Work Calculator: W = F × d × cos θ
Solve work, force, distance or angle with W = F·d·cos θ. Pick joules, kJ, ft·lbf or kWh and see every step of the substitution. Free and private in your browser.
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About the Work Calculator
The Work Calculator solves the physics work formula W = F × d × cos θ. Fill in any three of Work, Force, Distance and Angle and the fourth appears as you type. The page shows the formula in standard notation with the solved symbol highlighted and a one-line substitution such as 100 N × 10 m × cos 60° = 500 J.
In physics, work is the energy a force transfers to an object as it moves the object. Only the part of the force that points along the direction of motion does work. Carrying a heavy bag horizontally at constant speed does no work on the bag, because the lifting force is at right angles to the movement, even though your arms get tired.
How to use the work calculator
Steps
- Enter three of the four values and leave the one you want to find empty.
- Enter the angle in degrees between the force and the direction of motion. Use 0 when the force pushes straight along the path.
- Choose units. Work offers J, kJ, kWh, ft·lbf and cal, force offers N, kN, lbf and dyn, and distance offers m, cm, km, ft, in and mi.
- Read the answer under the formula.
Force and work can be negative and any angle is accepted, but distance cannot be negative. If you solve for the angle, the answer is between 0 and 180 degrees. When the target value is impossible, for example a work larger than F × d, an inline message explains why instead of a number.
Formula
W is work in joules, F is the force, d is the distance moved and θ is the angle between the force and the motion. The cosine of 0° is 1, of 90° is exactly 0, and of 180° is −1, so a force that opposes the motion does negative work.
Worked examples
Example 1: force along the motion
Pushing with 10 N over 5 m at an angle of 0° gives W = 10 × 5 × 1 = 50 J.
Example 2: force at an angle
Pulling a sled with 100 N along a rope at 60° to the ground over 10 m gives W = 100 × 10 × 0.5 = 500 J.
Example 3: find the angle
A 100 N force moves an object 10 m and does 500 J of work. Then cos θ = 500 ÷ (100 × 10) = 0.5, so θ = 60°.
Example 4: opposing force
Friction of 10 N acts against a 5 m slide, so θ = 180° and W = 10 × 5 × (−1) = −50 J. The friction removes 50 J of energy.
Good to know
- The unit of work is the joule: 1 J = 1 N·m.
- 90° always gives zero work, because nothing moves in the direction of the force.
- Work and energy share the same units, which is why work changes kinetic energy. See the Kinetic Energy Calculator.
Privacy
Nothing is uploaded; every calculation runs in your browser and never leaves your device.
Related tools
Use the Force Calculator to find the force from mass and acceleration, the Length Converter to prepare a distance, and the Kinetic Energy Calculator to see where the energy goes.
Frequently asked questions
What is work in physics?
Work is the energy a force transfers to an object while moving it. Only the part of the force along the direction of motion counts, which is why the formula includes cos θ.
How do I calculate work done by a force?
Use W = F × d × cos θ. A 100 N force over 10 m at 60° to the motion does 100 × 10 × 0.5 = 500 J. Enter three values and leave the fourth empty.
What angle should I enter?
Enter the angle between the force and the direction of motion, in degrees. Use 0 for a force pushing along the path, 90 for a force at right angles, and 180 for a force that opposes the motion.
Why is the work zero or negative?
At 90° the cosine is exactly 0, so the force does no work. At angles above 90° the cosine is negative, so the force removes energy from the object, like friction on a slide: 10 N over 5 m at 180° is −50 J.
Can I solve for the angle?
Yes. Fill in work, force and distance. The answer is between 0 and 180 degrees. If the work is larger than force times distance, no angle exists, and the calculator says so with a range.
Which units are supported?
Work offers J, kJ, kWh, ft·lbf and cal. Force offers N, kN and lbf. Distance offers m, cm, km, ft, in and mi, and angle is in degrees.
Does it send my numbers anywhere?
No. The force, distance and angle you enter are worked out locally in your browser, and nothing is uploaded, stored or shared with a server, so your own figures never leave your device.