Question
Pom the separation ro. 8.14*** Consider a particle of reduced mass μ orbiting in a central force with U = kr" where kn > 0. (a) Explain what the condition kn > 0 tells us about the force. Sketch the effective potential energy Ueff for the cases that n = 2, -1, and -3. (b) Find the radius at which the particle (with given angular momentum () 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 Tosc = Torb/√√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.
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