The points \(A\) and \(B\) are at the same horizontal level and are \(4a\) apart. The ends of a light elastic string, of natural length \(4a\) and modulus of elasticity \(\lambda\), are attached to \(A\) and \(B\). A particle \(P\) of mass \(m\) is attached to the midpoint of the string.

The system is in equilibrium with \(P\) at a distance \(\dfrac32a\) below \(M\), the midpoint of \(AB\).

(a) Find \(\lambda\) in terms of \(m\) and \(g\).

The particle \(P\) is then pulled down vertically and released from rest at a distance \(\dfrac83a\) below \(M\).

(b) Find, in terms of \(a\) and \(g\), the speed of \(P\) as it passes through \(M\) in the subsequent motion.