7. A uniform square lamina \(P Q R S\), of mass \(m\) and side \(2 a\), is free to rotate about a fixed smooth horizontal axis which passes through \(P\) and \(Q\). The lamina hangs at rest in a vertical plane with \(S R\) below \(P Q\) and is given a horizontal impulse of magnitude \(J\) at the midpoint of \(S R\). The impulse is perpendicular to \(S R\).

1. Find the initial angular speed of the lamina.
2. Find the magnitude of the angular deceleration of the lamina at the instant when the lamina has turned through \(\frac { \pi } { 6 }\) radians.
3. Find the magnitude of the component of the force exerted on the lamina by the axis, in a direction perpendicular to the lamina, at the instant when the lamina has turned through \(\frac { \pi } { 6 }\) radians.
   \includegraphics[max width=\textwidth, alt={}, center]{f932d7cb-1299-41d1-8248-cfbf639795ed-12_2255_50_315_1978}