Work from Force and Displacement Components
Worked example: F = (12, −5) N through d = (3, 4) m → 16 J — press Try an example to run it live, then adjust anything.
Enter your known values, leave one input blank, and solves for the missing one. Tap a variable’s symbol to see what it means, with a typical value. Try different units for next level excitement!
Work is a dot product →
Grade 12Grade 12 Math
Test your skills in the Exam Room: new numbers every attempt — free lessons for students, no sign-up, just pure learning.
share your results
Work from Force and Displacement Components explained
Work is the dot product of force and displacement, so in components it is simply Fxdx + Fydy — no angle to measure and no cosine to look up. Each axis contributes independently, and the contributions can cancel: a force of (12, −5) N acting through a displacement of (3, 4) m does 36 − 20 = 16 J of work, the downward pull eating into what the forward push delivered. A component pair that sums to zero means the force was perpendicular to the motion and did no work at all, which is why a satellite in a circular orbit neither gains nor loses energy despite being pulled on the whole way around.
Gaspard-Gustave de Coriolis named the quantity travail in 1829 and fixed the modern convention of force times distance; the joule was attached to it later, honouring James Prescott Joule's paddle-wheel measurements of the mechanical equivalent of heat. The component form makes the sign convention obvious in a way the |F||d| cos θ form does not: negative work simply means the force opposed the motion, exactly what friction and braking do.
Work from Force and Displacement Components formula
- = Work done (J)
- = x-component of force (N)
- = y-component of force (N)
- = x-component of displacement (m)
- = y-component of displacement (m)
Missing one of these? Work it out first, then come back
- Work done — Work to Move a Charge (W = qΔV), Work (W = Fd cos θ)
- x-component of force — Newton's Second Law, Work (W = Fd cos θ)
- y-component of force — Newton's Second Law, Work (W = Fd cos θ)
- x-component of displacement — Displacement (Uniform Acceleration), Velocity-Displacement Relation (v² = v₀² + 2ad)
- y-component of displacement — Displacement (Uniform Acceleration), Velocity-Displacement Relation (v² = v₀² + 2ad)