A particle $$\(P\)$$ of mass $$\(0.5 \mathrm{~kg}\)$$ is attached to one end of a light elastic string of natural length $$\(2 \mathrm{~m}\)$$ and modulus of elasticity $$\(3 \mathrm{~N}\)$$. The other end of the string is attached to a fixed point $$\(O\)$$ on a rough plane. The plane is inclined at an angle $$\(\theta\)$$ to the horizontal, where $$\(\sin \theta=\frac{2}{7}\)$$ The coefficient of friction between $$\(P\)$$ and the plane is $$\(\frac{\sqrt{5}}{5}\)$$ The particle $$\(P\)$$ is initially at rest at the point $$\(O\)$$, as shown in Figure 8. The particle $$\(P\)$$ then receives an impulse of magnitude $$\(4 \mathrm{Ns}\)$$, directed up a line of greatest slope of the plane. The particle $$\(P\)$$ moves up the plane and comes to rest at the point $$\(A\)$$. (a) Find the extension of the elastic string when $$\(P\)$$ is at $$\(A\)$$. (8) (b) Show that the particle does not remain at rest at $$\(A\)$$. (3)

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