A particle of mass $$\(m\)$$ is held at rest at a point $$\(A\)$$ on a rough plane. The plane is inclined at an angle $$\(\alpha\)$$ to the horizontal, where $$\(\tan \alpha=\frac{5}{12}\)$$ The coefficient of friction between the particle and the plane is $$\(\frac{1}{5}\)$$ The points $$\(A\)$$ and $$\(B\)$$ lie on a line of greatest slope of the plane, with $$\(B\)$$ above $$\(A\)$$, and $$\(A B=d\)$$, as shown in Figure 1. The particle is pushed up the line of greatest slope from $$\(A\)$$ to $$\(B\)$$. (a) Show that the work done against friction as the particle moves from $$\(A\)$$ to $$\(B\)$$ is $$\(\frac{12}{65} m g d\)$$ (3) The particle is then held at rest at $$\(B\)$$ and released. (b) Use the work-energy principle to find, in terms of $$\(g\)$$ and $$\(d\)$$, the speed of the particle at the instant it reaches $$\(A\)$$. (4)

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