6. A light elastic string, of natural length \(l\) and modulus of elasticity \(4 m g\), has one end attached to a fixed point \(A\). The other end is attached to a particle \(P\) of mass \(m\). The particle hangs freely at rest in equilibrium at the point \(E\). The distance of \(E\) below \(A\) is \(( l + e )\).

1. Find \(e\) in terms of \(l\). At time \(t = 0\), the particle is projected vertically downwards from \(E\) with speed \(\sqrt { g l }\).
2. Prove that, while the string is taut, \(P\) moves with simple harmonic motion.
3. Find the amplitude of the simple harmonic motion.
4. Find the time at which the string first goes slack.