Pulley at edge of table, specific geometry

7 \includegraphics[max width=\textwidth, alt={}, center]{ceb367ee-4e12-4cb2-9020-078ea5724d6e-3_430_860_1585_641} Particle \(A\) of mass 1.26 kg and particle \(B\) of mass 0.9 kg are attached to the ends of a light inextensible string. The string passes over a small smooth pulley \(P\) which is fixed at the edge of a rough horizontal table. \(A\) is held at rest at a point 0.48 m from \(P\), and \(B\) hangs vertically below \(P\), at a height of 0.45 m above the floor (see diagram). The coefficient of friction between \(A\) and the table is \(\frac { 2 } { 7 } . A\) is released and the particles start to move.

1. Show that the magnitude of the acceleration of the particles is \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) and find the tension in the string.
2. Find the speed with which \(B\) reaches the floor.
3. Find the speed with which \(A\) reaches the pulley.

7 \includegraphics[max width=\textwidth, alt={}, center]{631ddcd9-17c0-4a15-8671-40788c3a84d3-3_565_828_1402_660} Particles \(A\) and \(B\) have masses 0.32 kg and 0.48 kg respectively. The particles are attached to the ends of a light inextensible string which passes over a small smooth pulley fixed at the edge of a smooth horizontal table. Particle \(B\) is held at rest on the table at a distance of 1.4 m from the pulley. \(A\) hangs vertically below the pulley at a height of 0.98 m above the floor (see diagram). \(A , B\), the string and the pulley are all in the same vertical plane. \(B\) is released and \(A\) moves downwards.

1. Find the acceleration of \(A\) and the tension in the string. \(A\) hits the floor and \(B\) continues to move towards the pulley. Find the time taken, from the instant that \(B\) is released, for
2. \(A\) to reach the floor,
3. \(B\) to reach the pulley.

2. \begin{figure}[h] \end{figure} A small ball, \(P\), of mass 0.8 kg , is held at rest on a smooth horizontal table and is attached to one end of a thin rope. The rope passes over a pulley that is fixed at the edge of the table.
The other end of the rope is attached to another small ball, \(Q\), of mass 0.6 kg , that hangs freely below the pulley. Ball \(P\) is released from rest, with the rope taut, with \(P\) at a distance of 1.5 m from the pulley and with \(Q\) at a height of 0.4 m above the horizontal floor, as shown in Figure 1. Ball \(Q\) descends, hits the floor and does not rebound.
The balls are modelled as particles, the rope as a light and inextensible string and the pulley as small and smooth. Using this model,

1. show that the acceleration of \(Q\), as it falls, is \(4.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\)
2. find the time taken by \(P\) to hit the pulley from the instant when \(P\) is released.
3. State one limitation of the model that will affect the accuracy of your answer to part (a).