M3 June 2014 Q3

3.

Figure 2 shows a container in the shape of a uniform right circular conical shell of height \(6r\). The radius of the open circular face is \(r\). The container is suspended by two vertical strings attached to two points at opposite ends of a diameter of the open circular face. It hangs with the open circular face uppermost and axis vertical. Molten wax is poured into the container. The wax solidifies and adheres to the container, forming a uniform solid right circular cone. The depth of the wax in the container is \(2r\). The container together with the wax forms a solid \(S\).

The mass of the container when empty is \(m\) and the mass of the wax in the container is \(3m\).

One of the strings is now removed and the solid \(S\) hangs freely in equilibrium suspended by the remaining vertical string.