M4 June 2012 Q5

5.

A uniform rod \(AB\), of length \(4a\) and weight \(W\), is free to rotate in a vertical plane about a fixed smooth horizontal axis which passes through the point \(C\) of the rod, where \(AC = 3a\). One end of a light inextensible string of length \(L\), where \(L \gt 10a\), is attached to the end \(A\) of the rod and passes over a small smooth fixed peg at \(P\) and another small smooth fixed peg at \(Q\). The point \(Q\) lies in the same vertical plane as \(P\), \(A\) and \(B\). The point \(P\) is at a distance \(3a\) vertically above \(C\) and \(PQ\) is horizontal with \(PQ = 4a\). A particle of weight \(\dfrac{1}{2}W\) is attached to the other end of the string and hangs vertically below \(Q\). The rod is inclined at an angle \(2\theta\) to the vertical, where \(-\pi \lt 2\theta \lt \pi\), as shown in Figure 1.