7. One end of a light elastic string, of natural length \(3 l \mathrm {~m}\), is attached to a fixed point \(O\). A particle of mass \(m \mathrm {~kg}\) is attached to the other end of the string. When the particle hangs freely in equilibrium, the string is extended by a length of \(l \mathrm {~m}\). The particle is then pulled down through a further distance \(2 l \mathrm {~m}\) and released from rest.

1. Prove that as long as the string is taut, the particle performs simple harmonic motion about its equilibrium position.
2. Show that the time between the release of the particle and the instant when the string becomes slack is \(\frac { 2 } { 3 } \pi \sqrt { \frac { l } { g } } \mathrm {~s}\).
3. Find the greatest height reached by the particle above its point of release.
4. Show that the time \(T\) s taken to reach this greatest height from the moment of release is given by \(T = \left( \frac { 2 \pi } { 3 } + \sqrt { 3 } \right) \sqrt { \frac { l } { g } }\).
   (4 marks)