An object is composed of a hemispherical shell of radius \(2 a\) attached to a closed hollow circular cylinder of height \(h\) and base radius \(a\). The hemispherical shell and the hollow cylinder are made of the same uniform material. The axes of symmetry of the shell and the cylinder coincide. \(A B\) is a diameter of the lower end of the cylinder (see diagram).
(a) Find, in terms of \(a\) and \(h\), an expression for the distance of the centre of mass of the object from \(A B\).

The object is placed on a rough plane which is inclined to the horizontal at an angle \(\theta\), where \(\tan \theta=\frac{2}{3}\). The object is in equilibrium with \(A B\) in contact with the plane and lying along a line of greatest slope of the plane.
(b) Find the set of possible values of \(h\), in terms of \(a\).