A particle p of mass 3m is suspended from a fixed point o

Question

A particle P of mass 3m is suspended from a fixed point O by a light inextensible string of length a. A second particle Q of mass m is in turn suspended from P by a second string of length a. The system moves in a vertical plane through O. Show that the linearised equations of motion for small oscillations near the downward vertical are

4θ^·· + Φ^·· +4n²θ = 0,

θ^·· + Φ^·· + n²Φ = 0.

where θ and φ are the angles that the two strings make with the downward vertical, and n² = g/a. Find the normal frequencies and the forms of the normal modes for this system.