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\includegraphics[max width=\textwidth, alt={}, center]{ab760a4b-e0ec-4256-838f-ed6c762ff18b-4_783_783_255_641} A uniform circular disc with centre \(C\) has mass \(m\) and radius \(a\). The disc is free to rotate in a vertical plane about a fixed horizontal axis passing through a point \(A\) on the disc, where \(A C = \frac { 1 } { 2 } a\). The disc is slightly disturbed from rest in the position with \(C\) vertically above \(A\). When \(A C\) makes an angle \(\theta\) with the upward vertical the force exerted by the axis on the disc has components \(R\) parallel to \(A C\) and \(S\) perpendicular to \(A C\) (see diagram).

1. Show that the angular speed of the disc is \(\sqrt { \frac { 4 g ( 1 - \cos \theta ) } { 3 a } }\).
2. Find the angular acceleration of the disc, in terms of \(a , g\) and \(\theta\).
3. Find \(R\) and \(S\), in terms of \(m , g\) and \(\theta\).
4. Find the magnitude of the force exerted by the axis on the disc at an instant when \(R = 0\).