# How to Find Resultant Force of Two Forces: Problem and Examples

How to find the resultant force of two forces? In simple words, the addition and subtraction of all the forces present in a system are called resultant forces.

When an isolated system is considered to be in motion there are possibilities of more than two forces to act on the system. So the question is how to find resultant force of two forces in a particular system. The answer may be simple, but one needs to identify the forces present in the system or probably act on one such.

One also needs to be aware of the resultant force and how it works on a system under motion or sometimes even motionless. When the individual vector forces are added up together then the result got is regarded to be the resultant force.

It is said that resultant force is a combination of all the forces present in the system. So what are the other forces that could exist? The primary and default force existing in all systems is the gravitational force.

Generally, the gravitational force is a downward force, and to counter that, there is an upward force most called the normal force. In cases when asked to calculate the resultant force, these two forces wouldn’t suffice.

When an object is at rest, the force acting on it will be the normal force. Also, when the object is in motion, the gravitational force acts on the object due to acceleration. Gravitational and normal forces are assumed to be the same; indeed, this is a misconception since both forces act on the same object.

Now that the basics of considerable force are seen briefly let us see what other forces influence and provide to the motion of any object.

## What is the resultant of two forces?

The resultant of two forces is simply the vector sum of the individual forces in a system.

The term resultant force pertains to the result only if two exact vector quantities are added. There can also be resultant displacement, resultant velocity if two velocities are added, and could also be resultant momentum.

Now that we are dealing with resultant forces let’s use an example to clearly understand how to find resultant force of two forces.

A force vector acts directed towards the east, and the other force vector is also directed towards the east. The magnitude of the vectors is the size of the force per se, having values 100N and 120N, respectively.

Now we take two forces acting in two different directions, one being west and the other east with different magnitudes. Since the direction is changed, the resultant force appears to be smaller than the starting forces.

Hence the direction of the resultant vector force will be directed towards the force having a magnitude smaller than the other one.

Consider two vectors in right angles to each other and how to find resultant force of two forces?

When two forces are said to be perpendicular to each other, we must draw a hypotenuse line to find the system’s resultant force. By doing this, a triangle will be formed.

Using Pythagoras Theorem, the third value can be found, which also happens to be the value for the resultant force.

## How to calculate resultant force with angles?

Now that we know how to find resultant force of two forces using a free-body diagram let us dive into the area where the resultant force with angle is to be calculated.

In the above section, we discussed how to find resultant force of two forces which was basically the magnitude of the resultant force. The angle of the vector force made with the tangent gives the direction of that particular force.

The angle can be determined by the formula, ϴ = tan-1(y/x). Here, the letters x and y denoted the direction of the components, also being the direction of two different forces in the act.

Let’s consider an example using a free-body diagram to understand this in a better way.

If we have a vector force directed towards the west (50) and the other force towards the north (120), as we’ve worked out in the previous example using Pythagoras Theorem, the magnitude of the resultant force can be evaluated, and that is 130N.

Now with the given information about the angle direction could now be determined using the magnitude values. Let 40N be the y component, and 120N be the x component. Using the formula ϴ = tan-1(y/x) and applying the formula accordingly, we get the answer as 67.4⁰.

This angle ϴ=67.4⁰ is termed as the reference angle. Now the relative angle to this particular reference angle should be determined to form the free-body diagram. The relative angle is said to be 247.4⁰.

Hence the above made calculations are the results of direction of the vector force. Also they can change according to different cases when the kind of forces is mentioned.

## How to find the resultant force of three forces?

In this section, we shall work out numerical in order to find the resultant force of three forces.

## Problem:

Three vector forces act in three different directions making angles with their tangents, as shown in the figure below. Now calculate the resultant force magnitude and direction with the given data.

## Solution:

All the forces have their own x and y component. So first, let us calculate the F1 and F2 forces. Determining the x components of F1 and F2, we get the answer as, Fx= -30.84N. Next, determining the y components of the F1 and F2, we get the result as Fy= -0.0794N. Since the component value is almost equal to zero, it isn’t essential.

Now calculating for F,’ we get the calculations to be F’= -30.84Ni-0.794Nj and the third force F3=50N to be in x-direction since I has no y component present. Now F’+f3 = Fr (resultant force). Fr= 19.17, which is the magnitude, and 2.37⁰, which is the direction of the resultant force.

This is how usually the resultant force of three forces has been determined, and this applies to all the other problems with a similar questionnaire.

In order to calculate the total force or the resultant force of the entire system, which has three forces acting upon it, we need to know in which direction the vector force is acting along with the angle value.

## Resultant of two forces

In simple words, the resultant of two forces can be easily found by adding or subtracting the respective individual force that has been acting on the system.

When a system is considered to be under motion, we say that the force is responsible for that particular motion. A free-body diagram is essential to determine the resultant force acting upon the system which is under constant motion.

From the free-body diagram that has been drawn and the values of forces that have been applied, it becomes easier to determine the forces present in the system theoretically.

## Problem 1:

Now let us consider a system having forces acting upon them in two different directions. Say one vector force acts eastwards, and the other vector force acts westwards. The magnitudes of the force are 10N and 30N, respectively. Now find the resultant force acting on the system.

## Solution:

The resultant force is denoted by Fr, so

Fr= -10N+30N

Fr=20N

The direction of the resultant force is said to act in the direction of the more significant force that is the one acting westwards.

## Problem 2:

Now lets us consider an isolated system having two forces acting upon them. The magnitude of the forces is 50N and 30N. Both the forces tend to act in the same direction that is eastwards, so the values will turn out to be positive. Calculate the resultant force of two forces with the given values.

## Solution:

Fr= 50N+30N

Fr= 80N

The direction of the force will be eastwards only since both the forces act eastwards.

## How to find the resultant of two concurrent forces?

How to find resultant force of two forces if they are concurrent? Meaning, how to find resultant of the forces if they lie on the same plane.

We all must be aware of the parallelogram law, which depicts and explains that two or more forces traveling in the same direction will pass through a common point.

## Problem:

Two forces are said to be concurrent, where the forces diverge from a common point. The magnitudes for the given forces are 100N and 70N. Find the resultant force acting on the system.

## Solution:

According to the sign convention, the forces are said to be positive and are supposed to be added in order to find the resultant force.

Fr=F1+F2

Fr= 100N + 70N

Fr= 170N.

In this way, when we are well aware of the sign convention, we can calculate the resultant force.

## How to find the resultant of two perpendicular forces?

When two forces are said to be perpendicular to each other, the resultant forces can be found using the parallelogram law and by determining the angle between them.

When two vector forces are perpendicular to each other and the resultant of those forces can be found using different mathematical methods.

All the x components of the forces can be added, which are parallel to them, and by adding all the y components of the forces that are parallel to them.

The tail-to-tail method is one of the least used methods to find the resultant force of two forces that are right angles to each other.

Keerthana Srikumar

Hi...I am Keerthana Srikumar, currently pursuing Ph.D. in Physics and my area of specialization is nano-science. I completed my Bachelor's and Master's from Stella Maris College and Loyola College respectively. I have a keen interest in exploring my research skills and also have the ability to explain Physics topics in a simpler manner. Apart from academics I love to spend my time in music and reading books. Let's connect through LinkedIn-https://www.linkedin.com/in/keerthana-s-91560920a/