Work Calculator
Work measures the energy a force transfers to an object as it moves. This calculator applies W = F·d·cos(θ), reverse-solves for force, distance, or the angle, converts between metric and imperial units, turns work into power when you add a time, and can find work from a velocity change using the work-energy theorem.
Formula
Worked example
50 N over 4 m at 0°: W = 50 × 4 × cos(0°) = 50 × 4 × 1 = 200 J. Reverse it: to get 200 J over 4 m at 0° you need 200 ÷ (4 × 1) = 50 N.
How It Works
In physics, work is the energy a force transfers to an object as it moves. The calculator multiplies the force you enter by the distance the object travels and then by the cosine of the angle between the force and the direction of motion: W = F·d·cos(θ). The cosine factor matters because only the part of the force that points along the path actually does work. When force and motion are aligned, cos(0°) equals 1 and you get the maximum possible work; when they are perpendicular, cos(90°) equals 0 and no work is done at all. The result is given in joules, the SI unit of energy, and also in foot-pounds and kilocalories so you can use whichever unit fits your problem.
Reverse-Solve and Unit Switches
Use the "What do you want to find?" menu to solve for any one of the four quantities. Leave it on Work for the direct calculation, or switch it to Force, Distance, or Angle to rearrange the formula and back out the missing value: F = W / (d·cos θ), d = W / (F·cos θ), or θ = arccos(W / (F·d)). Every field accepts a unit, so you can mix newtons or pounds-force with metres, feet, or centimetres and the calculator converts to SI internally before computing, then converts a reverse-solved force or distance back into the unit you chose. When you solve for the angle, the cosine ratio must fall between minus one and one; outside that range no real angle can produce the work you asked for, and the result stays blank.
Power and the Work-Energy Theorem
Work and power are linked by time. Turn on "Also find power" and enter how long the force acted, and the calculator divides the work by the time to give power in watts (P = W / t), along with the equivalent in mechanical horsepower. Separately, the "Work from a velocity change" mode applies the work-energy theorem, which says the net work done on an object equals its change in kinetic energy: W = ½·m·(v₁² − v₀²). Enter a mass and the starting and ending speeds in any supported unit and the calculator returns the net work, which is positive when the object speeds up and negative when it slows down. This is the cleanest way to find work when you know how the speed changed but not the force.
What Affects the Result and Limitations
Work grows in direct proportion to both force and distance, so doubling either doubles the work, while the angle has a non-linear effect because it passes through the cosine. Between 0° and 90° the cosine shrinks from 1 to 0; at 90° the work vanishes; beyond 90° the cosine turns negative and the force removes energy, as friction and braking do. The calculator assumes a constant force over a straight-line displacement, the standard textbook case. It does not handle forces that vary along the path, curved trajectories, or rotational work (torque through an angle), all of which need integration or a different formula. The displacement is the straight-line distance moved, not the total path length.
Angle, cosine, and the work done
| Angle θ | cos(θ) | Effect on work |
|---|---|---|
| 0° | 1.00 | Maximum positive work, force fully along motion |
| 30° | 0.87 | 87% of the maximum |
| 45° | 0.71 | 71% of the maximum |
| 60° | 0.50 | Half of the maximum |
| 90° | 0.00 | Zero work, force is perpendicular |
| 120° | -0.50 | Negative work, force opposes motion |
| 180° | -1.00 | Maximum negative work, force directly opposes motion |
How the angle between force and motion scales the work, for a fixed force and distance.
Frequently asked questions
Can this calculator solve for force or distance, not just work?
Yes. Use the "What do you want to find?" menu at the top. Choosing Force rearranges the formula to F = W / (d·cos θ), Distance gives d = W / (F·cos θ), and Angle gives θ = arccos(W / (F·d)). You supply the work plus the other two known values and the calculator returns the missing one, converted into the unit you selected.
Why does the angle reduce the amount of work done?
Only the component of the force that lies along the direction of motion contributes to work, and that component equals F·cos(θ). At 0° the entire force acts along the path and cos(θ) is 1; as the angle increases toward 90° the useful component shrinks to zero, which is why the work falls off following the cosine curve rather than linearly.
Can work be zero or negative even when a force is applied?
Yes. If the force is perpendicular to the motion (90°), cos(90°) is 0 and no work is done, which is why carrying a box horizontally at constant height does no work against gravity. Above 90° the cosine is negative, so the force opposes the motion and does negative work, removing energy. Friction, air resistance, and braking are common examples.
How do I get power from work?
Power is the rate of doing work, so divide the work by the time taken: P = W / t. Turn on "Also find power" and enter the time, and the calculator shows the average power in watts and in horsepower. One watt is one joule per second.
What is the work-energy theorem mode for?
When you know how an object's speed changed but not the force, the work-energy theorem gives the net work directly: W = ½·m·(v₁² − v₀²). Enter the mass and the initial and final speeds and the calculator returns the net work, positive if the object sped up and negative if it slowed down.