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Forces & Equilibrium (AQA A Level Maths: Mechanics): Flashcards

Exam code: 7357

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FrontForces in 2D & Vector Notation
A force is $(- 2 i - 3 j) N$ . Find its magnitude.

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Cards in this collection (6)

A force is $(- 2 i - 3 j) N$ . Find its magnitude.
The magnitude is $\sqrt{13} N$ , which is $3.61 N$ to 3 significant figures.
The two components are perpendicular, so Pythagoras' theorem gives:
$| F | = \sqrt{(- 2)^{2} + (- 3)^{2}} = \sqrt{13} N$
The signs disappear when the components are squared, so a magnitude is never negative.
How is the direction of a force in two dimensions usually given, and what is it measured from?
The direction is given as an angle, usually in degrees, measured anticlockwise from the positive horizontal direction.
So a direction of $216.9 °$ describes a force pointing down and to the left.
True or False?
A force given as $(3 i + 8 j) N$ tells you its magnitude and its direction straight away.
False.
Component form separates a force into its horizontal and vertical parts, so neither the magnitude nor the direction is shown directly.
The magnitude comes from Pythagoras' theorem, and the direction from a sketch and trigonometry.
A force is $(- 8 i - 6 j) N$ . Why does putting $\tan^{- 1} (\frac{- 6}{- 8})$ into a calculator not give its direction?
A calculator returns an angle between $- 90 °$ and $90 °$ , so it cannot distinguish between two opposite directions.
Sketch the components instead, find the acute angle from the sizes of the components, then adjust it. Here $\tan^{- 1} (\frac{6}{8}) = 36.9 °$ , and the force points down and to the left, so the direction is $180 ° + 36.9 ° = 216.9 °$ , measured anticlockwise from the positive horizontal direction.
What does the resultant force on a particle in equilibrium look like when the forces are written in component form?
The resultant force is the zero vector, written $0$ N.
In component form that is $0 i + 0 j$ , or as a column vector $(\begin{matrix} 0 \\ 0 \end{matrix})$ . Both components have to be zero separately, which is the vector form of the two equilibrium equations.
Two forces act on a particle and their resultant is $(- 2 i - 3 j) N$ . What third force would bring the particle into equilibrium?
The third force is $(2 i + 3 j) N$ .
All three forces must add to the zero vector, so the third force has both components of the existing resultant reversed.

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