Container or open-topped solid

4. \begin{figure}[h] \end{figure} A container is formed by removing a right circular solid cone of height \(4 l\) from a uniform solid right circular cylinder of height \(6 l\). The centre \(O\) of the plane face of the cone coincides with the centre of a plane face of the cylinder and the axis of the cone coincides with the axis of the cylinder, as shown in Figure 3. The cylinder has radius \(2 l\) and the base of the cone has radius \(l\).

1. Find the distance of the centre of mass of the container from \(O\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{049ea68c-d15f-41f8-860e-0816d36a2748-07_460_588_1254_676} \captionsetup{labelformat=empty} \caption{Figure 4}
   \end{figure} The container is placed on a plane which is inclined at an angle \(\theta ^ { \circ }\) to the horizontal. The open face is uppermost, as shown in Figure 4. The plane is sufficiently rough to prevent the container from sliding. The container is on the point of toppling.
2. Find the value of \(\theta\).

6 In this question you may use the fact that the volume of a sphere of radius \(r\) is \(\frac { 4 } { 3 } \pi r ^ { 3 }\).
Fig. 6.1 shows a container in the shape of an open-topped cylinder. The cylinder has height 18 cm and radius 4 cm . The curved surface and the base can be modelled as uniform laminae with the same mass per unit area. The container rests on a horizontal surface. \begin{figure}[h] \end{figure}

1. Show that the centre of mass of the container lies 8.1 cm above its base. The mass of the container is 400 grams. Water is poured into the container to reach a height of \(h \mathrm {~cm}\) above the base. The centre of mass of the combined container and water lies \(y \mathrm {~cm}\) above the base. Water has a density of 1 gram per \(\mathrm { cm } ^ { 3 }\).
2. In this question you must show detailed reasoning. By formulating an expression for \(y\) in terms of \(h\), determine the value of \(h\) for which \(y\) is lowest. More water is now poured into the container. A sphere of radius 3 cm is placed into the container, where it sinks to the bottom. The surface of the water is now 4.5 cm from the top of the container, as shown in Fig. 6.2. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Fig. 6.2} \includegraphics[alt={},max width=\textwidth]{cad8805d-59f6-4ed2-81f4-9e8c749461f5-6_432_355_2001_255}
   \end{figure}
3. Show that the centre of mass of the water in the container lies 7.5 cm above the base of the container. The sphere has a density of 4 grams per \(\mathrm { cm } ^ { 3 }\).
   The centre of mass of the combined container, water and sphere lies \(z \mathrm {~cm}\) above the base.
4. Determine the value of \(z\). \section*{END OF QUESTION PAPER}