| Exam Board | Edexcel |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2008 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Centre of Mass 1 |
| Type | Particle attached to lamina - find mass/position |
| Difficulty | Standard +0.3 This is a standard M2 centre of mass question with three straightforward parts: (a) uses the centre of mass formula with one unknown to show k=7 (routine algebra), (b) combines particles and a uniform lamina using standard formulas, and (c) applies the equilibrium condition that the centre of mass hangs vertically below the suspension point. All techniques are textbook exercises requiring no novel insight, making it slightly easier than average. |
| Spec | 6.04b Find centre of mass: using symmetry6.04c Composite bodies: centre of mass6.05a Angular velocity: definitions |
| Answer | Marks | Guidance |
|---|---|---|
| \(M(Oy)\): \((8 + k)m \times 6.4 = 5m \times 8 + km \times 8\) | M1 A1 | |
| \(1.6k = 11.2 \Rightarrow k = 7\) ★ | cso DM1 A1 | [4] |
| Answer | Marks | Guidance |
|---|---|---|
| \(M(Oy)\): \(27m\bar{x} = 12m \times 4 + 5m \times 8 + 7m \times 8\) | M1 A1 | |
| \(\bar{x} = \frac{16}{3}\) | 5.3 or better | |
| A1 | ||
| \(M(Ox)\): \(27m\bar{y} = 12m \times 2.5 + 8m \times 5\) | M1 A1 | [6] |
| \(\bar{y} = \frac{70}{27}\) | 2.6 or better | |
| A1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(\tan\theta = \frac{\bar{y}}{\bar{x}} = \frac{35}{72}\) | M1 A1 ft | |
| \(\theta \approx 26°\) | awrt 25.9° A1 | [3] [13] |
**Part (a):**
$M(Oy)$: $(8 + k)m \times 6.4 = 5m \times 8 + km \times 8$ | M1 A1 |
$1.6k = 11.2 \Rightarrow k = 7$ ★ | cso DM1 A1 | [4]
**Part (b):**
$M(Oy)$: $27m\bar{x} = 12m \times 4 + 5m \times 8 + 7m \times 8$ | M1 A1 |
$\bar{x} = \frac{16}{3}$ | | 5.3 or better
| A1 |
$M(Ox)$: $27m\bar{y} = 12m \times 2.5 + 8m \times 5$ | M1 A1 | [6]
$\bar{y} = \frac{70}{27}$ | | 2.6 or better
| A1 |
**Part (c):**
$\tan\theta = \frac{\bar{y}}{\bar{x}} = \frac{35}{72}$ | M1 A1 ft |
$\theta \approx 26°$ | awrt 25.9° A1 | [3] [13]6.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{a2738ce4-4dc5-4cd1-ac3d-0c3fcf21ea71-09_600_968_292_486}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{center}
\end{figure}
Figure 3 shows a rectangular lamina $O A B C$. The coordinates of $O , A , B$ and $C$ are ( 0,0 ), $( 8,0 ) , ( 8,5 )$ and $( 0,5 )$ respectively. Particles of mass $k m , 5 m$ and $3 m$ are attached to the lamina at $A , B$ and $C$ respectively.
The $x$-coordinate of the centre of mass of the three particles without the lamina is 6.4.
\begin{enumerate}[label=(\alph*)]
\item Show that $k = 7$.
The lamina $O A B C$ is uniform and has mass $12 m$.
\item Find the coordinates of the centre of mass of the combined system consisting of the three particles and the lamina.
The combined system is freely suspended from $O$ and hangs at rest.
\item Find the angle between $O C$ and the horizontal.
\end{enumerate}
\hfill \mbox{\textit{Edexcel M2 2008 Q6 [13]}}