Work Done Formula with Examples

In our daily lives, we do many kinds of work like lifting books or pushing a table, and so on. But in science, the word "work" has a different meaning. In physics, work is done only when a force moves an object from one place to another.

For example, if we push a wall and the wall does not move, no work is done. But if we push a chair and the chair slides across the floor, then work is done. The amount of work done depends on how much force we use and how far the object moves. To understand this better, we use the work done formula.

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Work Done in Science

Work done in physics means the amount of work that happens when a force is applied and the object moves or changes its place. If there is no movement, then no work done happens, even if we use a lot of force.

Furthermore, work done always changes the energy of the object. When the force is applied in the opposite direction of movement, negative work done takes place. This means energy is taken away from the object.

Is Work Done a Scalar or Vector?

Even though force and displacement are vector quantities, as they have size and direction, work done is a scalar quantity. This means work done has only size (magnitude) and no direction.

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Work Done Formula

The work formula helps us to calculate how much work is done when a force is used to move something. In science, work is done only when the object moves from its place. If the object does not move, then no work is done. Let us learn about the work done formula now.

Formula of Work Done

1. The work formula is Force × Distance. Or simply, W = F × d

2. Sometimes, the force is not in the same direction as the movement. The work done formula in such situations is W = F × d × cos θ, or Work = Force × Distance × cos θ.  Here:

• W means work done.

• F means force applied.

• d is for the distance the object moves.

• θ (theta) is for the angle between the direction of force and the direction of movement.

Additionally, it is important for students to remember that:

• Formula of work done is used to calculate work done, force, or distance.

• The unit of work is Joule (J).

• 1 Joule of work means applying 1 Newton of force to move something by 1 meter.

The formula of work done shows that the amount of work depends on how much force is used and how far the object moves.

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What Factors Affect Work Done?

There are a few important factors that affect work done. Let’s understand each of them one by one.

1. Force Applied: Force means a push or pull on an object. The size and direction of the force both matter. If no force is applied, there is no work done. If the force is too small to move the object, then also no work done happens.

2. Displacement: Displacement means how far an object moves from its starting place to its final place. If the object does not move, the displacement is zero, and so the work done is also zero. For example, if we push a wall with full force but the wall does not move, then no work done happens.

3. Angle Between Force and Displacement: The angle between the direction of the force and the movement also affects the work done.

• If the force and movement are in the same direction, positive work done happens.

• If the force is in the opposite direction of movement (like friction), it is called negative work done.

• If the force is applied at a 90-degree angle (perpendicular) to the movement, no work done happens.

Types of Work Done in Science

In science, work done can be of three types. These types depend on the angle between the force applied and the direction of movement (displacement). Let’s learn about each one.

1. Positive Work Done

Positive work done happens when the force and the movement are in the same direction. For example, when a ball falls to the ground, the force of gravity and the ball’s movement are both in the same direction.

• When the angle between the force and the displacement is between 0° and less than 90°, positive work is done.

2. Negative Work Done

Negative work done happens when the force and the movement are in opposite directions. For example, when we throw a ball upward, the ball moves up, but the force of gravity pulls it down.

• When the angle between the force and the displacement is between 90° and 180°, negative work is done.

3. Zero Work Done

Zero work done happens in two cases:

• Case 1: When the object does not move (displacement is zero). Example: Pushing a wall that does not move.

• Case 2: When the force and the movement are at a 90° angle (perpendicular) to each other. Example: When we carry a load on our head and walk forward. The force is upwards, but the movement is forward, so the angle is 90° and no work is done.

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Work Done Examples with Solutions

Work done examples with solutions help students understand how to use the work formula in different situations. Let’s see some solved examples to better understand this concept.

Example 1: A boy pulls a cart on the ground by applying a force of 80 N at an angle of 40° with the ground. The cart moves 15 meters. Calculate the work done by the boy.

Solution:

Given:

• Force, F = 80 N

• Displacement, d = 15 m

• Angle, θ = 40°

Work Done Formula: W = F × d × cos θ

• W = 80 × 15 × cos 40°

• W = 80 × 15 × 0.766

• W = 918.4 J

So, the work done by the boy is 918.4 Joules.

Example 2: A woman applies a force of 60 N to push a box across the floor. The box moves 4 meters. Find the work done by the woman.

Solution:

Given:

• Force, F = 60 N

• Displacement, d = 4 m

Formula of Work Done:

• W = F × d = 60 × 4 = 240 J

So, the work done by the woman is 240 Joules.

Example 3: A suitcase is pulled up an inclined plane using a force of 4,000 N. The suitcase moves 6 m along the incline, which makes a 30° angle with the ground. Find the work done.

Solution:

• Force applied, F = 4,000 N

• Angle of incline, θ = 30°

• Displacement, d = 6 m

Formula of Work: W = F × d × cos θ

• W = 4,000 × 6 × cos 30°

• W = 4,000 × 6 × 0.866

• W = 20,784 J

So, the work done is 20,784 Joules.

Example 4: A person pulls a box with a force of 50 N at an angle of 40° to the ground. The box moves a distance of 10 meters along the ground. Calculate the work done by the person.

Solution:

Given:

• Force (F) = 50 N

• Displacement (d) = 10 m

• Angle (θ) = 40°

Formula of Work done (W) = F × d × cos(θ)

• W = 50 × 10 × cos(40°)

• W = 500 × 0.766 = 383 J

So, the work done by the person is 383 joules.

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