Consider a particle of reduced mass µ orbiting in a central force with U = krⁿ where kn > 0. (a) Explain what the condition kn > 0 tells us about the force. Sketch the effective potential energy U_eff for the cases that n = 2, -1, and -3. (b) Find the radius at which the particle (with given angular momentum l) can orbit at a fixed radius. For what values of n is this circular orbit stable? Do your sketches confirm this conclusion? (c) For the stable case, show that the period of small oscillations about the circular orbit is τ_osc = τ_orb/√n+2. Argue that if, √n+2 is a rational number, these orbits are closed. Sketch them for the cases that n = 2, -1, and 7.
