A particle $$\(P\)$$ of mass $$\(m\)$$ is attached to one end of a light inextensible string. Another particle $$\(Q\)$$, also of mass $$\(m\)$$, is attached to the other end of the string. The string passes over a small smooth pulley which is fixed at the edge of a rough horizontal table. Particle $$\(Q\)$$ is held at rest on the table and particle $$\(P\)$$ hangs vertically below the pulley with the string taut, as shown in Figure 4. The pulley, $$\(P\)$$ and $$\(Q\)$$ all lie in the same vertical plane. The coefficient of friction between $$\(Q\)$$ and the table is $$\(\mu\)$$, where $$\(\mu< 1\)$$ Particle $$\(Q\)$$ is released from rest. The tension in the string before $$\(Q\)$$ hits the pulley is $$\(k m g\)$$, where $$\(k\)$$ is a constant. (a) Find $$\(k\)$$ in terms of $$\(\mu\)$$. (7) Given that $$\(Q\)$$ is initially a distance $$\(d\)$$ from the pulley, (b) find, in terms of $$\(d, g\)$$ and $$\(\mu\)$$, the time taken by $$\(Q\)$$, after release, to reach the pulley. (4) (c) Describe what would happen if $$\(\mu \geqslant 1\)$$, giving a reason for your answer. (2)

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