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.

​

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 $$\underline{6N}$$.