5 The region bounded by the curve \(y = \sqrt { a x }\) for \(a \leqslant x \leqslant 4 a\) (where \(a\) is a positive constant), the \(x\)-axis, and the lines \(x = a\) and \(x = 4 a\), is rotated through \(2 \pi\) radians about the \(x\)-axis to form a uniform solid of revolution of mass \(m\).

1. Show that the moment of inertia of this solid about the \(x\)-axis is \(\frac { 7 } { 5 } m a ^ { 2 }\). The solid is free to rotate about a fixed horizontal axis along the line \(y = a\), and makes small oscillations as a compound pendulum.
2. Find, in terms of \(a\) and \(g\), the approximate period of these small oscillations.
   \includegraphics[max width=\textwidth, alt={}, center]{a9e010ce-c3a8-4f95-a154-fd16ef3e5e5b-3_734_862_813_644} A uniform rectangular lamina \(A B C D\) has mass \(m\) and sides \(A B = 2 a\) and \(B C = 3 a\). The mid-point of \(A B\) is \(P\) and the mid-point of \(C D\) is \(Q\). The lamina is rotating freely in a vertical plane about a fixed horizontal axis which is perpendicular to the lamina and passes through the point \(X\) on \(P Q\) where \(P X = a\). Air resistance may be neglected. When \(Q\) is vertically above \(X\), the angular speed is \(\sqrt { \frac { 9 g } { 10 a } }\). When \(X Q\) makes an angle \(\theta\) with the upward vertical, the angular speed is \(\omega\), and the force acting on the lamina at \(X\) has components \(R\) parallel to \(P Q\) and \(S\) parallel to \(B A\) (see diagram).