A force can transfer energy when its point of application moves. For a constant force acting through a displacement $\Delta\mathbf r$, the mechanical work done by that force is

$$W=\mathbf F\cdot\Delta\mathbf r =F,\Delta r\cos\theta,$$

where $\theta$ is the angle between force and displacement.

Sign of work

If the force has a component in the direction of motion, then

$$W>0.$$

If it opposes the motion,

$$W<0.$$

If force and displacement are perpendicular,

$$W=0.$$

A centripetal force in ideal circular motion, for example, is perpendicular to the instantaneous displacement and therefore does no work.

Example

A constant force of $10,\mathrm N$ pulls an object $3,\mathrm m$ in the same direction. The work is

$$W=10\cdot3=30,\mathrm J.$$

If the force were directed opposite the displacement, the work would be $-30,\mathrm J$.

Work is a scalar

Although force and displacement are vectors, their dot product is a scalar. Work records energy transfer rather than a direction of motion.

When force varies along a path, the same idea is extended through a line integral.