Problem

The object shown is called a speed governor, a mechanical device for the regulation and co...

The object shown is called a speed governor, a mechanical device for the regulation and control of the speed of mechanisms. The system consists of two arms of negligible mass at the end of which are attached two spheres, each of mass m. The upper end of each arm is attached to a fixed collar A. The system is then made to spin with a given angular speed ω0 at a set opening angle θ0. Once it is in motion, the opening angle of the governor can be varied by adjusting the position of the collar C (by the application of some force). Let θ represent the generic value of the governor opening angle. If the arms are free to rotate, that is, if no moment is applied to the system about the spin axis after the system is placed in motion, determine the expression of the angular velocity ω of the system as a function of ω0, θ0, m, d, and L, where L is the length of each arm and d is the distance of the top hinge point of each arm from the spin axis. Neglect any friction at A and C.