A point mass m slides without friction on a horizontal table at one end of a mass less spring of natural length a and spring constant k as shown in Figure 3-C. The spring is attached to the table so it can rotate freely without friction. The net force on the mass is the central force F(r) = €“ k (r €“ a).
(a) Find and sketch both the potential energy U(r) and the effective potential Ueff(r).
(b) What angular velocity w0 is required for a circular orbit with radius r0?
(c) Derive the frequency of small oscillations w about the circular orbit with radius r0. Express your answers for (b) and (c) in terms of k, m, r0, and a.