2 A light elastic string has natural length \(a\) and modulus of elasticity 4 mg . One end of the string is fixed to a point \(O\) on a smooth horizontal surface. A particle \(P\) of mass \(m\) is attached to the other end of the string. The particle \(P\) is projected along the surface in the direction \(O P\). When the length of the string is \(\frac { 5 } { 4 } a\), the speed of \(P\) is \(v\). When the length of the string is \(\frac { 3 } { 2 } a\), the speed of \(P\) is \(\frac { 1 } { 2 } v\).

1. Find an expression for \(v\) in terms of \(a\) and \(g\).
2. Find, in terms of \(g\), the acceleration of \(P\) when the stretched length of the string is \(\frac { 3 } { 2 } a\).
   \includegraphics[max width=\textwidth, alt={}, center]{7febbd80-4cbb-4b2e-b022-d6a20e7e13aa-04_552_1059_264_502} A smooth cylinder is fixed to a rough horizontal surface with its axis of symmetry horizontal. A uniform rod \(A B\), of length \(4 a\) and weight \(W\), rests against the surface of the cylinder. The end \(A\) of the rod is in contact with the horizontal surface. The vertical plane containing the rod \(A B\) is perpendicular to the axis of the cylinder. The point of contact between the rod and the cylinder is \(C\), where \(A C = 3 a\). The angle between the rod and the horizontal surface is \(\theta\) where \(\tan \theta = \frac { 3 } { 4 }\) (see diagram). The coefficient of friction between the rod and the horizontal surface is \(\frac { 6 } { 7 }\). A particle of weight \(k W\) is attached to the rod at \(B\). The rod is about to slip. The normal reaction between the rod and the cylinder is \(N\).