How to Find Resultant Force of Two Forces at an Angle

Introduction

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

R² = F₁²+ F₂²

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

R² = F₁²+ F₂² – 2 F₁ F₂ Cos ( 180⁰ – θ )

Where Forces F₁ and F₂ 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 F₁ and F₂ in the same direction where F₁ = 5N and F₂ = 7N

R = F₁ + F₂ = 5N + 7N = 12N

Forces F₁, and F₂ in the Same Direction While F₃ is in Opposite Direction and F₁ = 5N, F₂ = 10N and F₃ = 3N

R = F₁ + F₂ + F₃ = 5N + (-3N) = 12N

F₃ 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 120⁰ 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

F₁ = 5N

F₂ = 12N

The resultant ( R ) = ?

Angle between the two forces ( θ ) = 120⁰

Now, if we apply the formula

R² = F₁²+ F₂² – 2 F₁ F₂ Cos ( 180⁰ – θ )

Now, substitute the values of F₁, F₂ and θ into the above equation.

We now haveR² = 5² + 12² – 2 x 5 x 12 Cos ( 180⁰ – 120⁰ )

which shows that R² = 25 + 144 – 120 Cos 60⁰

Hence, R² = 169 – ( 120 x 0.5 )

We will now arrive at R² = 169 – 60

After subtraction, we will have R² = 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 120⁰ is Ten ( 10N ) newton.

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