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An object is formed from a square frame \(A B C D\) with a square lamina attached in one corner of the frame. The frame consists of four identical thin rods, each of mass \(M\) and length \(2 a\). The lamina has mass \(k M\) and edges of length \(a\). It has one vertex at \(C\) and adjacent sides in contact with \(C B\) and \(C D\) (see diagram).
- Show that the moment of inertia of the object about an axis \(l\) through \(A\) perpendicular to the plane of the object is \(\frac { 2 } { 3 } M a ^ { 2 } ( 7 k + 20 )\).
The object is released from rest with the edge \(A B\) horizontal and \(D\) vertically above \(A\). The object rotates freely about the fixed axis \(l\). The angular speed of the object is \(\frac { 1 } { 2 } \sqrt { } \left( \frac { 5 g } { a } \right)\) when \(D\) is first vertically below \(A\). - Find the value of \(k\).