One end of a light elastic string, of natural length \(a\) and modulus of elasticity \(k m g\), is attached to a fixed point \(A\). The other end of the string is attached to a particle \(P\) of mass \(4 m\). The particle \(P\) hangs in equilibrium a distance \(x\) vertically below \(A\).
(a) Show that \(k=\frac{4 a}{x-a}\).

An additional particle, of mass \(2 m\), is now attached to \(P\) and the combined particle is released from rest at the original equilibrium position of \(P\). When the combined particle has descended a distance \(\frac{1}{3} a\), its speed is \(\frac{1}{3} \sqrt{g a}\).
(b) Find \(x\) in terms of \(a\).