5. A uniform square lamina \(A B C D\), of mass \(m\) and side \(2 a\), is free to rotate in a vertical plane about a fixed smooth horizontal axis \(L\) which passes through \(A\) and is perpendicular to the plane of the lamina. The moment of inertia of the lamina about \(L\) is \(\frac { 8 m a ^ { 2 } } { 3 }\).
Given that the lamina is released from rest when the line \(A C\) makes an angle of \(\frac { \pi } { 3 }\) with the downward vertical,
- find the magnitude of the vertical component of the force acting on the lamina at \(A\) when the line \(A C\) is vertical.
Given instead that the lamina now makes small oscillations about its position of stable equilibrium,
- find the period of these oscillations.
(5)
(Total 12 marks)