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\includegraphics[max width=\textwidth, alt={}, center]{337dd1f9-a691-4e99-9aa7-7a93d8bb13be-3_479_1225_1484_461}
A uniform rectangular block of mass \(m\) and cross-section \(A B C D\) has \(A B = C D = 6 a\) and \(A D = B C = 2 a\). The point \(X\) is on \(A B\) such that \(A X = a\) and \(G\) is the centre of \(A B C D\). The block is placed with \(A B\) perpendicular to the straight edge of a rough horizontal table. \(A X\) is in contact with the table and \(X B\) overhangs the edge (see diagram). The block is released from rest in this position, and it rotates without slipping about a horizontal axis through \(X\).
- Find the moment of inertia of the block about the axis of rotation.
For the instant when \(X G\) is horizontal,
- show that the angular acceleration of the block is \(\frac { 3 \sqrt { 5 } g } { 25 a }\),
- find the angular speed of the block,
- show that the force exerted by the table on the block has magnitude \(\frac { 2 \sqrt { 70 } } { 25 } m g\).