One end of a light elastic string, of natural length \(12 a\) and modulus of elasticity \(k m g\), is attached to a fixed point \(O\). The other end of the string is attached to a particle of mass \(m\). The particle moves with constant speed \(\frac{3}{2} \sqrt{3 a g}\) in a horizontal circle with centre at a distance \(12 a\) below \(O\). The string is inclined at an angle \(\theta\) to the downward vertical through \(O\).
(a) Find, in terms of \(a\), the extension of the string.
(b) Find the value of \(k\).