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\includegraphics[max width=\textwidth, alt={}, center]{ea62d6d9-ac2f-44e7-8bfb-ae9aeea7109b-4_524_732_258_705} The diagram shows a uniform rectangular lamina \(A B C D\) with \(A B = 6 a , A D = 8 a\) and centre \(G\). The mass of the lamina is \(m\). The lamina rotates freely in a vertical plane about a fixed horizontal axis passing through \(A\) and perpendicular to the lamina.

1. Find the moment of inertia of the lamina about this axis. The lamina is released from rest with \(A D\) horizontal and \(B C\) below \(A D\).
2. For an instant during the subsequent motion when \(A D\) is vertical, show that the angular speed of the lamina is \(\sqrt { \frac { 3 g } { 50 a } }\) and find its angular acceleration. At an instant when \(A D\) is vertical, the force acting on the lamina at \(A\) has magnitude \(F\).
3. By finding components parallel and perpendicular to \(G A\), or otherwise, show that \(F = \frac { \sqrt { 493 } } { 20 } \mathrm { mg }\).
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