A uniform rod \(P Q\), of mass \(m\) and length \(3 a\), is free to rotate about a fixed smooth horizontal axis \(L\), which passes through the end \(P\) of the rod and is perpendicular to the rod. The rod hangs at rest in equilibrium with \(Q\) vertically below \(P\). One end of a light inextensible string of length \(2 a\) is attached to the rod at \(P\) and the other end is attached to a particle of mass \(3 m\). The particle is held with the string taut, and horizontal and perpendicular to \(L\), and is then released. After colliding, the particle sticks to the rod forming a body \(B\).
Show that the moment of inertia of \(B\) about \(L\) is \(15 m a ^ { 2 }\).
Show that \(B\) first comes to instantaneous rest after it has turned through an angle \(\arccos \left( \frac { 9 } { 25 } \right)\).