| Exam Board | Edexcel |
|---|---|
| Module | FM2 (Further Mechanics 2) |
| Year | 2022 |
| 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 centre of mass formulas with standard coordinate geometry. Part (a) requires direct substitution into the formula, while part (b) involves solving a quadratic equation from the distance condition. The multi-step nature and algebraic manipulation elevate it slightly above pure recall, but it remains a routine Further Maths question with no novel insight required. |
| Spec | 6.04b Find centre of mass: using symmetry6.04c Composite bodies: centre of mass |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| Moments about \(y\)-axis | M1 | Moments equation to find \(\bar{x}\) – need all terms and dimensionally correct. Allow with \(m\) cancelled throughout. Allow if they have a common factor of \(g\) |
| \(\left((5+k)m\bar{x} = -3kma + 6ma + 3ma\right) \quad \bar{x} = \dfrac{(9-3k)a}{5+k}\) | A1 | Correct expression for \(\bar{x}\). Any equivalent form. Allow recovery |
| Moments about \(x\)-axis | M1 | Moments equation to find \(\bar{y}\) – need all terms and dimensionally correct. Allow with \(m\) cancelled throughout. Allow if they have a common factor of \(g\) |
| \(\left((5+k)m\bar{y} = 4kma + 4ma - 12ma\right) \quad \bar{y} = \dfrac{(4k-8)a}{5+k}\) | A1 | Correct expression for \(\bar{y}\). Any equivalent form. Allow recovery |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| \(\Rightarrow 9\left[(9-3k)^2 + (4k-8)^2\right] = (5+k)^2\) leading to \(\left(224k^2 - 1072k + 1280 = 0\right)\) | M1 | Use their moments equations to form a quadratic equation in \(k\) only with no square root (need not simplify) |
| \(\Rightarrow k = \dfrac{5}{2}\), or \(k = \dfrac{16}{7}\) | A1 | Obtain both correct values. Accept 2.5 and 2.3 or better (2.2857…) |
## Question 1:
### Part (a):
| Working/Answer | Mark | Guidance |
|---|---|---|
| Moments about $y$-axis | M1 | Moments equation to find $\bar{x}$ – need all terms and dimensionally correct. Allow with $m$ cancelled throughout. Allow if they have a common factor of $g$ |
| $\left((5+k)m\bar{x} = -3kma + 6ma + 3ma\right) \quad \bar{x} = \dfrac{(9-3k)a}{5+k}$ | A1 | Correct expression for $\bar{x}$. Any equivalent form. Allow recovery |
| Moments about $x$-axis | M1 | Moments equation to find $\bar{y}$ – need all terms and dimensionally correct. Allow with $m$ cancelled throughout. Allow if they have a common factor of $g$ |
| $\left((5+k)m\bar{y} = 4kma + 4ma - 12ma\right) \quad \bar{y} = \dfrac{(4k-8)a}{5+k}$ | A1 | Correct expression for $\bar{y}$. Any equivalent form. Allow recovery |
### Part (b):
| Working/Answer | Mark | Guidance |
|---|---|---|
| $\Rightarrow 9\left[(9-3k)^2 + (4k-8)^2\right] = (5+k)^2$ leading to $\left(224k^2 - 1072k + 1280 = 0\right)$ | M1 | Use their moments equations to form a quadratic equation in $k$ only with no square root (need not simplify) |
| $\Rightarrow k = \dfrac{5}{2}$, or $k = \dfrac{16}{7}$ | A1 | Obtain both correct values. Accept 2.5 and 2.3 or better (2.2857…) |
**Total: 6 marks**\begin{enumerate}
\item Three particles of masses $2 m , 3 m$ and $k m$ are placed at the points with coordinates (3a, 2a), (a, -4a) and (-3a, 4a) respectively.
\end{enumerate}
The centre of mass of the three particles lies at the point with coordinates $( \bar { x } , \bar { y } )$.\\
(a) (i) Find $\bar { x }$ in terms of $a$ and $k$\\
(ii) Find $\bar { y }$ in terms of $a$ and $k$
Given that the distance of the centre of mass of the three particles from the point ( 0,0 ) is $\frac { 1 } { 3 } a$\\
(b) find the possible values of $k$
\hfill \mbox{\textit{Edexcel FM2 2022 Q1 [6]}}