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Forces and friction

Resolving forces

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Summary

Resolving forces

In a nutshell

When a force is applied at an angle, it can be resolved to find the component which acts in the direction of motion.

Components of the force

When a force acts at an angle, the vertical and horizontal components can be identified using trigonometry, and used to find resultant forces. Moreover, when given horizontal and vertical components of a force, trigonometry can be used to identify the direction of the resultant force. Pythagoras' theorem can be used to work out the magnitude of the resultant force.

Example 1

A force of $aN$ is applied to a body at an angle $\theta$ to the horizontal. Calculate the horizontal and vertical components of the force applied.

Maths; Forces and friction; KS5 Year 13; Resolving forces

The force creates a right-angled triangle. Using trigonometry, you know that:

$\cos(\theta) = \dfrac{adjacent}{hypotenuse}$ 

Substitute the values given:

$\cos(\theta) = \dfrac{x}{a}$ 

Therefore:

$x=a\cos(\theta)$ 

For the vertical component:

$\sin(\theta)=\dfrac{opposite}{hypotenuse}$ 

Substitute the values given:

$\sin(\theta)=\dfrac{y}{a}$ 

Therefore:

$y=a\sin(\theta)$ 

The horizontal component of the force is $\underline{a\cos(\theta)}$ and the vertical component is $\underline{a\sin(\theta)}$ .

Example 2

A force of $12N$ is applied to an object at an angle of $30\degree$ . Calculate the horizontal and vertical components of the force.

Vertical component:

$\begin{aligned}b&=12\sin(30)\\ b&=6N\end{aligned}$ 

Horizontal component:

$\begin{aligned}a&=12\cos(30)\\a&=10.4N \ (1\ d.p.)\end{aligned}$ 

The horizontal component is $\underline{10.4N}$ and the vertical component is $\underline{6N}$ .

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Unit 1

Resultant forces - Higher

Unit 2