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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.

(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.

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