One end of a light inextensible string is attached to a particle $$\(A\)$$ of mass $$\(2 m\)$$. The other end of the string is attached to a particle $$\(B\)$$ of mass $$\(3 \mathrm{~m}\)$$. The string passes over a small, smooth, light pulley $$\(P\)$$ which is fixed at the top of a rough inclined plane. The plane is inclined to the horizontal at angle $$\(\alpha\)$$, where $$\(\tan \alpha=\frac{3}{4}\)$$ Particle $$\(A\)$$ is held at rest on the plane with the string taut and $$\(B\)$$ hanging freely below $$\(P\)$$, as shown in Figure 4. The section of the string $$\(A P\)$$ is parallel to a line of greatest slope of the plane. The coefficient of friction between $$\(A\)$$ and the plane is $$\(\frac{1}{2}\)$$ Particle $$\(A\)$$ is released and begins to move up the plane. For the motion before $$\(A\)$$ reaches the pulley, (a) (i) write down an equation of motion for $$\(A\)$$, (ii) write down an equation of motion for $$\(B\)$$, (4) (b) find, in terms of $$\(g\)$$, the acceleration of $$\(A\)$$, (5) (c) find the magnitude of the force exerted on the pulley by the string. (4) (d) State how you have used the information that $$\(P\)$$ is a smooth pulley. (1)

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