Work Calculator (W = Fd·cosθ)
Calculate mechanical work done W = F·d·cos(θ). Enter force, distance and angle between force and displacement to get work in Joules.
How to use this tool
- Enter force, distance and angle between f and d in the fields above.
- Results update instantly as you type — or click Calculate.
- Read your work done and the full breakdown beneath it.
Mechanical work is done when a force moves an object: W = F · d · cos(θ), where θ is the angle between the force vector and the direction of displacement. No work is done by a force perpendicular to motion (θ = 90°).
Formula
W = F × d × cos(θ)
Force component along motion = F × cos(θ)
Where F = force (N), d = distance (m), θ = angle between force vector and displacement direction (°).
How it works
This calculator computes mechanical work — the energy transferred by a force acting over a displacement. Only the component of force parallel to the displacement does work, which is captured by the cosine of the angle between them. When force and displacement are in the same direction (θ = 0°), cos(θ) = 1 and all the force contributes. At θ = 90° the force is perpendicular and does zero work. The result is signed: angles above 90° give negative work (force opposes motion).
Worked example
- Given: force F = 10 N, distance d = 5 m, angle θ = 0° (force parallel to motion).
- Calculate force component: F × cos(0°) = 10 × 1 = 10 N.
- Calculate work: W = 10 × 5 × cos(0°) = 10 × 5 × 1 = 50 J.
Work done = 50 J; Force component along motion = 10 N
Common mistakes to avoid
- Leaving theta = 0 degrees when the force is perpendicular to displacement -- perpendicular force does zero work (cos 90 = 0), so entering 0 degrees instead of 90 degrees grossly overstates the result.
- Confusing work (a scalar, in Joules) with force (a vector, in Newtons) -- entering force magnitude as the work value skips the displacement and angle entirely.
- Using the angle between the force and the surface instead of the angle between force and the displacement direction, which are only the same when displacement is horizontal.
Key terms
- Mechanical work
- Energy transferred to or from an object by a force acting over a distance. Measured in Joules (J). Positive work adds energy; negative work removes it.
- Displacement
- The straight-line distance moved by the object in a specific direction, measured in metres. Only displacement parallel to the force counts toward work done.
- Angle of application (θ)
- The angle between the force vector and the direction of displacement. At 0° the full force is effective; at 90° the force does no work.
- cos(θ)
- The cosine function extracts the component of force along the direction of motion. It ranges from 1 (0°, fully aligned) to 0 (90°, perpendicular) to −1 (180°, opposing).
Frequently asked questions
- What if the force is applied at an angle?
- Only the component along the direction of motion does work. A force at 60° contributes cos(60°) = 0.5 of its magnitude.
- How is work related to energy?
- The work-energy theorem states that net work done on an object equals its change in kinetic energy: W_net = ΔKE.