Work (W = Fd cos θ)
Also known as W = Fd · work done by a force
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In physics, work is force applied through a distance — and only the component of force along the motion counts, which is where the cos θ comes from. Pull a sled with 100 N on a rope angled 30° above the snow for 20 m, and you do W = 100 × 20 × cos 30° ≈ 1732 J, not the full 2000 J. The term itself was coined by the French engineer Gaspard-Gustave de Coriolis in 1826, precisely to compare what steam engines and horses could deliver.
Two useful edge cases: a force perpendicular to the motion (θ = 90°) does no work at all — the Moon's orbit costs gravity nothing — and a force opposing the motion, like friction, does negative work, showing up as cos θ below zero. Solving for θ uses the arccos principal branch, 0° to 180°, which conveniently covers the entire physical range of angles between two directions.
- = Work
- = Force
- = Displacement
- = Angle between force and motion
- Work — Power (P = W/t), Work from Force and Displacement Components
- Force — Newton's Second Law, Power from Force and Velocity (P = Fv)
- Displacement — Displacement (Uniform Acceleration), Velocity-Displacement Relation (v² = v₀² + 2ad)
- Angle between force and motion — Torque, Torque with a Lever Arm (τ = rF sin θ)