5. A light elastic string of natural length \(l\) has one end attached to a fixed point \(A\). A particle \(P\) of mass \(m\) is attached tot he other end of the string and hangs in equilibrium at the point \(O\), where \(A O = \frac { 5 } { 4 } l\).

1. Find the modulus of the elasticity of the string. The particle \(P\) is then pulled down and released from rest. At time \(t\) the length of the string is \(\frac { 5 l } { 4 } + x\).
2. Prove that, while the string is taut, $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = - \frac { 4 g x } { l }$$ When \(P\) is released, \(A P = \frac { 7 } { 4 } l\). The point \(B\) is a distance \(l\) vertically below \(A\).
3. Find the speed of \(P\) at \(B\).
4. Describe briefly the motion of \(P\) after it has passed through \(B\) for the first time until it next passes through \(O\).