The hub of a wheel is attached to a spring with spring constant k and negligible mass. The wheel has a radius R and total mass M. The mass of the spokes is negligible. The wheel rolls without slipping, i. e., the wheel translates by the same distance that its circumference rotates. The center of mass of the wheel oscillates (simple harmonic motion) in the horizontal direction about its equilibrium point x = 0.
Find an expression for the total energy in terms of k, M, R, and x(t). Since the spokeshave negligible mass, assume that the moment of inertia for rotation about the axle is MRsquare and using conservation of energy derive the differential equation of motion, what is the period of small oscillations about equilibrium?