A particle \(P\) of mass \(m\) is attached to one end of a light elastic string of natural length \(a\) and modulus of elasticity \(2mg\). A particle \(Q\) of mass \(km\) is attached to the other end.

Particle \(P\) lies on a smooth horizontal table. The string has part of its length in contact with the table and then passes through a small smooth hole \(H\) in the table.

Particle \(P\) moves in a horizontal circle on the surface of the table with constant speed \(\sqrt{\dfrac12ga}\). Particle \(Q\) hangs in equilibrium vertically below the hole with \(HQ=\dfrac14a\).

(a) Find, in terms of \(a\), the extension in the string.

(b) Find \(k\).