3. A uniform lamina of mass \(m\) is in the shape of a rectangle \(P Q R S\), where \(P Q = 8 a\) and \(Q R = 6 a\).
- Find the moment of inertia of the lamina about the edge \(P Q\).
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The flap on a letterbox is modelled as such a lamina. The flap is free to rotate about an axis along its horizontal edge \(P Q\), as shown in Fig. 1. The flap is released from rest in a horizontal position. It then swings down into a vertical position. - Show that the angular speed of the flap as it reaches the vertical position is \(\sqrt { \left( \frac { g } { 2 a } \right) }\).
- Find the magnitude of the vertical component of the resultant force of the axis \(P Q\) on the flap, as it reaches the vertical position.