Challenging +1.2 This is a standard M3/Further Mechanics question requiring solid integration techniques for centre of mass calculations (Pappus's theorem for solids of revolution, standard lamina formulas) and straightforward moment equilibrium. While technically demanding with multiple integration steps, it follows well-established procedures without requiring novel insight or particularly complex problem-solving.
A uniform solid of revolution \(S\) is formed by rotating the region enclosed between the \(x\)-axis and the curve \(y = x \sqrt { 4 - x }\) for \(0 \leqslant x \leqslant 4\) through \(2 \pi\) radians about the \(x\)-axis, as shown in Fig. 4.1. O is the origin and the end A of the solid is at the point \(( 4,0 )\).
\begin{figure}[h]
Find the \(x\)-coordinate of the centre of mass of the solid \(S\).
The solid \(S\) has weight \(W\), and it is freely hinged to a fixed point at O . A horizontal force, of magnitude \(W\) acting in the vertical plane containing OA , is applied at the point A , as shown in Fig. 4.2. \(S\) is in equilibrium.
\begin{figure}[h]
Find the angle that OA makes with the vertical. [0pt]
[Question 4(b) is printed overleaf]
Fig. 4.3 shows the region bounded by the \(x\)-axis, the \(y\)-axis, the line \(y = 8\) and the curve \(y = ( x - 2 ) ^ { 3 }\) for \(0 \leqslant y \leqslant 8\).
\begin{figure}[h]
4
\begin{enumerate}[label=(\alph*)]
\item A uniform solid of revolution $S$ is formed by rotating the region enclosed between the $x$-axis and the curve $y = x \sqrt { 4 - x }$ for $0 \leqslant x \leqslant 4$ through $2 \pi$ radians about the $x$-axis, as shown in Fig. 4.1. O is the origin and the end A of the solid is at the point $( 4,0 )$.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{db60e7d9-bec5-47f7-9e27-38b7d112851e-5_520_625_408_703}
\captionsetup{labelformat=empty}
\caption{Fig. 4.1}
\end{center}
\end{figure}
\begin{enumerate}[label=(\roman*)]
\item Find the $x$-coordinate of the centre of mass of the solid $S$.
The solid $S$ has weight $W$, and it is freely hinged to a fixed point at O . A horizontal force, of magnitude $W$ acting in the vertical plane containing OA , is applied at the point A , as shown in Fig. 4.2. $S$ is in equilibrium.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{db60e7d9-bec5-47f7-9e27-38b7d112851e-5_346_512_1361_781}
\captionsetup{labelformat=empty}
\caption{Fig. 4.2}
\end{center}
\end{figure}
\item Find the angle that OA makes with the vertical.\\[0pt]
[Question 4(b) is printed overleaf]
\end{enumerate}\item Fig. 4.3 shows the region bounded by the $x$-axis, the $y$-axis, the line $y = 8$ and the curve $y = ( x - 2 ) ^ { 3 }$ for $0 \leqslant y \leqslant 8$.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{db60e7d9-bec5-47f7-9e27-38b7d112851e-6_631_695_370_683}
\captionsetup{labelformat=empty}
\caption{Fig. 4.3}
\end{center}
\end{figure}
Find the coordinates of the centre of mass of a uniform lamina occupying this region.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI M3 2013 Q4 [18]}}