*The single force that generates the same effect as multiple forces is the magnitude of the resultant force. It is the sum of all the forces acting on a body.*

**To understand the motion of an object it is important to know how to find magnitude of net force.** **All the forces that a body experience sum up to find the magnitude of the net force. Force being a vector quantity, it is important to consider direction.**

The individual forces’ magnitude should be known before calculating the net force acting on a body. The force is calculated by using the formula derived from Newton’s Second Law of Motion.

F=ma

Here,

m = mass of the body

a = acceleration produced by force.

For instance, a body of mass 3 kg accelerates at the rate of 4 m/s. To find the force, we substitute the values of mass and acceleration in the above formula.

Hence, we get,

F = 3* 4

F = 12 N

This is the case to find the magnitude of a single force. To find the magnitude of net force all such individual forces are summed up. The forces that act in the same direction are taken positive, while those acting in opposite directions are negative.

Understand how to find magnitude of net force by examining the different situations;

- Forces acting parallel and in the same direction
- Forces acting parallel and in opposite directions
- Non-parallel forces

**Forces acting parallel and in the same direction**

When the two forces act in the same direction, their magnitude adds up to find the resultant force.

This is the simplest form of magnitude of the resultant force. The resultant force acts in the is the same as the direction of the other forces.

F = f_{1} + f_{2}

Suppose two forces of 4 N and 2 N act in the same direction and parallel each other. The net force becomes;

F = f_{1} + f_{2}

F = 4+2 = 6N

Hence the magnitude of net force becomes 6 N.

**Forces acting parallel and in opposite directions**

The forces acting parallel and in opposite directions get subtracted to find the magnitude of the net force. The resultant force acts in the same direction as the force with greater magnitude.

The formula for the resultant magnitude if given as;

F = f_{1} – f_{2}

For example, a force of 7 N acts forwards, whereas the force of 4 N acts backward. We take 7 N force to be positive and 4 N force to be negative. Therefore, the resultant force will be;

F = f_{1} – f_{2} = 7-4 = 3N

The net force here is of 3 N and acts in the forward direction same as 7 N force.

In this other example, the 7 N and 3 N force acts in forward direction and are taken positive. whereas the 4 N force acts in backward direction and gets negative sign. So, the resultant force is calculated as;

F = f_{1 }+ f_{2} – f_{3} = 7+ 3 – 4 = 6 N

The resultant force of 6 N acts in the forward direction.

**Non-parallel forces**

When the forces are non-parallel or in a 2-dimensional or 3-dimensional plane, we cannot directly add the forces. The magnitude of the net force is calculated by using any of the three vector addition laws;

**Triangle Law**

When the two vector forces can be represented as the sides of a triangle, then the third side gives the magnitude of the resultant force. This is known to be the triangle law of finding the resultant vector.

Suppose two forces of P and Q are given as shown. Now the first step is to draw the line of a force and then, from its end, draw the second force. On completing the triangle, the third side will give the magnitude of the resultant force.

vec{R} =vec{P}+vec{Q}

To derive the formula for resultant force, see the diagram above.

Using Pythagoras theorem, we have,

BO^{2} = OC^{2} + CB^{2}

Lets this be the equation 1.

Using trigonometry, we get;

cos θ = AC/BA

AC = BA cos θ

Again;

sin θ = BC/AB

Substituting values of AC and BC in (eqn. 1);

R^{2} = (P+Q cos θ)^{2} + (Q sin θ)^{2}

Therefore,

R = P^{2} +2 PQ cos θ + Q^{2}

This equation is used to calculate the magnitude of the net force.

For example, two vector forces are aligned to each other at an angle, as shown.

The resultant force will be given as;

R = √P^{2}+ 2PQ cos θ + Q^{2} = 100 + 2 (10*5) cos 60 + 25 = 13.2 N

**Parallelogram Law**

When two vector forces can be represented as the adjacent sides of a parallelogram, then the addition is done according to the parallelogram law.

In the above figure, the two forces, P and Q, are given. These two forces are arranged and represented as the parallelogram ABCD. The diagonal AC of the parallelogram will be the magnitude of the resultant force of P and Q.

To derive the formula for the parallelogram law of addition, look at the above figure.

Proceeding in the same way as the triangle law of addition, we get,

**Polygon Law**

Polygon law of forces states that when multiple forces act on a given point, then resultant force is calculated by representing a polygon. On joining the forces, the last side that encloses the polygon provides the magnitude and direction of the net force. It is applicable for the forces acting in a three-dimensional plane. The resultant force for the given situation is;

R^{– }= F_{1} + F_{2} + F_{3 }+ F_{4}

**Frequently Asked Question (FAQs)**

**What is the magnitude of net force?**

The measure or strength of the force is known as its magnitude.

**The resultant force that is the sum of all the forces acting on a body is the magnitude of the net force. The single resultant force has the same effect on an object as the combination of all the others. For example, two men are needed to lift a box, but a bodybuilder can alone lift the box. The force exerted by him would be the resultant force of the other two men.**

**How to find the magnitude of net force?**

The total sum of forces that acts on a particular body is known to be the magnitude of the net force.

**All the forces that the body experiences are added up to find the magnitude of the net force. The forward forces are taken as positive, at the same time one that acts in the opposite direction are negative.**

**For example, in the above figure, the force of 10 N is acting in a forward direction; therefore, it would have a positive sign. The 5 N force will have a negative sign. On adding the forces, we get;**

F = f_{1} – f_{2} = 10-5 = 5N

**The net force will be 5 N and acts in the direction same as that of 10 N.**

**When is a net force equal to zero?**

The resultant force is differentiated as a balanced and unbalanced force.

**When the magnitude of resultant force equals zero, it is known as balanced force, and in other cases, it is unbalanced. The forces get balanced when they act in opposite direction but have an equal magnitude. Hence, the forces get balanced, and the object remains in a state of rest or motion. There will be no acceleration. For example, the book kept on a table has a net force equal to zero.**

**What are the** **three laws of vector force addition?**

The vector quantities are those that have direction along with the magnitude.

**The vector law of addition of forces are;**

**Triangle Law: When two vectors can be represented as the side of the triangles, the third side generates the magnitude of the resultant force.**

**Parallelogram Law: When the two vectors are represented as the adjacent side of the parallelogram, the diagonal represents the direction and magnitude of forces.**

**Polygon Law: When multiple forces act on a point, then the resultant force is generated by completing the polygon.**

**Is there always** **more than one force acting on a body?**

According to the third law of motion, every action has an equal reaction.

**Therefore, for every force, there will be an equal opposing or counterforce. Hence according to Newton’s law, it is proven that all the forces act in pairs. There is always more than one force acting on an object. For example, a box kept on a table experiences gravity that pulls it downwards, but at the same time, normal force emerges that acts upwards and keeps the box stable on the box.**