1 A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\). The string is held taut with \(O P\) making an angle \(\alpha\) with the downward vertical, where \(\cos \alpha = \frac { 2 } { 3 }\). The particle \(P\) is projected perpendicular to \(O P\) in an upwards direction with speed \(\sqrt { 3 a g }\). It then starts to move along a circular path in a vertical plane. Find the cosine of the angle between the string and the upward vertical when the string first becomes slack.
\includegraphics[max width=\textwidth, alt={}, center]{9b3f3add-17fd-4597-bd5d-27e3abb527be-03_671_455_255_845} A uniform lamina is in the form of a triangle \(A B C\) in which angle \(B\) is a right angle, \(\mathrm { AB } = 9 \mathrm { a }\) and \(\mathrm { BC } = 6 \mathrm { a }\). The point \(D\) is on \(B C\) such that \(\mathrm { BD } = \mathrm { x }\) (see diagram). The region \(A B D\) is removed from the lamina. The resulting shape \(A D C\) is placed with the edge \(D C\) on a horizontal surface and the plane \(A D C\) is vertical. Find the set of values of \(x\), in terms of \(a\), for which the shape is in equilibrium.