7. Particles \(P\) and \(Q\) have mass \(3 m\) and \(m\) respectively. Particle \(P\) is attached to one end of a light inextensible string and \(Q\) is attached to the other end. The string passes over a circular pulley which can freely rotate in a vertical plane about a fixed horizontal axis through its centre \(O\). The pulley is modelled as a uniform circular disc of mass \(2 m\) and radius \(a\). The pulley is sufficiently rough to prevent the string slipping. The system is at rest with the string taut. A third particle \(R\) of mass \(m\) falls freely under gravity from rest for a distance \(a\) before striking and adhering to \(Q\). Immediately before \(R\) strikes \(Q\), particles \(P\) and \(Q\) are at rest with the string taut.

1. Show that, immediately after \(R\) strikes \(Q\), the angular speed of the pulley is \(\frac { 1 } { 3 } \sqrt { \left( \frac { g } { 2 a } \right) }\). When \(R\) strikes \(Q\), there is an impulse in the string attached to \(Q\).
2. Find the magnitude of this impulse. Given that \(P\) does not hit the pulley,
3. find the distance that \(P\) moves upwards before first coming to instantaneous rest.