## Introduction

The formula below will help you understand how to find resultant force of two forces

R^{2} = F_{1}^{2}+ F_{2}^{2}

When two forces are acting at an angle, we will use the formula

R^{2} = F_{1}^{2}+ F_{2}^{2} – 2 F_{1} F_{2} Cos ( 180^{0} – θ )

*Where**Forces F _{1} and F_{2} are acting at each other at an angle*

## Explanation

Therefore, when you have two forces say 5N and 7N heading towards north ( the same ) direction and you are required to find the resultant of the two forces ( total force ). What you need to do is to add the two forces together to get the resultant force.

*Forces F _{1} and F_{2} in the same direction where F_{1} = 5N and F_{2} = 7N*

R = F_{1} + F_{2} = 5N + 7N = 12N

*Forces F _{1}, and F_{2} in the Same Direction While F_{3}*

*is in Opposite Direction*

*and F*_{1}= 5N,**and F***F*_{2}= 10N_{3}= 3NR = F_{1} + F_{2} + F_{3} = 5N + (-3N) = 12N

F_{3} is in opposite direction and this is the reason we have (-3N) which shows that any force that is in negative direction is to be subtracted.

## Problem

An object is acted upon by two forces of 5N and 12N. Calculate the resultant of the forces acting at an angle of 120^{0} to each other.

[UTME 2019]

Solution

To calculate the resultant of the two forces, we need to extract our data from the above question

Data:

From the question above

We have two forces and they are as follows

F_{1} = 5N

F_{2} = 12N

The resultant ( R ) = ?

Angle between the two forces ( θ ) = 120^{0}

Now, if we apply the formula

R^{2} = F_{1}^{2}+ F_{2}^{2} – 2 F_{1} F_{2} Cos ( 180^{0} – θ )

Now, substitute the values of F_{1}, F_{2} and θ into the above equation.

We now haveR^{2 }= 5^{2 }+ 12^{2 }– 2 x 5 x 12 Cos ( 180^{0 }– 120^{0 })

which shows that R^{2 }= 25 + 144 – 120 Cos 60^{0}

Hence, R^{2 }= 169 – ( 120 x 0.5 )

We will now arrive at R^{2 }= 169 – 60

After subtraction, we will have R^{2 }= 109

By taking square root of each side we will have R = √109

And our final answer is R = 10.44N

Therefore, the resultant of the two forces acting between an angle of 120^{0} is Ten ( 10N ) newton.

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