| Exam Board | OCR MEI |
|---|---|
| Module | M1 (Mechanics 1) |
| Year | 2013 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors 3D & Lines |
| Type | Forces as vectors |
| Difficulty | Moderate -0.8 This is a straightforward mechanics question requiring basic vector operations: calculating three magnitudes using Pythagoras (part i), then applying F=ma with vector addition including weight (part ii). All steps are routine procedures with no problem-solving insight needed, making it easier than average A-level material. |
| Spec | 1.10b Vectors in 3D: i,j,k notation1.10c Magnitude and direction: of vectors3.03d Newton's second law: 2D vectors |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \( | \mathbf{p} | = \sqrt{(-1)^2+(-1)^2+5^2} = \sqrt{27}\) |
| \( | \mathbf{q} | = \sqrt{1+16+4} = \sqrt{21}\), \( |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Weight \(= 0.4 \times 10 = 4\) N, i.e. \(\begin{pmatrix}0\\0\\-4\end{pmatrix}\) | B1 | |
| Resultant \(= \mathbf{p}+\mathbf{q}+\mathbf{r}+\mathbf{w} = \begin{pmatrix}-1+(-1)+2+0\\-1+(-4)+5+0\\5+2+0-4\end{pmatrix} = \begin{pmatrix}0\\0\\3\end{pmatrix}\) | M1 A1 | |
| \(a = \frac{3}{0.4} = 7.5\) m s\(^{-2}\) vertically upwards | A1 |
# Question 3:
## Part (i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $|\mathbf{p}| = \sqrt{(-1)^2+(-1)^2+5^2} = \sqrt{27}$ | M1 | Correct method for at least one magnitude |
| $|\mathbf{q}| = \sqrt{1+16+4} = \sqrt{21}$, $|\mathbf{r}| = \sqrt{4+25+0} = \sqrt{29}$ | A1 | $\mathbf{r}$ has greatest magnitude |
## Part (ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Weight $= 0.4 \times 10 = 4$ N, i.e. $\begin{pmatrix}0\\0\\-4\end{pmatrix}$ | B1 | |
| Resultant $= \mathbf{p}+\mathbf{q}+\mathbf{r}+\mathbf{w} = \begin{pmatrix}-1+(-1)+2+0\\-1+(-4)+5+0\\5+2+0-4\end{pmatrix} = \begin{pmatrix}0\\0\\3\end{pmatrix}$ | M1 A1 | |
| $a = \frac{3}{0.4} = 7.5$ m s$^{-2}$ vertically upwards | A1 | |
---3 In this question take $\boldsymbol { g } = \mathbf { 1 0 }$.\\
The directions of the unit vectors $\left( \begin{array} { l } 1 \\ 0 \\ 0 \end{array} \right) , \left( \begin{array} { l } 0 \\ 1 \\ 0 \end{array} \right)$ and $\left( \begin{array} { l } 0 \\ 0 \\ 1 \end{array} \right)$ are east, north and vertically upwards.\\
Forces $\mathbf { p } , \mathbf { q }$ and $\mathbf { r }$ are given by $\mathbf { p } = \left( \begin{array} { r } - 1 \\ - 1 \\ 5 \end{array} \right) \mathrm { N } , \mathbf { q } = \left( \begin{array} { r } - 1 \\ - 4 \\ 2 \end{array} \right) \mathrm { N }$ and $\mathbf { r } = \left( \begin{array} { l } 2 \\ 5 \\ 0 \end{array} \right) \mathrm { N }$.\\
(i) Find which of $\mathbf { p } , \mathbf { q }$ and $\mathbf { r }$ has the greatest magnitude.\\
(ii) A particle has mass 0.4 kg . The forces acting on it are $\mathbf { p } , \mathbf { q } , \mathbf { r }$ and its weight.
Find the magnitude of the particle's acceleration and describe the direction of this acceleration.
\hfill \mbox{\textit{OCR MEI M1 2013 Q3 [6]}}