7. A uniform lamina of mass \(m\) is in the shape of an equilateral triangle \(A B C\) of perpendicular height \(h\). The lamina is free to rotate in a vertical plane about a fixed smooth horizontal axis \(L\) through \(A\) and perpendicular to the lamina.
- Show, by integration, that the moment of inertia of the lamina about \(L\) is \(\frac { 5 m h ^ { 2 } } { 9 }\).
The centre of mass of the lamina is \(G\). The lamina is in equilibrium, with \(G\) below \(A\), when it is given an angular speed \(\sqrt { \left( \frac { 6 g } { 5 h } \right) }\).
- Find the angle between \(A G\) and the downward vertical, when the lamina first comes to rest.
- Find the greatest magnitude of the angular acceleration during the motion.
(Total 17 marks)