Work

Work is done when a force acts over a distance. Its units are given in Newton-metres, or Joules (J). If force is variable and given as a function $\vec{F} = f(x)$ (with x being the position), and b - a is the interval over which the force acts, work is given by

$W = \int^b_a f(x)dx$

or more generally (as a vector line integral)

$\int_a^b \vec{F} \cdot d \vec{r}$

If the force is constant and always parallel to the displacement vector, this simplifies to

$W = | \vec{F} | d$ or

Formulae for work

• $W = -\Delta U$, where U is potential energy
• $W = \int^b_a P(t)dt = P_\mathrm{avg} t$ where P is power
• $W = \int_{\alpha}^{\beta} \tau\, d \theta$ where τ is torque and θ is the angle across which torque is applied
• $W = \tau \theta$ (if τ is constant)
• $W = \frac{1}{2} k x^2$ for a spring obeying Hooke's law
• $W = \int^b_a P \ dV$ where P is pressure and a and b are initial and final volumes