Definition

Work

W = \int F \cdot d s = F d cos θ

Statement

The component of force along the direction of motion, integrated over the displacement; for a constant force, simply Fd cos θ.

Symbols and units

$W$	work	J
$F$	applied force	N
$d$	displacement	m
$θ$	angle between force and displacement	degrees

When it applies

Any force and displacement; for a varying force or curved path, the integral form is required.

Derivation

Only the component of force along the displacement does work; the perpendicular component changes direction, not speed:

d W = F \cdot d s = F cos θ d s

For a constant force over a straight displacement, this integrates to:

W = F d cos θ

Worked examples

Pulling a sled at an angle

A 50 N pull at 30° above horizontal, over 10 m:

W = F d cos θ = (50) (10) (0.866) \approx 433 J

Watch out for

A force perpendicular to the motion — such as the normal force on a level surface, or tension in circular motion — does zero work, however large it is.