| Exam Board | Edexcel |
|---|---|
| Module | FM2 (Further Mechanics 2) |
| Year | 2023 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Centre of Mass 1 |
| Type | Particles at coordinate positions |
| Difficulty | Moderate -0.5 This is a straightforward application of the centre of mass formula for particles in 2D. Students set up two equations (one for x-coordinate, one for y-coordinate) using the standard formula and solve the resulting linear system. While it involves algebraic manipulation with parameters, it requires only direct recall and routine calculation with no conceptual challenges or problem-solving insight. |
| Spec | 6.04c Composite bodies: centre of mass |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| Moments about \(y\)-axis | M1 | Dimensionally correct equation with all terms. Allow with \(\bar{x}\) in place of \(2a\) |
| \(2a(7+k) = 3 \times 2a + 4 \times a + k \times 2\mu a\) leading to \((14 + 2k = 10 + 2\mu k)\) | A1 | Correct unsimplified equation |
| Moments about \(x\)-axis | M1 | Dimensionally correct equation with all terms. Allow with \(\bar{y}\) in place of \(4a\) |
| \(4a(7+k) = 3 \times 3a + 4 \times 5a + k \times \mu a\) leading to \((28 + 4k = 29 + \mu k)\) | A1 | Correct unsimplified equation |
| These two equations can be combined as a single vector equation | ||
| Obtain \(k = 1\) or \(\mu = 3\) | A1 | One correct value |
| Obtain \(k = 1\) and \(\mu = 3\) | A1 | Both correct values |
| Total: (6) | (6 marks) |
## Question 1:
| Working/Answer | Mark | Guidance |
|---|---|---|
| Moments about $y$-axis | M1 | Dimensionally correct equation with all terms. Allow with $\bar{x}$ in place of $2a$ |
| $2a(7+k) = 3 \times 2a + 4 \times a + k \times 2\mu a$ leading to $(14 + 2k = 10 + 2\mu k)$ | A1 | Correct unsimplified equation |
| Moments about $x$-axis | M1 | Dimensionally correct equation with all terms. Allow with $\bar{y}$ in place of $4a$ |
| $4a(7+k) = 3 \times 3a + 4 \times 5a + k \times \mu a$ leading to $(28 + 4k = 29 + \mu k)$ | A1 | Correct unsimplified equation |
| | | These two equations can be combined as a single vector equation |
| Obtain $k = 1$ or $\mu = 3$ | A1 | One correct value |
| Obtain $k = 1$ and $\mu = 3$ | A1 | Both correct values |
| **Total: (6)** | | **(6 marks)** |\begin{enumerate}
\item Three particles of masses $3 m , 4 m$ and $k m$ are positioned at the points with coordinates ( $2 a , 3 a$ ), ( $a , 5 a$ ) and ( $2 \mu a , \mu a$ ) respectively, where $k$ and $\mu$ are constants.
\end{enumerate}
The centre of mass of the three particles is at the point with coordinates $( 2 a , 4 a )$.\\
Find (i) the value of $k$\\
(ii) the value of $\mu$
\hfill \mbox{\textit{Edexcel FM2 2023 Q1 [6]}}