[Note: Post restored from archive - gneill, PF Mentor] 1. The problem statement, all variables and given/known data A pendulum consists of a uniform circular disk of radius r which is free to turn about a horizontal axis perpendicular to it's plane. Find the position of the axis for which the periodic time ( for small amplitude oscillations) is a minimum. τ: torque Ι (i): moment of inertia m: mass g:gravity θ: angle between force mg and the radius length r: radius ω: angular frequency T: period Sinθ=θ for small amplitudes. L: length (offset for parallel axis theorem.) (unknown?) 2. Relevant equations Equation (1): τ= -mgL θ Equation (2): θ'' + (mgL/ I ) θ = 0 Equation (3): ω = (gL / (.5 r^2 + L^2)) ^.5 3. The attempt at a solution So I solved for ω and my reasoning was that in order to have the minimum periodic time ω has to be at it's maximum. So I used applied optimization and set dω/dL = 0. However, I am not sure if this was the right way to do it since I squared both sides of the equation 3 to get rid of the square root and then used implicit differentiation. But I think this might be wrong because it's like if I differentiates the inside of Eq. 1 and that differentiation does not take into account the square root. So I am not sure if I did it right. I differentiated directly but the derivative was a monster and I couldn't factor out the L to solve for it. Is this a right way to do it correctly?