Work Done by a Force at an Angle Calculator
Find the work a force does when it pushes at an angle to the motion, using W = F·d·cosθ.
How it works
W = F·d·cosθ · component along motion = F·cosθ · component perpendicular = F·sinθ (no work)
Only the part of a force that lies along the direction of motion does work, so the work is F·d·cosθ. When the force is perpendicular to the displacement, θ = 90° and cosθ = 0 — that is why the tension in a whirling string or the normal force under a sliding box does no work at all. Push behind the motion (θ > 90°) and cosθ turns negative, meaning the force removes energy, as friction does.
Worked example
Pull a sled with a 50 N rope angled 30° above the ground for 10 m: W = 50 × 10 × cos30° = 500 × 0.866 ≈ 433 J. The horizontal component doing the work is 50 × cos30° ≈ 43.3 N; the 25 N vertical component lifts nothing along the path and contributes no work.
Sources: Work (physics) — Wikipedia
Frequently asked questions
Why does the angle reduce the work?
Work counts only the force component in the direction of motion, which is F·cosθ. As the angle grows the useful component shrinks, reaching zero at 90° where the force is entirely sideways to the path.
Can work be negative?
Yes. When the angle exceeds 90°, cosθ is negative and the force opposes the motion, taking energy away. Friction and air resistance do negative work in exactly this way.
Why does a perpendicular force do no work?
At 90° the force has no component along the displacement, so F·d·cos90° = 0. The centripetal force on an orbiting body and the normal force on a level floor are common examples — they change direction, not speed.
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This calculator is for educational purposes. Double-check important results before acting on them.