Edexcel M3 — Question 6

Exam BoardEdexcel
ModuleM3 (Mechanics 3)
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Mark schemeDownload PDF ↗
TopicCentre of Mass 2
TypeComposite solid with standard shapes - calculation only
DifficultyStandard +0.3 This is a standard M3 centre of mass question involving a solid of revolution and composite body. Part (a) uses routine integration formulas for volume and centre of mass of a solid of revolution (which students typically memorize or have on formula sheets). Part (b) combines this with the standard hemisphere result using the composite body formula. While it requires careful calculation and multiple steps, it follows a completely standard template with no novel problem-solving required—slightly easier than average for M3 level.
Spec4.08d Volumes of revolution: about x and y axes6.04b Find centre of mass: using symmetry6.04c Composite bodies: centre of mass6.04d Integration: for centre of mass of laminas/solids
6. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 4} \includegraphics[alt={},max width=\textwidth]{ab85ec29-b1fc-45a9-9343-09feb33ab6c5-010_515_1015_319_477}
\end{figure} The shaded region \(R\) is bounded by the curve with equation \(y = \frac { 1 } { 2 x ^ { 2 } }\), the \(x\)-axis and the lines \(x = 1\) and \(x = 2\), as shown in Figure 4. The unit of length on each axis is 1 m . A uniform solid \(S\) has the shape made by rotating \(R\) through \(360 ^ { \circ }\) about the \(x\)-axis.
  1. Show that the centre of mass of \(S\) is \(\frac { 2 } { 7 } \mathrm {~m}\) from its larger plane face. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 5} \includegraphics[alt={},max width=\textwidth]{ab85ec29-b1fc-45a9-9343-09feb33ab6c5-010_616_431_1420_778}
    \end{figure} A sporting trophy \(T\) is a uniform solid hemisphere \(H\) joined to the solid \(S\). The hemisphere has radius \(\frac { 1 } { 2 } \mathrm {~m}\) and its plane face coincides with the larger plane face of \(S\), as shown in Figure 5. Both \(H\) and \(S\) are made of the same material.
  2. Find the distance of the centre of mass of \(T\) from its plane face.
Question 6(c): (by reference circle)
AnswerMarks Guidance
Answer/WorkingMark Guidance
Finding the required angle in radiansM1
Using the period \(\left(\frac{2\pi}{\omega}\right)\) and their angle to find the required timeM1
Correct timeA1 
## Question 6(c): (by reference circle)

| Answer/Working | Mark | Guidance |
|---|---|---|
| Finding the required angle in radians | M1 | |
| Using the period $\left(\frac{2\pi}{\omega}\right)$ and their angle to find the required time | M1 | |
| Correct time | A1 | |
6.

\begin{figure}[h]
\begin{center}
\captionsetup{labelformat=empty}
\caption{Figure 4}
  \includegraphics[alt={},max width=\textwidth]{ab85ec29-b1fc-45a9-9343-09feb33ab6c5-010_515_1015_319_477}
\end{center}
\end{figure}

The shaded region $R$ is bounded by the curve with equation $y = \frac { 1 } { 2 x ^ { 2 } }$, the $x$-axis and the lines $x = 1$ and $x = 2$, as shown in Figure 4. The unit of length on each axis is 1 m . A uniform solid $S$ has the shape made by rotating $R$ through $360 ^ { \circ }$ about the $x$-axis.
\begin{enumerate}[label=(\alph*)]
\item Show that the centre of mass of $S$ is $\frac { 2 } { 7 } \mathrm {~m}$ from its larger plane face.

\begin{figure}[h]
\begin{center}
\captionsetup{labelformat=empty}
\caption{Figure 5}
  \includegraphics[alt={},max width=\textwidth]{ab85ec29-b1fc-45a9-9343-09feb33ab6c5-010_616_431_1420_778}
\end{center}
\end{figure}

A sporting trophy $T$ is a uniform solid hemisphere $H$ joined to the solid $S$. The hemisphere has radius $\frac { 1 } { 2 } \mathrm {~m}$ and its plane face coincides with the larger plane face of $S$, as shown in Figure 5. Both $H$ and $S$ are made of the same material.
\item Find the distance of the centre of mass of $T$ from its plane face.
\end{enumerate}

\hfill \mbox{\textit{Edexcel M3  Q6}}
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