# Net Force Calculator

Created by Purnima Singh, PhD
Reviewed by Steven Wooding
Last updated: May 27, 2022

Omni's net force calculator allows you to determine the resultant force on a body when several forces act simultaneously on it.

Continue reading this article to know what net force is and how to find the net force on an object. You will also see some examples of net force calculations.

## What is net force? – Net force definition

When a number of forces act simultaneously on an object or a system, the net force or resultant force is the vector sum of all these forces.

We know that the effect of a force $F$ acting on an object of mass $m$ is to accelerate the object according to the equation:

$\quad a = \frac{F}{m}$

The effect of net force would accelerate the object by the same amount as all the actual forces acting on the object. So we can say that the net force is a single force that would produce the same effect as all the forces working together.

Now that we know what net force is, let us understand how to find the net force on an object.

## How to find net force – Net force equation

We know that force is a vector quantity, i.e., we need to specify both the magnitude and direction of a force to give its complete description. This means that we should add the individual forces to find the net force just like we add other vectors.

To understand this, let us consider a simple scenario where two forces $F_1$ and $F_2$ are acting on a body from two different directions. We can represent these two forces as two vectors $\overrightarrow{F_1}$ and $\overrightarrow{F_2}$ acting at angles $\theta_1$ and $\theta_2$ (see figure 1).

Using the triangle law of vector addition, we know that if two vectors acting simultaneously on a body can be represented in magnitude and direction by the two sides of a triangle taken in one order, the resultant of these two vectors can be represented in magnitude and direction by the third side of the triangle taken in the opposite order (see the net force diagram in figure 1).

Hence, we get the resultant vector as:

$\ \ \overrightarrow{F} = \overrightarrow{F_1} + \overrightarrow{F_2}$

If several forces are acting together (see figure 2), we can apply the polygon rule and write a more general net force formula as:

\begin{aligned} \overrightarrow{F} &= \overrightarrow{F_1} + \overrightarrow{F_2} + \overrightarrow{F_3} + ... + \overrightarrow{F_n}\\ \text{ or:}\\ \overrightarrow{F} &= \sum_{i=1}^\infty \overrightarrow{F_i} \end{aligned}

In the next section, we will see a step-by-step guide to how to find the magnitude and direction of the resultant force?

## How to calculate net force

To get the magnitude of the resultant force $F$, we will use the net force equation. We will first resolve each of the forces $F_1$ and $F_2$ into their respective rectangular components:

$\quad F_{1x} = F_1\cos \theta_1 \\\ \\ \quad F_{1y} = F_1\sin \theta_1$

Similarly, for the force $F_2$, we can write:

$\quad F_{2x} = F_2\cos \theta_2\\\ \\ \quad F_{2y} = F_2\sin \theta_2$

Since the components of a vector are scalar quantities, now we can add them. To find the horizontal component $F_x$ of the resultant force, we will sum all the horizontal components of the individual forces:

$\quad F_x = F_{1x} + F_{2x}$

In the same way, the vertical component will be:

$\quad F_y = F_{1y} + F_{2y}$

Finally, we will calculate the magnitude of the resultant force using:

$\quad F = \sqrt{F_x^2 + F_y^2}$

And the angle of the resultant force with respect to the horizontal axis with:

$\quad \theta = \tan^{-1}\frac{F_y}{F_x}$

For $n$ number of forces, we can write a general formula as:

$\quad F_x = \sum_{i=1}^\infty F_{ix}\\\ \\ \quad F_y = \sum_{i=1}^\infty F_{iy}$

Using the net force formula, we can calculate the magnitude of the resultant force using:

$\quad F = \sqrt{F_x^2 + F_y^2}$

And the angle of the resultant force with respect to the horizontal axis using:

$\quad \theta = \tan^{-1}\frac{F_y}{F_x}$

## Examples of net force calculation

To further understand how to find the resultant force, we will consider two simple examples.

First we will consider a simple case where two forces $\overrightarrow{F_1}$ and $\overrightarrow{F_2}$ are applied to an object in opposite directions such that;

• $|F_1| = 10\ N$ and $\theta_1 = 0 \degree$; and
• $|F_2| = 15\ N$ and $\theta_2 = 180 \degree$.

An example of this case would be when you and your friend are sitting in front of each other and trying to push a book towards each other.

Let us see how to calculate the net force.

1. First, we will find the vertical and horizontal components for both the forces:

• $F_{1x} = 10 \cdot \cos 0 \degree$ and $F_{1y} =10 \cdot \sin 0 \degree$.
$\implies F_{1x} = 10$ and $F_{1y} =0$

• $F_{2x} = 15 \cdot \cos 180 \degree$ and $F_{2y} =15 \cdot \sin 180 \degree$.
$\implies F_{1x} = -15$ and $F_{1y} =0$

2. As the vertical components, $F_{1y}$ and $F_{2y}$ are zero, the vertical component of the resultant force will be zero. We can calculate the horizontal component as:

• $F_x = 10 + (-15)$, or
$F_x = -5\ N$
3. According to the definition of the net force, the magnitude of the resultant force will be:

• $F = \sqrt{F_x^2} = 5\ N$

and the direction will be along the direction of the larger force, i.e., $180 \degree$.

Now let us consider another case where the magnitude of the forces is the same as in the previous example, but now both the forces act in the same direction, i.e., $\theta_1 = \theta_2 = 180 \degree$. For example, when you and your friend try to push a heavy box together. In this case, the net force would be $25\ N$ along the direction of both the forces, i.e., $180 \degree$.

In the next section, we will see how to solve the same problem using our resultant force calculator.

## How to use the net force calculator

Let us see how to use the net force calculator:

1. Enter the magnitude ($|F_1| = 10\ N$ and $|F_2| = 15\ N$) and direction ($\theta_1 = 0 \degree$ and $\theta_2 = 180 \degree$) of different forces acting on the body. The directions are measured with respect to the positive x-axis.

2. You can add data for up to 10 forces; fields will appear as you need them.

3. The resultant force calculator will display the magnitude ($F = 5\ N$) and direction ($\theta = 180 \degree$) of the net force. It will also show the values of the horizontal and vertical components of the resultant force.

## FAQ

### How do I find the resultant force acting on an object?

To find the resultant force or net force acting on an object, follow the given instructions:

1. Determine the horizontal and vertical components of all the individual forces by using the formula;

• Horizontal component –
Fₓ = F cos θ
• Vertical component –
Fᵧ = F sin θ
2. Add the individual horizontal and vertical components to get the horizontal and vertical components of the resultant force.

3. Sum the square of the horizontal and vertical components of the resultant force and take the square root of the result. You will get the magnitude of the resultant force.

### When the net force on an object is zero?

According to Newton's first law of motion, if an object is either at rest or moving in a straight line with constant velocity (no acceleration), the net force on the object is zero.

### What are the units of net force?

The units for net force are the same as the unit of force. The SI unit of the net force is the newton (N), and the cgs unit of the net force is dyne.

### How does net force affect the speed of an object?

From Newton's second law of motion, we know that the acceleration or rate of change of speed of an object is directly proportional to the net force acting on the object.

Purnima Singh, PhD
Force 1 (F₁)
N
Angle 1 (θ₁)
deg
Force 2 (F₂)
N
Angle 2 (θ₂)
deg
You can add up to 10 different forces. Each angle is measured with respect to the positive x-axis.
Resultant force
Horizontal component (Fx)
0.97
N
Vertical component (Fy)
0.26
N
Magnitude of resultant force (F)
1
N
Direction of resultant force (θ)
15
deg
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