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.
The system is governed by the equation:
(M+m)a^2 (d^2(x)/dt^2)= a*g*(m -(2Msin(x))/(pi))
where x is the angle between the downward vertical and the diameter through the center of mass of the heavy rim.
Find the equilibrium points and the conditions for their existence and stability.
Let k = m/M. how does the wheel behave when k is large?
I have some thoughts on the problem, but I am not sure if they are right:
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 points as well.
another observation is when k = 2/pi, sin (x) = 1. i think bifurcation occurs here, but i am not sure what would happen when k>(2/pi), becuase it seems like that means sin(x) >1...