A uniform lamina \(P Q R\) of mass \(m\) is in the shape of an isosceles triangle, with \(P Q = P R = 5 a\) and \(Q R = 6 a\). The midpoint of \(Q R\) is \(T\).
Show, using integration, that the moment of inertia of the lamina about an axis which passes through \(P\) and is parallel to \(Q R\), is \(8 m a ^ { 2 }\).
Show, using integration, that the moment of inertia of the lamina about an axis which passes through \(P\) and \(T\), is \(1.5 m a ^ { 2 }\). [0pt]
[You may assume without proof that the moment of inertia of a uniform rod, of mass \(m\) and length \(2 l\), about an axis perpendicular to the rod through its midpoint is \(\frac { 1 } { 3 } m l ^ { 2 }\) ]
(4)
The lamina is now free to rotate in a vertical plane about a fixed smooth horizontal axis \(A\) which passes through \(P\) and is perpendicular to the plane of the lamina. The lamina makes small oscillations about its position of stable equilibrium.
By writing down an equation of rotational motion for the lamina as it rotates about \(A\), find the approximate period of these small oscillations.