The diagram shows a uniform thin rod \(A B\) of length \(3 a\) and mass \(8 m\). The end \(A\) is rigidly attached to the surface of a sphere with centre \(O\) and radius \(a\). The rod is perpendicular to the surface of the sphere. The sphere consists of two parts: an inner uniform solid sphere of mass \(\frac { 3 } { 2 } m\) and radius \(a\) surrounded by a thin uniform spherical shell of mass \(m\) and also of radius \(a\). The horizontal axis \(l\) is perpendicular to the rod and passes through the point \(C\) on the rod where \(A C = a\).

1. Show that the moment of inertia of the object, consisting of rod, shell and inner sphere, about the axis \(l\) is \(\frac { 289 } { 15 } m a ^ { 2 }\).
   The object is free to rotate about the axis \(l\). The object is held so that \(C A\) makes an angle \(\alpha\) with the downward vertical and is released from rest.
2. Given that \(\cos \alpha = \frac { 1 } { 6 }\), find the greatest speed achieved by the centre of the sphere in the subsequent motion.