Work in Physics: Formula, Units, and Examples

This calculator finds the work done by a constant force when you enter the force, the distance moved, and the angle between them. It returns work in joules, using the standard physics formula that counts only the part of the force acting along the displacement.

Formula

For a constant force, work is the force multiplied by displacement and by the cosine of the angle between the force and the direction of motion:

W = Fd cos θ

• W is work, measured in joules (J).
• F is the magnitude of the force, measured in newtons (N).
• d is the magnitude of the displacement, measured in metres (m).
• θ is the angle between the force vector and displacement vector, measured in degrees in this calculator.

One joule equals one newton-metre: 1 J = 1 N·m. The calculator also shows the equivalent value in kilojoules, where 1 kJ = 1,000 J. The cosine factor makes the result depend on direction, not just on the force and distance.

Worked Example

Using the calculator’s default inputs, a 20 N force acts along a 5 m displacement, so the angle is 0°.

1. Substitute into the formula: W = 20 N × 5 m × cos(0°).
2. Since cos(0°) = 1, W = 20 × 5 × 1 = 100 N·m.
3. Convert units: 100 N·m = 100.00 J, or 0.100 kJ.

The answer is positive because the force points in the direction of the displacement.

What Work Means in Physics

In everyday speech, “work” often means effort or time spent on a task. In physics, work has a narrower meaning: energy is transferred when a force causes displacement. A person can exert a force without doing mechanical work on an object if that object does not move. For example, holding a stationary box requires muscular effort, but the box’s displacement is zero, so the mechanical work done on the box is zero.

Work is a scalar quantity: it has a magnitude and sign, but no direction of its own. It is closely connected to energy. Positive work by a force transfers energy to an object; negative work removes energy from it. This relationship is summarized by the work–energy theorem: the net work on an object equals its change in kinetic energy.

How the Angle Changes the Answer

The angle determines how much of a force contributes to motion along the displacement. The force component parallel to the displacement is F cos θ. A perpendicular force has no component along the path, so it does no work on the object in that instant.

• 0°: The force points with the displacement, so work is positive and at its maximum for the given force and distance.
• 90°: The force is perpendicular to the displacement, so cos 90° is zero and the work is zero.
• 180°: The force points against the displacement, so work is negative. Friction opposing a sliding object is a common example.

The angle is between the force and displacement vectors, not necessarily the angle between the object’s path and the horizontal. If a force is applied at an upward tilt while a box moves horizontally, use the angle between that tilted force and the horizontal displacement.

When the Simple Formula Applies

Use W = Fd cos θ when the force is constant in magnitude and direction over a straight displacement. It is a useful model for a steady push, pull, or friction force acting along a known path. Make sure the force and distance refer to the same object and the same part of its motion.

If the force changes as the object moves, a single force value and angle may not represent the whole process. The general calculation adds the work over small parts of the path, expressed as the line integral W = ∫ F · dr. In one dimension, this is the area under a graph of force versus position. For a spring, for example, force changes with stretch, so its work is not generally found by multiplying the final force by the full stretch.

Units, Signs, and Common Mistakes

In SI units, enter force in newtons and displacement in metres to obtain joules. If your measurements use other units, convert before calculating: 1 foot = 0.3048 m, and 1 pound-force is approximately 4.448 N. A foot-pound of work is approximately 1.356 J. Do not put a mass in kilograms or pounds into the force field; mass and force are different quantities. If you know mass, you may need additional information, such as acceleration, to determine force.

• Use displacement, which is the change in position, rather than total distance traveled when direction changes.
• Use the angle between force and displacement. A calculator set to degrees expects degrees, not radians.
• Keep the sign. Negative work is a meaningful result, not a calculation error.
• Do not round intermediate values unnecessarily, especially when several forces or stages are involved.

This calculator assumes the input describes one constant force. If several forces act on an object, calculate the work done by each force and add the signed results to find net work. A force can do positive work while another does negative work; the net result determines the change in kinetic energy.

Frequently Asked Questions

What is the formula for work in physics?
For a constant force, use W = Fd cos θ, where force is in newtons, displacement is in metres, and θ is the angle between them. The result is in joules.

Can work be negative?
Yes. Work is negative when a force has a component opposite the displacement. Friction often does negative work on a moving object because it acts against the direction of sliding.

Is work zero if an object does not move?
The mechanical work done on that object is zero when its displacement is zero, even if someone is exerting a force. This definition concerns energy transferred through the object’s displacement.

Does a perpendicular force do work?
Not on an object moving perpendicular to that force at that instant. The angle is 90°, and cos 90° = 0. For example, an ideal centripetal force perpendicular to circular motion changes direction rather than speed, so it does no work.

What is the difference between work and power?
Work measures energy transferred and is measured in joules. Power measures how quickly work is done, or energy is transferred, and is measured in watts; one watt equals one joule per second.

Related

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in f | Force | N | 20 | 0
in d | Displacement | m | 5 | 0
in a | Angle between force and displacement | deg | 0 | 0 | 360
out w | Work | J | f * d * cosd(a) | 2
out kj | Work | kJ | w / 1000 | 3