Consider a mass m moving in a circular orbit of radius a due to a central force

Question

Consider a mass m moving in a circular orbit of radius a due to a central force F = - mkr^n r (a) Show that the orbital frequency is ? = (ka^n - 1)^0.5 (b) A slightly eccentric orbit can be considered a slight perturbation on a circular orbit - that is, r will undergo small oscillations about the "equilibrium" radius a. Find the frequency of these oscillations. Answer: ? = ( (3 + n)k a^(n-1) )^0.5 (c) The (eccentric) orbits will be closed orbits if the oscillation frequency found in part (b) is an integer multiple of the orbital frequency found in part (a). Under what circumstances (i.e., what values of n) does this occur? What are the most physically significant examples?

Explanation / Answer

F=-mkr^n

m(w^2)*a = F (centripetal force)

w= (ka^n - 1)^0.5

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