| Exam Board | Edexcel |
|---|---|
| Module | M2 (Mechanics 2) |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Centre of Mass 1 |
| Type | Lamina with removed circle/semicircle |
| Difficulty | Standard +0.8 This is a composite body centre of mass problem requiring subtraction of a semicircle from a rectangle, followed by applying equilibrium conditions for a suspended lamina. It involves multiple steps: finding individual centres of mass, using the composite body formula with negative mass, then applying trigonometry for the hanging equilibrium. While the techniques are standard M2 content, the combination of geometric reasoning, careful coordinate work, and the two-part structure makes this moderately challenging, above average difficulty. |
| Spec | 6.04c Composite bodies: centre of mass6.04e Rigid body equilibrium: coplanar forces |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(360(15) = 12 \cdot 5\pi(25) + (360 - 12 \cdot 5\pi)\bar{x}\) | M1 A1 A1 | |
| \(\bar{x} = 13.8\) | M1 A1 A1 A1 | |
| \(360(6) = 12 \cdot 5\pi(20/3\pi) + (360 - 12 \cdot 5\pi)\bar{y}\) | ||
| \(\bar{y} = 6.47\) | M1 M1 A1 A1 | |
| (b) \(\tan\alpha = 13.78 \div (12 - 6.475) = 2.494\) | M1 A1 M1 A1 | |
| \(\alpha = 68.2°\) | M1 A1 M1 A1 | Total: 11 marks |
**(a)** $360(15) = 12 \cdot 5\pi(25) + (360 - 12 \cdot 5\pi)\bar{x}$ | M1 A1 A1 |
$\bar{x} = 13.8$ | M1 A1 A1 A1 |
$360(6) = 12 \cdot 5\pi(20/3\pi) + (360 - 12 \cdot 5\pi)\bar{y}$ | |
$\bar{y} = 6.47$ | M1 M1 A1 A1 |
**(b)** $\tan\alpha = 13.78 \div (12 - 6.475) = 2.494$ | M1 A1 M1 A1 |
$\alpha = 68.2°$ | M1 A1 M1 A1 | **Total: 11 marks**A rectangular piece of cardboard $ABCD$, measuring $30$ cm by $12$ cm, has a semicircle of radius $5$ cm removed from it as shown.
\includegraphics{figure_6}
\begin{enumerate}[label=(\alph*)]
\item Calculate the distances of the centre of mass of the remaining piece of cardboard from $AB$ and from $BC$. [7 marks]
\end{enumerate}
The remaining cardboard is suspended from $A$ and hangs in equilibrium.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Find the angle made by $AB$ with the vertical. [4 marks]
\end{enumerate}
\hfill \mbox{\textit{Edexcel M2 Q6 [11]}}