Show that the moment of inertia of the lamina about the axis through \(X\) is \(\frac { 4 } { 3 } m a ^ { 2 }\).
At an instant when \(\cos \theta = \frac { 3 } { 5 }\), show that \(\omega ^ { 2 } = \frac { 6 g } { 5 a }\).
At an instant when \(\cos \theta = \frac { 3 } { 5 }\), show that \(R = 0\), and given also that \(\sin \theta = \frac { 4 } { 5 }\) find \(S\) in terms of \(m\) and \(g\).