Particles \(P\) and \(Q\) have mass \(3m\) 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 \(2m\) 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.
- Show that, immediately after \(R\) strikes \(Q\), the angular speed of the pulley is \(\frac{1}{3}\sqrt{\frac{g}{2a}}\). [5]
When \(R\) strikes \(Q\), there is an impulse in the string attached to \(Q\).
- Find the magnitude of this impulse. [3]
Given that \(P\) does not hit the pulley,
- find the distance that \(P\) moves upwards before first coming to instantaneous rest. [6]