Question

M A spring connects a fixed wall to the top of the uniform cylinder on a horizontal surface. Originally the spring is horizontal, and it is neither extended nor compressed. The cylinder is rolled slightly to one side, and released from rest. The cylinder rolls without slipping on the surface, and the centre of mass follows a simple harmonic oscillator. You can treat the spring as being horizontal throughout the motion. The mass of the cylinder is M, the radius is R. The spring constant is k. (a) What is rotational inertia of the cylinder about the point of contact? (b) What is the frequency of the oscillation? (c) If M = 1.5 kg, R = 0.25 m, k = 12 N m-1, and the maximum displacement is Xm = 0.020 m, what is the maximum speed of the centre of mass? (d) Suppose the energy loss is actually not negligible, but it's also not too large. Let's say the speed is dropped by about 10% between the first two consecutive passages of the equilibrium position. Sketch, on the same graph, (i) the potential energy, and (ii) the total kinetic energy. Show at least two periods in the graph.

Answer 1

a) I- MR b11 axes theorem MR2 1-5x MR Ans So I 5 MR abont Point P b U: Totnl mechanical eny o hm en dr mone te ight b der 2 Kt Mu + + 2KxMu du 0 4Kxi +M U= Conctant E O 4K 3Ma D a 3 M 3M So w 8K 3M Am 8X12 0.012 m/s 3x.5 X 02 3м t