A uniform lamina \(A B C\) of mass \(m\) is in the shape of an isosceles triangle with \(A B = A C = 5 a\) and \(B C = 8 a\).
Show, using integration, that the moment of inertia of the lamina about an axis through \(A\), parallel to \(B C\), is \(\frac { 9 } { 2 } m a ^ { 2 }\).
The foot of the perpendicular from \(A\) to \(B C\) is \(D\). The lamina is free to rotate in a vertical plane about a fixed smooth horizontal axis which passes through \(D\) and is perpendicular to the plane of the lamina. The lamina is released from rest when \(D A\) makes an angle \(\alpha\) with the downward vertical. It is given that the moment of inertia of the lamina about an axis through \(A\), perpendicular to \(B C\) and in the plane of the lamina, is \(\frac { 8 } { 3 } m a ^ { 2 }\).
Find the angular acceleration of the lamina when \(D A\) makes an angle \(\theta\) with the downward vertical.
Given that \(\alpha\) is small,
find an approximate value for the period of oscillation of the lamina about the vertical.