2
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5addd79d-d502-455c-936f-27005483164e-2_655_334_440_861}
\captionsetup{labelformat=empty}
\caption{Fig. 1}
\end{figure}
A child's toy is a uniform solid consisting of a hemisphere of radius \(r \mathrm {~cm}\) joined to a cone of base radius \(r \mathrm {~cm}\). The curved surface of the cone makes an angle \(\alpha\) with its base. The two shapes are joined at the plane faces with their circumferences coinciding (see Fig. 1). The distance of the centre of mass of the toy above the common circular plane face is \(x \mathrm {~cm}\).
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[The volume of a sphere is \(\frac { 4 } { 3 } \pi r ^ { 3 }\) and the volume of a cone is \(\frac { 1 } { 3 } \pi r ^ { 2 } h\).]
- Show that \(x = \frac { r \left( \tan ^ { 2 } \alpha - 3 \right) } { 8 + 4 \tan \alpha }\).
The toy is placed on a horizontal surface with the hemisphere in contact with the surface. The toy is released from rest from the position in which the common plane circular face is vertical (see Fig. 2).
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5addd79d-d502-455c-936f-27005483164e-2_193_670_1827_699}
\captionsetup{labelformat=empty}
\caption{Fig. 2}
\end{figure} - Find the set of values of \(\alpha\) such that the toy moves to the upright position.