Edexcel M5 2018 June — Question 1 5 marks

Exam BoardEdexcel
ModuleM5 (Mechanics 5)
Year2018
SessionJune
Marks5
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicWork done and energy
DifficultyStandard +0.3 This is a straightforward M5 mechanics question requiring students to find the direction vector of the wire, calculate the displacement vector, and apply the work formula W = F·s. While it involves vectors and the dot product, it's a direct application of standard techniques with no conceptual subtlety—slightly easier than average for A-level but appropriate for M5 material.
Spec1.10f Distance between points: using position vectors6.02b Calculate work: constant force, resolved component
  1. A small bead is threaded on a smooth straight horizontal wire. The wire is modelled as a line with vector equation \(\mathbf { r } = ( 2 + \lambda ) \mathbf { i } + ( 2 \lambda - 1 ) \mathbf { j }\), where the unit of length is the metre. The bead is moved a distance of \(\sqrt { 80 } \mathrm {~m}\) along the wire by a force \(\mathbf { F } = ( 4 \mathbf { i } - 3 \mathbf { j } ) \mathrm { N }\). Find the magnitude of the work done by \(\mathbf { F }\).
    (5)
Question 1:
AnswerMarks Guidance
[answer/working][mark] [guidance notes]
```
Question 1:
Alternative 1:
AnswerMarks Guidance
Answer/WorkingMarks Guidance
Direction of line is \((\mathbf{i} + 2\mathbf{j})\)B1 For \((\mathbf{i} + 2\mathbf{j})\) or multiple seen
\(\mathbf{d} = \frac{\sqrt{80}}{\sqrt{1^2+2^2}}(\mathbf{i}+2\mathbf{j}) = (4\mathbf{i}+8\mathbf{j})\)M1 A1 M1 for attempt to find displacement vector d; A1 for \((4\mathbf{i}+8\mathbf{j})\)
W.D.\(= \(4\mathbf{i}-3\mathbf{j}).(4\mathbf{i}+8\mathbf{j})\ = 8\) J
Alternative 2:
AnswerMarks Guidance
Answer/WorkingMarks Guidance
Direction of line is \((\mathbf{i} + 2\mathbf{j})\)B1
\(\cos\theta = \frac{(4\mathbf{i}-3\mathbf{j}).(\mathbf{i}+2\mathbf{j})}{5\sqrt{5}} = \frac{-2}{5\sqrt{5}}\)M1 A1 M1 for attempt to find angle between d (multiple of \((\mathbf{i}+2\mathbf{j})\)) and \((4\mathbf{i}-3\mathbf{j})\); A1 for \(\cos\theta = \frac{-2}{5\sqrt{5}}\) oe
W.D.\(= \(4\mathbf{i}-3\mathbf{j})\ \cos\theta\
Alternative 3:
AnswerMarks Guidance
Answer/WorkingMarks Guidance
Direction of line is \((\mathbf{i} + 2\mathbf{j})\)B1
\(\hat{\mathbf{d}} = \frac{(\mathbf{i}+2\mathbf{j})}{\sqrt{5}}\)M1 A1 M1 for attempt to find unit vector; A1 oe
W.D.\(= \(4\mathbf{i}-3\mathbf{j}).\hat{\mathbf{d}}\ \times \sqrt{80} = 8\) J
Total: [5]Answer must be positive. Ignore units. Column vectors allowed throughout. 
Question 1:
[answer/working] | [mark] | [guidance notes]
```

## Question 1:

**Alternative 1:**

| Answer/Working | Marks | Guidance |
|---|---|---|
| Direction of line is $(\mathbf{i} + 2\mathbf{j})$ | B1 | For $(\mathbf{i} + 2\mathbf{j})$ or multiple seen |
| $\mathbf{d} = \frac{\sqrt{80}}{\sqrt{1^2+2^2}}(\mathbf{i}+2\mathbf{j}) = (4\mathbf{i}+8\mathbf{j})$ | M1 A1 | M1 for attempt to find displacement vector **d**; A1 for $(4\mathbf{i}+8\mathbf{j})$ |
| W.D.$= \|(4\mathbf{i}-3\mathbf{j}).(4\mathbf{i}+8\mathbf{j})\| = 8$ J | M1 A1 | M1 for WD $= (4\mathbf{i}-3\mathbf{j}).\mathbf{d}$, **d** must be multiple of $(\mathbf{i}+2\mathbf{j})$; A1 for 8 (J) |

**Alternative 2:**

| Answer/Working | Marks | Guidance |
|---|---|---|
| Direction of line is $(\mathbf{i} + 2\mathbf{j})$ | B1 | |
| $\cos\theta = \frac{(4\mathbf{i}-3\mathbf{j}).(\mathbf{i}+2\mathbf{j})}{5\sqrt{5}} = \frac{-2}{5\sqrt{5}}$ | M1 A1 | M1 for attempt to find angle between **d** (multiple of $(\mathbf{i}+2\mathbf{j})$) and $(4\mathbf{i}-3\mathbf{j})$; A1 for $\cos\theta = \frac{-2}{5\sqrt{5}}$ oe |
| W.D.$= \|(4\mathbf{i}-3\mathbf{j})\|\cos\theta\| \times \sqrt{80} = 8$ J | M1 A1 | A1 for 8 (J) |

**Alternative 3:**

| Answer/Working | Marks | Guidance |
|---|---|---|
| Direction of line is $(\mathbf{i} + 2\mathbf{j})$ | B1 | |
| $\hat{\mathbf{d}} = \frac{(\mathbf{i}+2\mathbf{j})}{\sqrt{5}}$ | M1 A1 | M1 for attempt to find unit vector; A1 oe |
| W.D.$= \|(4\mathbf{i}-3\mathbf{j}).\hat{\mathbf{d}}\| \times \sqrt{80} = 8$ J | M1 A1 | $\hat{\mathbf{d}}$ must be multiple of $(\mathbf{i}+2\mathbf{j})$; A1 for 8 (J) |

**Total: [5]** | Answer must be positive. Ignore units. Column vectors allowed throughout.

---
\begin{enumerate}
  \item A small bead is threaded on a smooth straight horizontal wire. The wire is modelled as a line with vector equation $\mathbf { r } = ( 2 + \lambda ) \mathbf { i } + ( 2 \lambda - 1 ) \mathbf { j }$, where the unit of length is the metre. The bead is moved a distance of $\sqrt { 80 } \mathrm {~m}$ along the wire by a force $\mathbf { F } = ( 4 \mathbf { i } - 3 \mathbf { j } ) \mathrm { N }$. Find the magnitude of the work done by $\mathbf { F }$.\\
(5)
\end{enumerate}

\hfill \mbox{\textit{Edexcel M5 2018 Q1 [5]}}
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