Work Calculator

About the Work Calculator

In physics, work has a precise meaning: it is done when a force moves something. Push a trolley across a car park, lift a box onto a shelf or pull a sledge through snow, and you do work on it, transferring energy. Hold a heavy bag perfectly still, and — however tiring it feels — you do no work on it, because nothing moves.

This work calculator finds the work done by a force over a distance at any angle, or works backwards to the force or the distance. Forces can be in newtons, kilonewtons, pounds-force or kilograms-force, and distances in metres, kilometres, centimetres, miles or feet. The answer is given in joules and converted to kilojoules, kilowatt-hours, kilocalories and foot-pounds.

How to Use the Work Calculator

Choose Find work, Find force or Find distance.

Enter the known values and choose their units.

Enter the angle between the force and the direction of motion — 0° when pushing straight along the motion.

The Formula

  W = F × d × cos θ

  W = work (J)
  F = force (N)
  d = distance moved (m)
  θ = angle between the force and the motion

  F = W ÷ (d cos θ)        d = W ÷ (F cos θ)
  1 J = 1 N·m ≈ 0.7376 ft·lbf        1 kcal = 4,184 J

Step-by-Step Examples

Pushing a trolley. A force of 50 N pushes a trolley 10 m along the ground.

  W = 50 × 10 × cos 0° = 500 J ≈ 368.78 ft·lbf

Pulling a sledge at an angle. A rope pulls with 100 N at 30° to the ground over 20 m.

  W = 100 × 20 × cos 30° = 100 × 20 × 0.866 = 1,732.1 J

Carrying a bag along a corridor. The lifting force is at 90° to the motion, so W = F × d × cos 90° = 0.

Friction slowing an object. A friction force of 100 N acts against an object sliding 10 m: W = 100 × 10 × cos 180° = −1,000 J.

Finding force. Doing 500 J of work over 10 m needs an average force of 50 N.

Why the Angle Matters

Only the part of the force pointing along the motion does work. When you pull a suitcase by its handle at an angle, part of your effort lifts the suitcase slightly and only the horizontal part moves it forward. At 30°, about 87 percent of the force does work; at 60°, only half. A force at right angles to the motion does no work at all, which is why a satellite in a circular orbit keeps a constant speed: gravity pulls it towards the planet, at right angles to its motion.

Positive and Negative Work

Work is positive when the force has a component in the direction of motion, adding energy to the object — an engine speeding up a car. Work is negative when the force opposes the motion, taking energy away — brakes slowing the car, or friction and air resistance on anything that moves. The net work done by all the forces equals the change in the object's kinetic energy, the work–energy theorem.

Work and Energy

Work is the way energy is transferred by forces, so it is measured in the same unit as energy, the joule. One joule is the work done when a force of one newton moves something one metre in the direction of the force. Lifting a 100 g apple by one metre does about one joule of work. Work done against gravity is stored as potential energy; work done to speed something up becomes kinetic energy; work done against friction becomes heat.

Work in the Human Body

The work done on an object in the physics sense is often much less than the energy the body uses. Muscles are only about a quarter efficient at turning food energy into mechanical work, and holding a load still still consumes energy as muscle fibres repeatedly tense, even though no work is done on the load. That is why carrying shopping home is tiring even along a flat street, where the physics work done on the bags is almost zero.

Machines and Work

Simple machines such as levers, pulleys and ramps do not reduce the work needed; they trade force for distance. Pushing a 50 kg barrel up a long ramp onto a lorry takes less force than lifting it straight up, but the force must act over a longer distance, so the work done — the force times the distance — is the same, or slightly more because of friction.

A Brief History

The word "work" was given its scientific meaning by the French engineer Gaspard-Gustave Coriolis in the 1820s, who wanted a way to measure what machines actually accomplish. The joule is named after James Prescott Joule, whose experiments in the 1840s showed that mechanical work and heat are two forms of the same thing.

Understanding Your Result

The headline is the work, force or distance you asked for.

The work in other units line gives the work in J, kJ, kWh, kcal and ft·lbf.

The force and distance line restates the force, distance and angle.

The worth knowing line explains the effect of the angle.

When Should You Use This Calculator?

Use it for physics homework on work and energy.

Use it to estimate the energy needed to move or lift loads.

Use it to convert between joules, kilocalories and foot-pounds.

Common Mistakes

Ignoring the angle. Only the component of the force along the motion does work.

Counting holding as work. No movement means no work on the object.

Mixing units. Use newtons and metres for joules.

Forgetting the sign. Forces opposing motion do negative work.