Multi-stage motion: particle reaches ground/pulley causing string to go slack

\includegraphics{figure_6} Particles \(A\) and \(B\), of masses 0.2 kg and 0.45 kg respectively, are connected by a light inextensible string of length 2.8 m. The string passes over a small smooth pulley at the edge of a rough horizontal surface, which is 2 m above the floor. Particle \(A\) is held in contact with the surface at a distance of 2.1 m from the pulley and particle \(B\) hangs freely (see diagram). The coefficient of friction between \(A\) and the surface is 0.3. Particle \(A\) is released and the system begins to move.

1. Find the acceleration of the particles and show that the speed of \(B\) immediately before it hits the floor is 3.95 m s\(^{-1}\), correct to 3 significant figures. [7]
2. Given that \(B\) remains on the floor, find the speed with which \(A\) reaches the pulley. [4]

\includegraphics{figure_6} Particles \(A\) and \(B\), of masses \(0.2 \text{ kg}\) and \(0.45 \text{ kg}\) respectively, are connected by a light inextensible string of length \(2.8 \text{ m}\). The string passes over a small smooth pulley at the edge of a rough horizontal surface, which is \(2 \text{ m}\) above the floor. Particle \(A\) is held in contact with the surface at a distance of \(2.1 \text{ m}\) from the pulley and particle \(B\) hangs freely (see diagram). The coefficient of friction between \(A\) and the surface is \(0.3\). Particle \(A\) is released and the system begins to move.

1. Find the acceleration of the particles and show that the speed of \(B\) immediately before it hits the floor is \(3.95 \text{ m s}^{-1}\), correct to 3 significant figures. [7]
2. Given that \(B\) remains on the floor, find the speed with which \(A\) reaches the pulley. [4]

\includegraphics{figure_4} Two particles \(P\) and \(Q\) have masses \(3m\) and \(5m\) respectively. They are connected by a light inextensible string which passes over a small smooth light pulley fixed at the edge of a rough horizontal table. Particle \(P\) lies on the table and particle \(Q\) hangs freely below the pulley, as shown in Fig. 4. The coefficient of friction between \(P\) and the table is 0.6. The system is released from rest with the string taut. For the period before \(Q\) hits the floor or \(P\) reaches the pulley,

1. write down an equation of motion for each particle separately, [4]
2. find, in terms of \(g\), the acceleration of \(Q\), [4]
3. find, in terms of \(m\) and \(g\), the tension in the string. [2]

When \(Q\) has moved a distance \(h\), it hits the floor and the string becomes slack. Given that \(P\) remains on the table during the subsequent motion and does not reach the pulley,

1. find, in terms of \(h\), the distance moved by \(P\) after the string becomes slack until \(P\) comes to rest. [6]

A small package \(P\), of mass 1 kg, is initially at rest on the rough horizontal top surface of a wooden packing case which is 1.5 m long and 1 m high and stands on a horizontal floor. The coefficient of friction between \(P\) and the case is 0.2. \(P\) is attached by a light inextensible string, which passes over a smooth fixed pulley, to a weight \(Q\) of mass \(M\) kg which rests against the smooth vertical side of the case. The system is released from rest with \(P\) 0.75 m from the pulley and \(Q\) 0.5 m from the pulley. \(P\) and \(Q\) start to move with acceleration 0.4 ms\(^{-2}\). Calculate

1. the tension in the string, in N, [3 marks]
2. the value of \(M\), [3 marks]
3. the time taken for \(Q\) to hit the floor. [3 marks]

Given that \(Q\) does not rebound from the floor,

1. calculate the distance of \(P\) from the pulley when it comes to rest. [6 marks]

\includegraphics{figure_2}

Two particles \(P\) and \(Q\), of masses \(2m\) and \(3m\) respectively, are connected by a light string. Initially, \(P\) is at rest on a smooth horizontal table. The string passes over a small smooth pulley and \(Q\) rests on a rough plane inclined at an angle \(\theta\) to the horizontal, where \(\tan \theta = \frac{4}{3}\). The coefficient of friction between \(Q\) and the inclined plane is \(\frac{1}{6}\). \includegraphics{figure_7} The system is released from rest with \(Q\) at a distance of 0.8 metres above a horizontal floor.

1. Show that the acceleration of \(P\) and \(Q\) is \(\frac{21g}{50}\), stating a modelling assumption which you must make to ensure that both particles have the same acceleration. [7 marks]
2. Find the speed with which \(Q\) hits the floor. [2 marks]

After \(Q\) hits the floor and does not rebound, \(P\) moves a further 0.2 m until it hits the pulley.

1. Find the total time after the system is released before \(P\) hits the pulley. [5 marks]