Work Done by a Constant Force: Formula, Calculation, Examples & Applications

Work done by a constant force helps to describe how energy is transferred when a constant force moves an object through a certain displacement. Whether you are simply pushing a shopping cart, pulling a suitcase, or lifting a backpack, work is done only if the applied force causes the object to move. 

The amount of work depends on the force applied, the distance the object moves, and the angle between the force and the direction of displacement. In this article, you will learn what work done by a constant force is, its formula, calculation, conditions, and real-life applications by using simple examples. 

Table of Contents

• What is Work Done by a Constant Force
• Mathematical Formula for Work Done by a Constant Force
• Work Done When Force Acts in the Direction of Displacement
• Work Done When Force Acts at an Angle to Displacement
• Real-Life Applications of Work Done by a Constant Force
• Frequently Asked Questions on Work Done by a Constant Force

What is Work Done by a Constant Force

Work done by a constant force is the energy transferred to or from an object when a constant force acts on it and causes it to move through a displacement. The amount of work depends not only on the magnitude of the force but also on the displacement in the direction of the applied force.

You might wonder, after knowing that, why only the component of the force that acts parallel to the direction of displacement does work. If a force acts at an angle, the perpendicular component does not contribute to the work done. 

For example, while shovelling snow, the forward component of the force moves the snow, whereas the downward component only presses the shovel against the ground.

Mathematical Formula for Work Done by a Constant Force

The amount of work done depends on three factors. The magnitude of the force applied, the displacement of the object, and the angle between the force and the direction of displacement.

The mathematical expression for work done by a constant force is:

$W = Fd\cos\theta$

Where,

• W is work done (joules, J)
• F is the applied force (newtons, N)
• d is displacement (metres, m)
• θ is the angle between the force and the displacement

The SI unit of work is the joule (J). One joule is the work done when a force of 1 newton moves an object through a displacement of 1 metre in the direction of the force.

If the force acts in the same direction as the displacement (θ = 0°), then cos 0° = 1, and the equation becomes:

W=Fd

This shows that maximum work is done when the applied force and the displacement are in the same direction.

Work Done When Force Acts in the Direction of Displacement

When the force and displacement are in the same direction, the angle between them is 0°. Since,

$\cos 0^\circ = 1$

the formula becomes,

$W = F \times d$

This means the entire force contributes to moving the object, so the work done is maximum.

Example

Imagine pushing a box across a smooth floor with a constant force of 20 N. If the box moves 5 m in the same direction as the force,

$W=20×5=100 JW = 20 \times 5 = 100\ JW=20×5=100 J$

So, the work done is 100 J.

Have you ever noticed that pulling a suitcase straight ahead feels easier than pulling it sideways? That's because more of your force acts in the direction of displacement.

Work Done When Force Acts at an Angle to Displacement

In real life, force does not always act in exactly the same direction as the object's motion.

Imagine pulling a suitcase using its handle. The handle is usually tilted, so the applied force makes an angle with the ground. In this case, only the horizontal component of the force helps move the suitcase forward.

To calculate work, we use the complete equation:

$W = Fd\cos\theta$

Here, θ represents the angle between the force and the direction of displacement.

The larger the angle, the smaller the value of cos θ, and the less work is done in the direction of motion.

How Does the Angle Affect Work?

Angle (θ)	Value of cos θ	Work Done
0°	1	Maximum work is done.
30°	0.866	Most of the force contributes to the motion.
60°	0.5	Only half of the force contributes to the motion.
90°	0	No work is done in the direction of displacement.

Example

Suppose a person pulls a suitcase with a force of 50 N at an angle of 60°, moving it 4 m.

Using the formula:

$W = 50 \times 4 \times \cos60^\circ$

Since:

$\cos60^\circ = 0.5$

$W = 50 \times 4 \times 0.5 = 100\ J$

Therefore, the work done is 100 J.

The fact is that increasing the angle reduces the amount of force that helps move the object forward. When the force acts perpendicular (90°) to the displacement, no work is done in that direction, even though a force is applied.

Real-Life Applications of Work Done by a Constant Force

The concept of work done by a constant force is used in many everyday activities and engineering applications.

• Moving Furniture: When you push a table or chair across the floor with a steady force, work is done because the furniture moves in the direction of the applied force.
• Pulling Luggage: Pulling a suitcase using its handle applies force at an angle. Only part of the force helps move the suitcase forward, making it a practical example of angled work.
• Lifting Objects: Lifting a school bag or a bucket of water requires an upward force against gravity. As the object moves upward, work is done.
• Cycling: A cyclist applies force to the pedals continuously. This force helps the bicycle move forward by transferring energy to the wheels.
• Construction Work: Workers use cranes and lifting machines to move heavy materials. These machines apply force over a distance, making work easier and more efficient.
• Sports: In games like cricket, football, and golf, players apply force to move the ball. The distance travelled by the ball depends on the force applied.

Work done by a constant force explains how energy is transferred when a force moves an object through a displacement. It depends on the magnitude of the force, the distance moved, and the angle between the force and the direction of motion. From pushing furniture and pulling luggage to lifting objects and cycling, this concept helps explain many everyday activities.