7. A uniform solid sphere, of radius \(a\), is divided into two sections by a plane at a distance \(\frac { a } { 2 }\) from the centre and parallel to a diameter.
- Show that the centre of gravity of the smaller cap from its plane face is \(\frac { 7 a } { 40 }\). This smaller cap is now placed on an inclined plane whose angle of inclination to the horizontal is \(\theta\). The plane is rough enough to prevent slipping and the cap rests with its curved surface in contact with the plane.
- If the maximum value of \(\theta\) for which this is possible without the cap turning over is \(30 ^ { \circ }\), find the corresponding maximum inclination of the axis of symmetry of the cap to the vertical.
(6 marks)