oh i am sorry about that.
note that k in the question refers to m/M.
so i think when k is large, d^2(x)/dt^2 tends to g/a, which means the wheel behaves like a simple pendulum??
and one of the equilibrium points is, of course, (k,x) = (0,0). then (0, pi) and (0,2*pi) are equilibrium...
A light wheel of radius a has a uniform semicircular rim of mass M, and my rotate freely in a vertical plane about a horizontal axis through its center. A light string passes around the wheel and suspends a mass m. You may assume that this mechanical system is governed by the equation:
(M+m)a^2...