How to Calculate Work and Power

Work is defined as the energy transferred to or from an object via the application of force along a displacement. Work is done when a force causes an object to move. Power is defined as the rate at which work is done or energy is transferred over time. It measures how quickly work can be completed or energy is expended. In this article, we will learn how to calculate work and power using solved examples.

What is Work?

Work refers to the measure of energy transfer that occurs when an object is moved over a distance by an external force. Work is done when a force applied to an object moves that object.

The force must have a component along the direction of the object's displacement. If there is no movement, or if the force is perpendicular to the movement, no work is done.

• The unit of work is the joule (J). One joule is the work done when a force of one newton displaces an object by one meter in the direction of the force.
• Work has a scalar quantity, which means it does not have a direction. It involves only magnitude.

What is Power?

Power is defined as the rate at which work is done or energy is transferred. It indicates how quickly work can be completed or how fast energy is being used or generated.

Power measures the amount of energy transferred or converted per unit time. The standard unit of power is the watt (W), where one watt equals one joule per second (1 W = 1 J/s).

How to Calculate Work and Power?

The formula to calculate work (W) is written as:

W = F⋅d⋅cos(θ)

where,

• F is the magnitude of the force applied,
• d is the distance over which the force is applied,
• θ is the angle between the force and the direction of displacement.

The basic formula to calculate power (P) is:

P = W​ /t

where,

• P is the power,
• W is the work done or energy transferred,
• t is the time over which the work is performed or energy is transferred.

In addition to this, power can be expressed directly in terms of force and velocity, especially when constant force is considered:

P = F⋅v⋅cos(θ)

where,

• F is the force applied,
• v is the velocity of the object,
• θ is the angle between the force and the velocity vector.

Calculating Work and Power

Question: A block is pulled across a flat surface for 30 meters using a horizontal force of 50 Newtons. If this action takes 10 seconds, how much work is done and what is the power developed?

Solution :

Step 1 : Calculation of Work:

Work done is calculated using the formula,

W=F⋅d⋅cos(θ).

Here, F=50N, d=30m, and θ=0^∘ (since the force is horizontal and the displacement is also horizontal).

So, W=50 × 30 × cos(0^∘) =1500 joules.

Step 2 : Calculation of Power:

Power is found by dividing the work done by the time it takes to complete.

t=10 

P = W​/t = 1500​/ 10 =150 watts.

Thus, pulling the block with a force of 50 Newtons over 30 meters in 10 seconds results in a total of 1500 joules of work and develops a power output of 150 watts.

Example 2: A weight of 200 Newtons is lifted vertically 5 meters in 3 seconds. Calculate the work done and the power.

Solution:

Step 1 : Calculation of Work: The work done is given by the formula,

W=F⋅d⋅cos(θ).

We have,

F = 200N, d = 5m, and θ = 0^∘ because the force is in the same direction as the displacement.

W = 200 × 5 × cos(0^∘) = 1000 joules.

Step 2: Calculation of Power: Power is determined by the ratio of work to time.

t=3s.

P = W​/t =1000/3 ​≈ 333.33 watts

Therefore, lifting a 200 Newton weight vertically 5 meters in 3 seconds requires 1000 joules of work, producing a power output of approximately 333.33 watts.

Example 3: A cart is pushed with a force of 100 Newtons at an angle of 30 degrees to the horizontal over a distance of 20 meters in 8 seconds. Determine the work and power.

Solution:

Step 1 : Calculation of Work: The formula, W = F⋅d⋅cos(θ) is used.

We have,

F=100N, d=20m, and θ=30^∘.

W = 100 × 20 × cos(30^∘) ≈1732joules.

Step 2 : Calculation of Power: Power is evaluated as follows:

t = 8s.

P =W​/t = 1732​/8 ≈ 216.5 watts

Hence, pushing the cart with 100 Newtons at a 30-degree angle over 20 meters in 8 seconds leads to the performance of approximately 1732 joules of work and a power generation of about 216.5 watts.

Example 4: A hammer exerts a force of 500 Newtons to drive a nail 0.05 meters into a piece of wood. If the hammer strikes for 0.01 seconds, what are the work done and the power generated?

Solution:

Step 1 : Calculation of Work: Work done is calculated using the formula,

W = F⋅d⋅cos(θ).

F = 500N, d = 0.05m, and θ = 0^∘ (force and displacement are in the same direction).

W = 500 × 0.05 × cos(0^∘) = 25 joules

Step 2 : Calculation of Power: Power is determined using the formula:

P = W​/t

?=0.01 s

P = W​/t = 0.0125 ​= 2500watts

Hence, driving a nail using a force of 500 Newtons over a distance of 0.05 meters in just 0.01 seconds involves doing 25 joules of work and results in a very high power output of 2500 watts.

Also, Check

• How to Calculate Force of Gravity
• How to calculate the Impact Force?
• Work Done By Gravity Formula