1 A uniform solid rectangular prism has cross-section with width $w \mathrm {~cm}$ and height 24 cm . Another uniform solid prism has cross-section in the shape of an isosceles triangle with width $w \mathrm {~cm}$ and height $h \mathrm {~cm}$. The prisms are both placed with their axes vertical on a rough horizontal plane (see Fig. 1.1, which shows the cross-sections through the centres of mass of both solids).

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{6418c1b7-092a-4747-bc88-1b57815c6ad9-2_520_1123_520_246}
\captionsetup{labelformat=empty}
\caption{Fig. 1.1}
\end{center}
\end{figure}

The plane is slowly tilted and both solids remain in equilibrium until the angle of inclination of the plane reaches $\alpha$, when both solids topple simultaneously.
\begin{enumerate}[label=(\alph*)]
\item Determine the value of $h$.

It is given that $w = 12$.
\item Determine the value of $\alpha$.

Both prisms are made from the same material and are of uniform density. The triangular prism is now placed on top of the rectangular prism to form a composite body C such that the base of the triangular prism coincides with the top of the rectangular prism. A cross-section of C is shown in Fig. 1.2.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{6418c1b7-092a-4747-bc88-1b57815c6ad9-2_777_439_1784_258}
\captionsetup{labelformat=empty}
\caption{Fig. 1.2}
\end{center}
\end{figure}
\item Determine the height of the centre of mass of C from its base.
\end{enumerate}

\hfill \mbox{\textit{OCR MEI Further Mechanics Minor 2020 Q1 [6]}}