An object consists of two hollow spheres which touch each other, together with a thin uniform \(\operatorname{rod} A B\). The rod passes through small holes in the surfaces of the spheres. The rod is fixed to the spheres so that it passes through the centre of the smaller sphere. The end \(B\) of the rod is at the centre of the larger sphere. The larger sphere has radius \(2 a\) and mass \(M\), the smaller sphere has radius \(a\) and mass \(k M\), and the rod has length \(7 a\) and mass \(5 M\). A fixed horizontal axis \(L\) passes through \(A\) and is perpendicular to \(A B\) (see diagram).
(i) Find the moment of inertia of the object, consisting of the rod and two spheres, about \(L\).

The object is pivoted at \(A\) so that it can rotate freely about \(L\). The object is released from rest with the rod making an angle of \(60^{\circ}\) to the downward vertical. The greatest angular speed attained by the object in the subsequent motion is \(\frac{9}{20} \sqrt{ }\left(\frac{g}{a}\right)\).
(ii) Find the value of \(k\).