7. A particle \(P\) of mass 0.5 kg is attached to one end of a light elastic string. The string has natural length \(l\) metres and modulus of elasticity 29.4 N . The other end of the string is attached to a fixed point \(A\). The particle hangs freely in equilibrium at the point \(B\), where \(B\) is vertically below \(A\) and \(A B = 1.4 \mathrm {~m}\).

1. Show that \(l = 1.2\) The point \(C\) is vertically below \(A\) and \(A C = 1.8 \mathrm {~m}\). The particle is pulled down to \(C\) and released from rest.
2. Show that, while the string is taut, \(P\) moves with simple harmonic motion.
3. Calculate the speed of \(P\) at the instant when the string first becomes slack. The particle first comes to instantaneous rest at the point \(D\).
4. Find the time taken by \(P\) to return directly from \(D\) to \(C\).