# How to calculate the magnitude and direction of resultant force?

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Puneet Jain
A virtual SEO expert, content writer, and digital marketer.

The resultant force is one of the types of force. Force is simply defined as the pull and push of things. The resultant force is a force used to detect whether the object moves or remains constant. Consider the following scenario. A tug of war is taking place between two groups of individuals, with each group pulling on the rope as hard as they can.

Finally, who will emerge successfully? Another question that may arise is whether one group will be able to move the other. We need to know how to find the resultant force to appropriately answer these issues. The resultant force is the single force that has the same effect on an item as the original force. A rope is an object in this example. In this article, we will cover all the basics and methods of the resultant force.

## What is the Resultant Force?

When numerous forces act on an object, the resultant force is the force that creates the same acceleration as all of the other forces creates combinedly. When two or more forces act on a body, the resultant force or net force is the sum of all the forces that generate the resulting effect. Because force is a vector, we must calculate the resultant by taking the vector sum of all the forces.

The resultant force is useful because it allows us to think about several forces as if they were a single force. This means that determining the influence of several forces on an object is as simple as determining the effect of a single force.

As we are familiar with the equation of force as,

F = ma

The resultant force equation is,

R = ma

Here R is the vector sum of vector forces such as F1, F2, F3,

This formula is simply written as,

FR = F1 + F2 + F3

The net force acting on an object under several forces is known as the resultant force. The resultant force has both magnitude and direction because force is a vector quantity.

The following diagram is a vector representation of the resultant force:

## The formula of the Resultant Force

We can calculate the magnitude and direction of the forces by using resultant force. Both direction and magnitude have different equations.

The equation for magnitude in resultant force.

Magnitude of the resultant force = √ ((Σfx)2 + (Σfy)2 – cos(θ))

First of all, we have to calculate the angle to find the magnitude of the resultant force.

The equation for direction in resultant force.

Direction of the resultant force = arctan (Σfx / Σfy)

## How to find the resultant force magnitude and direction?

First of all, we calculate the resultant force.

Example 1
find the resultant force when 3 forces such as 90N, 30N, and 45N are acting on an object concurrently.

Solution

Step 1: Name the forces.

F1 = 90N

F2 = 30N

F3 = 45N

Step 2: Write the general formula.

FR = F1 + F2 + F3

Step 3: Put the values.

FR = 90N + 30N + 45N

FR = 165N

Now we have taken some examples to calculate the magnitude and direction of the resultant force. The direction and magnitude of the force can be easily be calculated by using an online tool like resultant force calculator.

Example 2
A bus is being pulled by 20N at 0 degrees and by 7N at 90 degrees in the forward direction calculate the direction and magnitude of the resultant force.

Solution

Step 1: Identify the values.

F1 = 20N

F2 = 7N

Angle of F1 = 0 degree

Angle of F2 = 90 degrees

Step 2: Calculate the components of forces.

X-complement of F1 = 20cos (0) = 20(1) = 20

Y-component of F2 = 20sin (0) = 20(0) = 0

X-complement of F2 = 7cos (90) = 7(0) = 0

Y-component of F2 = 7sin (90) = 7(1) = 7

Step 3: Find the sum of x components and y components.

Σfx = 20 + 0 = 20

Σfy = 0 + 7 = 7

Step 4: Find the direction of resultant force.

Direction of the resultant force = arctan (Σfx / Σfy)

θ = arctan (Σfx / Σfy)

= arctan (20/7)

= arctan (2.86)

θ = 71 degree

Step 5: Now calculate the magnitude of the resultant force.

Magnitude of the resultant force = √ ((Σfx)2 + (Σfy)2 – cos(θ))

= √ ((20)2 + (7)2 – cos (71))

= √ ((400+ 49 – 0.3256)

= √ ((449 – 0.3256)

= √ (448.6744)

= 21.1819

Example 3
A bus is being pulled by 19N at 90 degrees and by 17N at 0 degrees in the forward direction calculate the direction and magnitude of the resultant force.

Solution

Step 1: Identify the values.

F1 = 19N

F2 = 17N

Angle of F1 = 90 degree

Angle of F2 = 0 degrees

Step 2: Calculate the components of forces.

X-complement of F1 = 19cos (90) = 19(0) = 0

Y-component of F2 = 19sin (90) = 19(1) = 19

X-complement of F2 = 17cos (0) = 17(1) = 17

Y-component of F2 = 17sin (0) = 7(0) = 0

Step 3: Find the sum of x components and y components.

Σfx = 0 + 19 = 19

Σfy = 17 + 0 = 17

Step 4: Find the direction of resultant force.

Direction of the resultant force = arctan (Σfx / Σfy)

θ = arctan (Σfx / Σfy)

= arctan (19/17)

= arctan (1.1176)

θ = 48.1786 degree

Step 5: Now calculate the magnitude of the resultant force.

Magnitude of the resultant force = √ ((Σfx)2 + (Σfy)2 – cos(θ))

= √ ((20)2 + (7)2 – cos (48.1786))

= √ ((361+ 289 + 0.4934)

= √ ((650 + 0.4934)

= √ (650.4934)

= 25.5048

## Summary

The force which is used to find the combined relation is said to be the resultant force. When numerous forces act on an object, the resultant force is the force that creates the same acceleration as all of the other forces creates combinedly. This force also tells the magnitude and direction of the forces. We can easily solve resultant force-related problems, but just a little effort is required.