Work Calculator

Work is the energy expended when a force causes an object to move. If you push a box across the floor, you are doing work; your exertion of energy goes into the box.

Physicists write this as W = F x d x cos(theta), where the cosine accounts for the angle between the force and motion. This calculator can solve for any one of the four quantities (work, force, distance or angle) given the other three. Enter any three values, and the fourth will be calculated.

What is work?

Work is a transfer of energy. When a force acts on an object and causes it to move, energy is transferred into or out of the object, and the amount of this energy transferred is work.

The unit of work is the joule, named after James Prescott Joule. One joule is roughly equivalent to the amount of work required to lift a small apple about one metre, or to push with the force of one newton approximately one metre.

Work is a scalar quantity; that is, it is simply a number with no direction associated with it, even though both force and distance have directions. This is because work is the dot product of the two quantities, and the cosine in the formula combines the two directions into a single value.

How to use this calculator:

Enter the force and displacement to get the result of work done. The angle is set by default at 0 degrees, which means that the force acts exactly in the direction of movement. This is the most common case in everyday life and also leads to the greatest amount of work being done.

If the force is not acting in the direction of motion, adjust the angle. Each field has a separate unit selector switch so you can enter the force in pounds, distance in feet and read off work in joules or calories if desired.

To perform an inverse calculation leave one of the fields blank: enter the target value for work and two other values, then the calculator will calculate the missing force, displacement or angle.

Understanding of formula.

Work is equal to force times displacement, times cosine of the angle between those two vectors.

where W is work, F is force, d is the distance over which the object moves, and θ is the angle between the force and the direction of motion. Suppose you push a box with a force of 50 newtons along the floor for a distance of 10 meters, where the pushing force is at an angle of 0 degrees to the direction of motion:

Since the cosine of 0 degrees is 1, this formula can be simplified to force multiplied by distance. The table below lists each symbol and a sample value.

Why Angle Is Important

The cosine term plays a seemingly unimportant role. If you push in the direction of motion, the cosine is 1 so all your force does work. If you pull at an angle of 60 degrees, only half your force does any work. If you push perpendicular to the direction, no work is done because the cosine of 90 degrees is zero.

Only the component of force acting in the direction of motion, which is F times cos(θ), actually does work; the rest presses sideways and thus is wasted. If you are pulling a sled with a rope at an angle of 30 degrees to the ground, only the amount of force equal to cos(30) will move it forward, which is about 87% of your pull. Open the angle graph below to see the full change from 0 to 180 degrees.

Positive work, zero work and negative work.

The sign of work depends on cosine and reflects the actual physical process.

When you lift a bag of groceries from the ground onto your counter top both your force and direction of motion are acting upward so the work done is positive; you are transferring energy to the bag. When you move that same bag horizontally through your kitchen, the lifting force acts upward while the direction of motion is sideways, and the angle between them is 90 degrees, so the work done on the bag is zero no matter how tired your arms get. Your muscles do use up energy to hold their position but this energy isn't converted into work acting on the bag.

Now if you let the bag slide back down to the ground, friction is acting upwards while the bag is moving downwards, and the angle between them is 180 degrees, so the friction force does negative work: it takes energy out of the system. Forces like braking forces, resistive forces, and frictional forces are all forces that do negative work, which is why they slow things down.

Work and Kinetic Energy

The net work done on an object equals the change in its kinetic energy. This is called the work-energy principle:

It relates force to the speed at which that force can be applied. If you apply a net work of 100 joules to a small car with a mass of 2 kg initially at rest, it will accelerate to 10 m/s. A pitcher applies about 116 joules of work to a baseball with a mass of 145 grams, giving the ball a speed of about 40 m/s when it leaves the pitcher's hand.

Open the function layer, enter mass and initial velocity, then read off the speed that the object reaches. If the work is negative and its absolute value is larger than the initial energy of the object, the object will stop before it fully absorbs this part of the work. The calculator will then output a message saying there is no realistic final speed.

Work, Energy and Power

Work and energy are two sides of the same coin. Energy is the ability to do work; work is the process by which energy is transferred from one place or form to another. When you do work on an object, its energy increases by exactly that amount.

Power is a different beast. Power tells you how quickly work is done: P = W / t. Lifting an object with a mass of 100 N one meter always requires the same amount of work, 100 Joules; whether it happens in one second (then 100 Watts are required) or ten seconds (then only 10 Watts are needed). Open the power layer to convert work into watts or horsepower for a given time.

Units and conversions

Both work and energy use the joule as their SI unit. There are many other units that represent the same physical quantity, and this calculator can switch between these units in the "work" field, so this field itself is a real-time conversion table. The above example with the box shows 500 joules, and the converted results are as follows:

A common mistake is to confuse mass with weight. The weight given in pounds or kiloponds is already a force and can be entered directly; the mass given in kilograms, however, is not a force: it must first be multiplied by the acceleration due to gravity (about 9.8 newtons per kilogram) to obtain the weight. Another mistake is forgetting that this simple form assumes motion in a straight line and constant force; for curved paths or varying forces, a full definition is an integral.