OCR H240/03 2018 March — Question 7 3 marks

Exam BoardOCR
ModuleH240/03 (Pure Mathematics and Mechanics)
Year2018
SessionMarch
Marks3
TopicForces, equilibrium and resultants
TypeForces in vector form: equilibrium (find unknowns)
DifficultyModerate -0.8 This is a straightforward equilibrium problem requiring students to set the sum of forces equal to zero and solve two simultaneous linear equations for a and b. It's a standard textbook exercise with routine algebraic manipulation, making it easier than average for A-level mechanics.
Spec3.03b Newton's first law: equilibrium3.03m Equilibrium: sum of resolved forces = 0
Three forces \(\mathbf{F}_1\), \(\mathbf{F}_2\) and \(\mathbf{F}_3\) acting on a particle are given by $$\mathbf{F}_1 = (3\mathbf{i} - 2a\mathbf{j})\text{N}, \quad \mathbf{F}_2 = (2b\mathbf{i} + 3a\mathbf{j})\text{N} \quad \text{and} \quad \mathbf{F}_3 = (-2\mathbf{i} + b\mathbf{j})\text{N}.$$ The particle is in equilibrium under the action of these three forces. Find the value of \(a\) and the value of \(b\). [3]
AnswerMarks Guidance
\(\begin{pmatrix}\frac{3}{-2a}\end{pmatrix} + \begin{pmatrix}\frac{2b}{3a}\end{pmatrix} + \begin{pmatrix}\frac{-2}{b}\end{pmatrix} = 0\)M1 Equilibrium \(\Rightarrow \sum \mathbf{F} = 0\); Either correct equation in \(a\) and/or \(b\) can imply this mark
\(3 + 2b - 2 = 0 \Rightarrow b = -0.5\)A1
\(-2a + 3a + b = 0 \Rightarrow a = 0.5\)A1 
$\begin{pmatrix}\frac{3}{-2a}\end{pmatrix} + \begin{pmatrix}\frac{2b}{3a}\end{pmatrix} + \begin{pmatrix}\frac{-2}{b}\end{pmatrix} = 0$ | M1 | Equilibrium $\Rightarrow \sum \mathbf{F} = 0$; Either correct equation in $a$ and/or $b$ can imply this mark

$3 + 2b - 2 = 0 \Rightarrow b = -0.5$ | A1 | 

$-2a + 3a + b = 0 \Rightarrow a = 0.5$ | A1 |
Three forces $\mathbf{F}_1$, $\mathbf{F}_2$ and $\mathbf{F}_3$ acting on a particle are given by
$$\mathbf{F}_1 = (3\mathbf{i} - 2a\mathbf{j})\text{N}, \quad \mathbf{F}_2 = (2b\mathbf{i} + 3a\mathbf{j})\text{N} \quad \text{and} \quad \mathbf{F}_3 = (-2\mathbf{i} + b\mathbf{j})\text{N}.$$

The particle is in equilibrium under the action of these three forces.

Find the value of $a$ and the value of $b$. [3]

\hfill \mbox{\textit{OCR H240/03 2018 Q7 [3]}}
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