A pendulum is modelled as a uniform rod \(A B\), of mass \(3 m\) and length \(2 a\), which has a particle of mass \(2 m\) attached at \(B\). The pendulum is free to rotate in a vertical plane about a fixed smooth horizontal axis \(L\) which passes through \(A\). The vertical plane is perpendicular to the axis \(L\).
Find the period of small oscillations of the pendulum about its position of stable equilibrium.
The pendulum is hanging at rest in a vertical position, with \(B\) below \(A\), when it is given a horizontal impulse of magnitude \(J\). The impulse acts at \(B\) in a vertical plane which is perpendicular to the axis \(L\).
Given that the pendulum turns through an angle of \(60 ^ { \circ }\) before first coming to instantaneous rest,