A light elastic string, of natural length \(l\) and modulus of elasticity \(4 m g\), is attached at one end to a fixed point and has a particle \(P\) of mass \(m\) attached to the other end. When \(P\) is hanging in equilibrium under gravity it is given a velocity \(\sqrt { } ( g l )\) vertically downwards. At time \(t\) the downward displacement of \(P\) from its equilibrium position is \(x\). Show that, while the string is taut, $$\ddot { x } = - \frac { 4 g } { l } x .$$ Find the speed of \(P\) when the length of the string is \(l\). Show that the time taken for \(P\) to move from the lowest point to the highest point of its motion is $$\left( \frac { \pi } { 3 } + \frac { \sqrt { } 3 } { 2 } \right) \sqrt { } \left( \frac { l } { g } \right)$$