Serway & Jewett — Oscillatory Motion: Problem 48

A smaller disk of radius \(r\) and mass \(m\) is attached rigidly to the face of a second larger disk of radius \(R\) and mass \(M\) as shown in Figure P15.48. The center of the small disk is located at the edge of the large disk. The large disk is mounted at its center on a frictionless axle. The assembly is rotated through a small angle \(\theta\) from its equilibrium position and released. (a) Show that the speed of the center of the small disk as it passes through the equilibrium position is \(v = 2 \left[ \frac{Rg(1 - \cos \theta)}{(M/m) + (r/R)^2 + 2} \right]^{1/2}\). (b) Show that the period of the motion is \(T = 2\pi \left[ \frac{(M + 2m)R^2 + mr^2}{2mgR} \right]^{1/2}\).