| Exam Board | Edexcel |
|---|---|
| Module | FM2 (Further Mechanics 2) |
| Year | 2024 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Moments |
| Type | Resultant force on lamina |
| Difficulty | Standard +0.8 This is a multi-part Further Maths mechanics question requiring: (a) finding the center of mass of a composite framework using symmetry and geometric decomposition (involving trigonometry for the equilateral triangle geometry), and (b) applying moments about a pivot with forces in equilibrium. While systematic, it demands careful coordinate geometry, understanding of composite centers of mass, and moment calculations—significantly above standard A-level but routine for FM2. |
| Spec | 6.04c Composite bodies: centre of mass6.04d Integration: for centre of mass of laminas/solids6.04e Rigid body equilibrium: coplanar forces |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Moments about \(ED\) | M1 | Dimensionally correct equation with required terms. Accept use of a parallel axis. Accept equivalent mass ratio e.g. \(4a\) replaced by 1 |
| \(4 \times 4a \times 2a\cos 30° + 2 \times 4a \times 4a\cos 30° = 7 \times 4a \times d\) | A1 | Unsimplified equation with at most one error. Allow distances in terms of \(\sin 60°\) or \(\cos 30°\) or equivalent. N.B. Repeated use of an incorrect distance is only one error. |
| A1 | Correct unsimplified equation. Allow distances in terms of \(\sin 60°\) or \(\cos 30°\) or equivalent | |
| \(32\sqrt{3}a^2 = 28ad \Rightarrow d = \frac{8\sqrt{3}}{7}a\) | A1* | Obtain given answer from correct working including reference to \(d\). |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Moments about \(C\) | M1 | Dimensionally correct equation with required terms and no extras. Equation should be of the form \(\lambda F = (\mu - d)\,W\) |
| \(8a \times F = \left(4a\cos 30° - \frac{8\sqrt{3}}{7}a\right) \times W\) | A1 | Correct unsimplified equation |
| \(F = \frac{3\sqrt{3}}{28}W\) | A1 | \(0.19W\) or better \((0.185576\ldots W)\) |
# Question 2:
## Part 2a:
| Answer/Working | Mark | Guidance |
|---|---|---|
| Moments about $ED$ | M1 | Dimensionally correct equation with required terms. Accept use of a parallel axis. Accept equivalent mass ratio e.g. $4a$ replaced by 1 |
| $4 \times 4a \times 2a\cos 30° + 2 \times 4a \times 4a\cos 30° = 7 \times 4a \times d$ | A1 | Unsimplified equation with at most one error. Allow distances in terms of $\sin 60°$ or $\cos 30°$ or equivalent. N.B. Repeated use of an incorrect distance is only one error. |
| | A1 | Correct unsimplified equation. Allow distances in terms of $\sin 60°$ or $\cos 30°$ or equivalent |
| $32\sqrt{3}a^2 = 28ad \Rightarrow d = \frac{8\sqrt{3}}{7}a$ | A1* | Obtain given answer from correct working including reference to $d$. |
## Part 2b:
| Answer/Working | Mark | Guidance |
|---|---|---|
| Moments about $C$ | M1 | Dimensionally correct equation with required terms and no extras. Equation should be of the form $\lambda F = (\mu - d)\,W$ |
| $8a \times F = \left(4a\cos 30° - \frac{8\sqrt{3}}{7}a\right) \times W$ | A1 | Correct unsimplified equation |
| $F = \frac{3\sqrt{3}}{28}W$ | A1 | $0.19W$ or better $(0.185576\ldots W)$ |
---2.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{c14975b7-6afa-44ce-beab-1cba2e82b249-06_373_847_251_609}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}
A uniform rod of length $28 a$ is cut into seven identical rods each of length $4 a$. These rods are joined together to form the rigid framework $A B C D E A$ shown in Figure 1.
All seven rods lie in the same plane.\\
The distance of the centre of mass of the framework from $E D$ is $d$.
\begin{enumerate}[label=(\alph*)]
\item Show that $d = \frac { 8 \sqrt { 3 } } { 7 } a$
The weight of the framework is $W$.\\
The framework is freely pivoted about a horizontal axis through $C$.\\
The framework is held in equilibrium in a vertical plane, with $A C$ vertical and $A$ below $C$, by a horizontal force that is applied to the framework at $A$.
The force acts in the same vertical plane as the framework and has magnitude $F$.
\item Find $F$ in terms of $W$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel FM2 2024 Q2 [7]}}