7. A container consists of two sections made from the same material : a hollow portion formed by removing a cone (shaded in the figure) from a solid cylinder of radius \(r\) and height \(h\), and a solid hemisphere of radius \(r\). The vertex of the removed cone coincides with the centre \(O\) of the horizontal plane face of the hemisphere. \(C D\) is a
\includegraphics[max width=\textwidth, alt={}, center]{3321c06a-29c3-430a-99a8-ec3a245abf10-2_332_350_1581_1604}
diameter of this plane face.

1. Show that the distance of the centre of mass of the container from the plane face of the hemisphere is \(\left| \frac { 3 } { 8 } ( h - r ) \right|\) Explain why the modulus sign is necessary.
2. Find the ratio \(h : r\) in each of the following cases :
   1. When the container is suspended from the point \(C\), the angle made by \(C D\) with the vertical is equal to the angle which \(C D\) would make with the vertical if the hemisphere alone were suspended from \(C\).
   2. The container is able to stand without toppling in any position when it is placed with the surface of the hemispherical part in contact with a smooth horizontal table.
      (3 marks)