M1 June 2011 Q6

6.

Two particles \(P\) and \(Q\) have masses 0.3 kg and \(m\) kg respectively. The particles are attached to the ends of a light inextensible string. The string passes over a small smooth pulley which is fixed at the top of a fixed rough plane. The plane is inclined to the horizontal at an angle \(\alpha\), where \(\tan\alpha = \dfrac{3}{4}\). The coefficient of friction between \(P\) and the plane is \(\dfrac{1}{2}\).

The string lies in a vertical plane through a line of greatest slope of the inclined plane. The particle \(P\) is held at rest on the inclined plane and the particle \(Q\) hangs freely below the pulley with the string taut, as shown in Figure 2.

The system is released from rest and \(Q\) accelerates vertically downwards at 1.4 m s\(^{-2}\). Find

When the particles have been moving for 0.5 s, the string breaks. Assuming that \(P\) does not reach the pulley,