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# Resultant Force Formula

Consider an example. Two groups of people are doing a tug of war, and each group is pulling on the rope as hard as they can. Then finally which team will win? Another question that may arise that will one group be able to make the other group move? In order to answer these questions effectively, we need to know how to find the resultant force. The resultant force is the single force which will produce the same effect on an object. In this case, a rope is an object. So, we have to apply the concept of the resultant force. In this article, we will see this concept. Also, we will discuss the resultant force formula with relevant examples, Let us begin the concept!

## Resultant Force Formula

### Concept of Resultant Force

The Resultant force is defined as the total effective force acting on a body along with their directions. Also, when the object is at rest position or traveling with the same velocity, then the resultant force has to be zero.

As it can be seen here that all the forces are acting towards the same direction, hence the resultant force should be the same for all the forces.

Also, if one force is acting towards the perpendicular direction to others, then the resultant force can be determined by using the Pythagorean Theorem.

### The Formula for Resultant Force:

The Resultant force formula is given by,

$$F_R = F_1 + F_2 + F_3$$

Where,

$$F_1, F_2, F_3$$ are the forces acting in the same direction on a body.

If $$F_2$$ is perpendicular to $$F_1$$, then the resultant force formula is given as,

$$F_R$$ = $$\sqrt{(F_1^2 + F_2^2)}$$

Notice that this is not merely the sum of the magnitudes of the forces. But the sum of the forces taken as vectors, which is much suitable. This is because vectors have both the magnitude and the direction which we need to consider while finding the sum.

Thus, according to the above equation, if an object is having no force, then the resultant force will also be zero. On the other hand, if an object is subject to only one force, then the resultant force will be equal to that force. Therefore we have to perform the vector sum of the forces, instead of the regular sum.

## Solved Examples

Q.1: Let us have three forces as 50 N, 60 N and 20 N. These are acting on a body simultaneously. The direction of 20 N is opposite to the direction of the other two. Determine the resultant force.

Solution:

As given in the problem:

$$F_1$$ = 50 N,

$$F_2$$ = 60 N

$$F_3$$ = – 20 N

$$F_3$$ force is negative because it is opposite to the other two forces.

Resultant force can be computed by the given formula:

$$F_R = F_1 + F_2 + F_3$$

$$F_R$$ = 50 + 60 – 20

$$F_R$$ = 90 N

Therefore resultant force will be 90 N.

Example-2: If 4 N and 9 N forces are acting perpendicular to an object. Then determine the resultant force.

Solution:

As given in the problem,:

$$F_1$$ = 4 N

$$F_2$$ = 9 N

Since these two are in perpendicular directions. Thus the resultant force will be computed as:

$$F_R$$ = $$\sqrt{(F_1^2 + F_2^2)}$$

= $$\sqrt{(4^2 + 9^2)}$$

= $$\sqrt{(16 + 81)}$$

= $$\sqrt (97)$$

= 9.84 N

Therefore resultant force will be 9.84 N.

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