2.
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\includegraphics[alt={},max width=\textwidth]{9fa0d04e-ff7b-46f8-a8ec-44393e383cdf-03_504_584_267_671}
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\caption{Figure 1}
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A uniform circular disc has mass \(4 m\), centre \(O\) and radius \(4 a\). The line \(P O Q\) is a diameter of the disc. A circular hole of radius \(2 a\) is made in the disc with the centre of the hole at the point \(R\) on \(P Q\) where \(Q R = 5 a\), as shown in Figure 1.
The resulting lamina is free to rotate about a fixed smooth horizontal axis \(L\) which passes through \(Q\) and is perpendicular to the plane of the lamina.
- Show that the moment of inertia of the lamina about \(L\) is \(69 m a ^ { 2 }\).
The lamina is hanging at rest with \(P\) vertically below \(Q\) when it is given an angular velocity \(\Omega\). Given that the lamina turns through an angle \(\frac { 2 \pi } { 3 }\) before it first comes to instantaneous rest,
- find \(\Omega\) in terms of \(g\) and \(a\).