Definition

Work is done on a body when a force causes displacement of the body.

$W = F \, d \cos\theta \quad [\text{M}^1\text{L}^2\text{T}^{-2}]\ (\text{J})$

• F = magnitude of force (N)
• d = magnitude of displacement (m)
• θ = angle between the force vector and displacement vector

Work is a scalar quantity. SI unit: joule (J); 1 J = 1 N·m.

Sign Convention

Vector Form

$W = \vec{F} \cdot \vec{d} = F_x d_x + F_y d_y + F_z d_z$

Special Cases

• Centripetal force: always perpendicular to velocity → W = 0
• Normal force on horizontal surface: vertical, displacement horizontal → W = 0
• Gravity on horizontal motion: vertical, displacement horizontal → W = 0
• Friction: always θ = 180° with displacement → W = −fd (negative)

Variable Force

$W = \int_{x_1}^{x_2} F(x)\, dx = \text{area under the F-x graph}$

For a spring (F = kx): $W_{\text{spring}} = \int_0^x kx\, dx = \tfrac{1}{2}kx^2$