A particle $$\(P\)$$ of mass $$\(m\)$$ is attached to one end of a light inextensible string of length $$\(a\)$$. The other end of the string is attached to a fixed point $$\(O\)$$. The particle is held at the point $$\(A\)$$, where $$\(O A=a\)$$ and $$\(O A\)$$ is horizontal, as shown in Figure 4. The particle is projected vertically downwards with speed $$\(\sqrt{\frac{9 a g}{5}}\)$$ When the string makes an angle $$\(\theta\)$$ with the downward vertical through $$\(O\)$$ and the string is still taut, the tension in the string is $$\(S\)$$. (a) Show that $$\(S=\frac{3}{5} m g(5 \cos \theta+3)\)$$ (6) At the instant when the string becomes slack, the speed of $$\(P\)$$ is $$\(v\)$$ (b) Show that $$\(v=\sqrt{\frac{3 a g}{5}}\)$$ (3) (c) Find the maximum height of $$\(P\)$$ above the horizontal level of $$\(O\)$$ (4)

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