A uniform rod \(A B\), of mass \(m\) and length \(2 a\), is free to rotate in a vertical plane about a fixed smooth horizontal axis \(L\). The axis \(L\) is perpendicular to the rod and passes through the point \(P\) of the rod, where \(A P = \frac { 2 } { 3 } a\).
Find the moment of inertia of the rod about \(L\).
The rod is held at rest with \(B\) vertically above \(P\) and is slightly displaced.
Find the angular speed of the rod when \(P B\) makes an angle \(\theta\) with the upward vertical.
Find the magnitude of the angular acceleration of the rod when \(P B\) makes an angle \(\theta\) with the upward vertical.
Find, in terms of \(g\) and \(a\) only, the angular speed of the rod when the force acting on the rod at \(P\) is perpendicular to the rod.