A uniform disk of mass M and radius a can roll along a rough horizontal rail. A particle of mass m is suspended from the centre C of the disk by a light inextensible string of length b. The whole system moves in the vertical plane through the rail. Take as generalised coordinates x, the horizontal displacement of C, and θ, the angle between the string and the downward vertical. Obtain Lagrange's equations. Show that x is a cyclic coordinate and find the corresponding conserved momentum px. Is px the horizontal linear momentum of the system?
Given that θ remains small in the motion, find the period of small oscillations of the particle.