2 One end of a light elastic string of natural length 0.8 m and modulus of elasticity 36 N is attached to a fixed point \(O\) on a smooth plane. The plane is inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 3 } { 5 }\). A particle \(P\) of mass 2 kg is attached to the other end of the string. The string lies along a line of greatest slope of the plane with the particle below the level of \(O\). The particle is projected with speed \(\sqrt { 2 } \mathrm {~ms} ^ { - 1 }\) directly down the plane from the position where \(O P\) is equal to the natural length of the string. Find the maximum extension of the string during the subsequent motion.
\includegraphics[max width=\textwidth, alt={}, center]{6dcce6fe-7a19-4c5f-9361-20e7acda458f-04_380_1173_267_447} Particles \(A\) and \(B\), of masses \(3 m\) and \(m\) respectively, are connected by a light inextensible string of length \(a\) that passes through a fixed smooth ring \(R\). Particle \(B\) hangs in equilibrium vertically below the ring. Particle \(A\) moves in horizontal circles on a smooth horizontal surface with speed \(\frac { 2 } { 5 } \sqrt { \text { ga } }\). The angle between \(A R\) and \(B R\) is \(\theta\) (see diagram). The normal reaction between \(A\) and the surface is \(\frac { 12 } { 5 } \mathrm { mg }\).

1. Find \(\cos \theta\).
2. Find, in terms of \(a\), the distance of \(B\) below the ring.
   \includegraphics[max width=\textwidth, alt={}, center]{6dcce6fe-7a19-4c5f-9361-20e7acda458f-06_703_481_264_785} A particle 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 particle is initially held with the string taut at the point \(A\), where \(O A\) makes an angle \(\theta\) with the downward vertical through \(O\). The particle is then projected with speed \(u\) perpendicular to \(O A\) and begins to move upwards in part of a vertical circle. The string goes slack when the particle is at the point \(B\) where angle \(A O B\) is a right angle. The speed of the particle when it is at \(B\) is \(\frac { 1 } { 2 } u\) (see diagram). Find the tension in the string at \(A\), giving your answer in terms of \(m\) and \(g\).