| Exam Board | Edexcel |
|---|---|
| Module | M3 (Mechanics 3) |
| Marks | 16 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Centre of Mass 1 |
| Type | Solid with removed cylinder or hemisphere from solid |
| Difficulty | Standard +0.3 This is a standard M3 centre of mass question involving composite bodies with removed shapes. Part (a) is a 'show that' requiring systematic application of the centre of mass formula for composite bodies (volumes and distances). Part (b) extends this to include liquid, requiring similar methodology. Both parts follow well-established procedures taught in M3 with no novel insight required. Part (c) is a straightforward conceptual explanation. Slightly easier than average due to the structured 'show that' format and routine application of standard techniques. |
| Spec | 6.04c Composite bodies: centre of mass6.04d Integration: for centre of mass of laminas/solids |
| Answer | Marks | Guidance |
|---|---|---|
| portion | mass | \(y\) |
| large cylinder | \(\rho\pi r(\frac{3}{4}r)^2 \times \frac{1}{2}h = \frac{9}{32}\rho\pi r^2h\) | \(\frac{3}{4}h\) |
| small cylinder | \(\rho\pi r^2h\) | \(\frac{1}{2}h\) |
| flask | \(\frac{5}{3}\rho\pi r^2h\) | \(\bar{y}\) |
| \(\rho = \text{mass per unit volume} \quad y \text{ coords. taken vert. from open face}\) | M2 A3 | |
| \(\frac{5}{3}\rho\pi r^2h \times \bar{y} = \frac{3}{5}\rho\pi r^2h^2 \therefore \bar{y} = \frac{3}{5}h + \frac{3}{5} = \frac{9}{10}h\) | M1 A1 |
| Answer | Marks | Guidance |
|---|---|---|
| portion | mass | \(y\) |
| flask | \(\frac{5}{3}\rho\pi r^2h\) | \(\frac{9}{10}h\) |
| liquid | \(k\rho\pi r^2h\) | \(\frac{1}{2}h\) |
| full flask | \((\frac{5}{3}+k)\rho\pi r^2h\) | \(\frac{15}{22}h\) |
| \(\rho = \text{mass per unit volume} \quad y \text{ coords. taken vert. from open face}\) | M2 A2 | |
| \((\frac{5}{3}+k)\rho\pi r^2h \times \frac{15}{22}h = (\frac{3}{2}+\frac{1}{2}k)\rho\pi r^2h^2\) | M1 | |
| \(\therefore 15(\frac{5}{3}+k) = 22(\frac{3}{2}+\frac{1}{2}k), 25+15k = 33+11k, k=2\) | M1 A1 | |
| (c) e.g. when full the centre of mass is in top half so easily knocked over | B2 | (16) |
**(a)**
| portion | mass | $y$ | $my$ |
|---------|------|-----|------|
| large cylinder | $\rho\pi r(\frac{3}{4}r)^2 \times \frac{1}{2}h = \frac{9}{32}\rho\pi r^2h$ | $\frac{3}{4}h$ | $\frac{27}{128}\rho\pi r^2h^2$ |
| small cylinder | $\rho\pi r^2h$ | $\frac{1}{2}h$ | $\frac{1}{2}\rho\pi r^2h^2$ |
| flask | $\frac{5}{3}\rho\pi r^2h$ | $\bar{y}$ | $\frac{3}{5}\rho\pi r^2h^2$ |
$\rho = \text{mass per unit volume} \quad y \text{ coords. taken vert. from open face}$ | M2 A3 |
$\frac{5}{3}\rho\pi r^2h \times \bar{y} = \frac{3}{5}\rho\pi r^2h^2 \therefore \bar{y} = \frac{3}{5}h + \frac{3}{5} = \frac{9}{10}h$ | M1 A1 |
**(b)**
| portion | mass | $y$ | $my$ |
|---------|------|-----|------|
| flask | $\frac{5}{3}\rho\pi r^2h$ | $\frac{9}{10}h$ | $\frac{3}{2}\rho\pi r^2h^2$ |
| liquid | $k\rho\pi r^2h$ | $\frac{1}{2}h$ | $\frac{1}{2}k\rho\pi r^2h^2$ |
| full flask | $(\frac{5}{3}+k)\rho\pi r^2h$ | $\frac{15}{22}h$ | $(\frac{3}{2}+\frac{1}{2}k)\rho\pi r^2h^2$ |
$\rho = \text{mass per unit volume} \quad y \text{ coords. taken vert. from open face}$ | M2 A2 |
$(\frac{5}{3}+k)\rho\pi r^2h \times \frac{15}{22}h = (\frac{3}{2}+\frac{1}{2}k)\rho\pi r^2h^2$ | M1 |
$\therefore 15(\frac{5}{3}+k) = 22(\frac{3}{2}+\frac{1}{2}k), 25+15k = 33+11k, k=2$ | M1 A1 |
**(c)** e.g. when full the centre of mass is in top half so easily knocked over | B2 | **(16)**
---5.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{e0668f31-4b72-4dfd-9cf7-470acef0bfdb-4_693_554_196_717}
\captionsetup{labelformat=empty}
\caption{Fig. 3}
\end{center}
\end{figure}
A flask is modelled as a uniform solid formed by removing a cylinder of radius $r$ and height $h$ from a cylinder of radius $\frac { 4 } { 3 } r$ and height $\frac { 3 } { 2 } h$ with the same axis of symmetry and a common plane as shown in Figure 3.
\begin{enumerate}[label=(\alph*)]
\item Show that the centre of mass of the flask is a distance of $\frac { 9 } { 10 } h$ from the open end of the flask.
The flask is made from a material of density $\rho$ and is filled to the level of the open plane face with a liquid of density $k \rho$. Given that the centre of mass of the flask and liquid together is a distance of $\frac { 15 } { 22 } h$ from the open end of the flask,
\item find the value of $k$.
\item Explain why it may be advantageous to make the base of the flask from a more dense material.\\
(2 marks)
\end{enumerate}
\hfill \mbox{\textit{Edexcel M3 Q5 [16]}}