| Exam Board | Edexcel |
|---|---|
| Module | M3 (Mechanics 3) |
| Year | 2007 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Centre of Mass 1 |
| Type | Conical or hemispherical shell composite |
| Difficulty | Standard +0.3 This is a standard M3 centre of mass question requiring systematic application of the formula for composite bodies. Part (a) involves combining a circular base and cylindrical shell using standard results, while part (b) adds a uniform cylinder of liquid. The question is slightly easier than average as it follows a predictable template with clearly defined components and standard geometric centres, requiring methodical calculation rather than problem-solving insight. |
| Spec | 6.04c Composite bodies: centre of mass6.04d Integration: for centre of mass of laminas/solids |
| Answer | Marks | Guidance |
|---|---|---|
| Base | Cylinder | Container |
| Mass ratios | \(\pi Rh^2\) | \(2\pi h^2\) |
| \(\bar{y}\) | 0 | \(\frac{h}{2}\) |
| \(3\pi h^2 \times \bar{y} = 2\pi h^2 \times \frac{h}{2}\) | M1 A1 | |
| Leading to \(\bar{y} = \frac{1}{3}h\) | A1 (cso) | 5 marks |
| Answer | Marks | Guidance |
|---|---|---|
| Liquid | Container | Total |
| Mass ratios | \(M\) | \(M\) |
| \(\bar{y}\) | \(\frac{h}{2}\) | \(\frac{h}{3}\) |
| \(2M \times \bar{y} = M \times \frac{h}{2} + M \times \frac{h}{3}\) | M1 A1 | |
| \(\bar{y} = \frac{5}{12}h\) | A1 | 5 marks [10] |
**(a)**
| | Base | Cylinder | Container |
|---|---|---|---|
| Mass ratios | $\pi Rh^2$ | $2\pi h^2$ | $3\pi h^2$ | Ratio of 1:2:3 | B1 |
| $\bar{y}$ | 0 | $\frac{h}{2}$ | $\bar{y}$ | | B1 |
$3\pi h^2 \times \bar{y} = 2\pi h^2 \times \frac{h}{2}$ | M1 A1 |
Leading to $\bar{y} = \frac{1}{3}h$ | A1 (cso) | 5 marks
**(b)**
| | Liquid | Container | Total |
|---|---|---|---|
| Mass ratios | $M$ | $M$ | $2M$ | Ratio of 1:1:2 | B1 |
| $\bar{y}$ | $\frac{h}{2}$ | $\frac{h}{3}$ | $\bar{y}$ | | B1 |
$2M \times \bar{y} = M \times \frac{h}{2} + M \times \frac{h}{3}$ | M1 A1 |
$\bar{y} = \frac{5}{12}h$ | A1 | 5 marks [10]
---An open container $C$ is modelled as a thin uniform hollow cylinder of radius $h$ and height $h$ with a base but no lid. The centre of the base is $O$.
\begin{enumerate}[label=(\alph*)]
\item Show that the distance of the centre of mass of $C$ from $O$ is $\frac{1}{4}h$. [5]
\end{enumerate}
The container is filled with uniform liquid. Given that the mass of the container is $M$ and the mass of the liquid is $M$,
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item find the distance of the centre of mass of the filled container from $O$. [5]
\end{enumerate}
\hfill \mbox{\textit{Edexcel M3 2007 Q2 [10]}}