2. krn potential Consider two bodies which orbit one another under the influence of the potential U(r) = krn where k and n are real constants such that kn > 0 and is the relative separation. a) What does the condition kn > 0 tell us about the force? b) Determine the effective potential Uef (r) which describes motion about the center of mass. c) Sketch Ueff (r) for the cases of n = -3, -1, and 2. d) Within the effective one-dimensional description, determine the radius r = r₁ for which the particle (with given angular momentum ) can orbit at a fixed radius. e) For what values of n are these circular orbits stable under small perturbations? f) For the stable case, show that the period of small oscillations about the circular orbit Tosc satisfies Torb Tosc = √√n+2. (2) Here Torb is the orbital period for a circular orbit, Torb = 2π/0. g) Argue that if √√n+2 is a rational number, these slightly perturbed orbits are closed. Sketch these orbits for the cases of n = -1, 2, and 7.