An object is formed by removing a cylinder of radius \(\dfrac23a\) and height \(kh\), where \(0\lt k\lt1\), from a uniform solid cylinder of radius \(a\) and height \(h\). The vertical axes of symmetry of the two cylinders coincide, and the upper faces are in the same plane.

The points \(A\) and \(B\) are opposite ends of a diameter of the upper face of the object.

(a) Find, in terms of \(h\) and \(k\), the distance of the centre of mass of the object from \(AB\).

When the object is suspended from \(A\), the angle between \(AB\) and the vertical is \(\theta\), where \(\tan\theta=\dfrac32\).

(b) Given that \(h=\dfrac83a\), find the possible values of \(k\).