So you can tell if it's hollow by swinging it. For a unit n-cube hanging by a vertex, this "center of oscillation" is 2/3 down the main diagonal, for all n. Wikipedia claims the center of oscillation is also the center of percussion. So if you hang a bat by the knob, and match oscillations with a ball on an adjustably long string, the ball will hang down to the sweet spot. If you're the sort of person who swings bats by their knob. Mathematica 7 clams on the ODE of motion: {\[Theta]''[t] == -g*Sin[\[Theta][t]]/L, \[Theta][0] == 0} but Macsyma returns an integral for t[\Theta] which Mma knows is an EllipticF. Testing by taking the limit for small \[Theta]'[0] insists the period approaches 149/30 Sqrt[L/g] instead of 2 pi Sqrt[L/g]! The bug is in Limit. Mma then solves for Theta, \[Theta][t] -> 2*JacobiAmplitude[ (1/2)*t*Derivative[1][\[Theta]][0],(4*g)/(L*Derivative[1][\[Theta]][0]^2)] with period \[CapitalLambda] == 4*Sqrt[L/g]*EllipticK[Sin[Subscript[\[Theta], max]/2]^2] and amplitude Subscript[\[Theta], max] == 2*ArcSin[(1/2)*Sqrt[L/g]*Derivative[1][\[Theta]][0]] The differentiator and simplifiers eventually confess that these solve the ODE. But had I thought to search "pendulum" in the Mma doc, it would have led me to an "application" of JacobiAmplitude where it is claimed, without derivation, to describe an overswung pendulum. This would have been a clue. I'm surprised this all isn't more obvious in, e.g., Wikipedia or MathWorld. Or DSolve. --rwg