A particle $$\(P\)$$ of mass $$\(m\)$$ is attached to one end of a light inextensible string of length $$\(a\)$$. The other end of the string is attached to a fixed point $$\(O\)$$. The particle $$\(P\)$$ is held at rest vertically below $$\(O\)$$, with the string taut, as shown in Figure 4. The particle is then projected horizontally with speed $$\(u\)$$, where $$\(u> \sqrt{2 a g}\)$$ Air resistance is modelled as being negligible. At the instant when the string makes an angle $$\(\theta\)$$ with the upward vertical through $$\(O\)$$, the speed of $$\(P\)$$ is $$\(v\)$$ and the string goes slack. (a) Show that $$\(3 v^{2}=u^{2}-2 a g\)$$ (7) From the instant when the string goes slack to the instant when $$\(O P\)$$ is next horizontal, $$\(P\)$$ moves as a projectile. The time from the instant when the string goes slack to the instant when $$\(O P\)$$ is next horizontal is $$\(T\)$$ Given that $$\(\theta=30^{\circ}\)$$ (b) show that $$\(T=\frac{2 v}{g}\)$$ (4) (c) Hence, show that the string goes taut again when it is next horizontal. (2)

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