2 A particle \(P\) of mass \(m\) is attached to one end of a light elastic string of natural length \(a\) and modulus of elasticity \(2 m g\). A particle \(Q\) of mass \(k m\) is attached to the other end of the string. 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 \(\sqrt { \frac { 1 } { 2 } g a }\). Particle \(Q\) hangs in equilibrium vertically below the hole with \(H Q = \frac { 1 } { 4 } a\).

1. Find, in terms of \(a\), the extension in the string.
2. Find the value of \(k\).