A pendulum consists of a uniform rod \(A B\), of length \(4 a\) and mass \(2 m\), whose end \(A\) is rigidly attached to the centre \(O\) of a uniform square lamina \(P Q R S\), of mass \(4 m\) and side \(a\). The \(\operatorname { rod } A B\) is perpendicular to the plane of the lamina. The pendulum is free to rotate about a fixed smooth horizontal axis \(L\) which passes through \(B\). The axis \(L\) is perpendicular to \(A B\) and parallel to the edge \(P Q\) of the square.
Show that the moment of inertia of the pendulum about \(L\) is \(75 m a ^ { 2 }\).
The pendulum is released from rest when \(B A\) makes an angle \(\alpha\) with the downward vertical through \(B\), where \(\tan \alpha = \frac { 7 } { 24 }\). When \(B A\) makes an angle \(\theta\) with the downward vertical through \(B\), the magnitude of the component, in the direction \(A B\), of the force exerted by the axis \(L\) on the pendulum is \(X\).
Find an expression for \(X\) in terms of \(m , g\) and \(\theta\).
Using the approximation \(\theta \approx \sin \theta\),
find an estimate of the time for the pendulum to rotate through an angle \(\alpha\) from its initial rest position.