Work done by a force field

Builds on Line integral of a vector field Vector field

Summary

The work done by a force field $F$ on a particle that moves along a curve $\gamma$ is the line integral $W = \int_\gamma F \cdot d\mathbf{r}$ . Only the component of the force along the direction of motion contributes, positively when it points forward and negatively when it points backward.

Remarks

For a constant force $F$ and a straight displacement $d$ , the work is $F \cdot d$ . For a $C^1$ path, $W = \int_a^b F(\gamma(t)) \cdot \gamma'(t) \, dt$ . The integrand $F \cdot \gamma'$ is the power, the rate at which the force does work. If $F = -\nabla U$ for a potential energy $U$ , then $W = U(\gamma(a)) - U(\gamma(b))$ , and the work does not depend on the path. Gravity near the surface of the Earth is an example. A force that is always orthogonal to the velocity does no work.

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Tag	`0CDZ684TCD`
URI	`m2://0CDZ684TCD`
Aliases	`work-of-a-force`
Kind	concept
Writing	W0 stub
MSC	26B12
Specializes	Line integral of a vector field
Uses	Line integral of a vector field, Vector field

Work done by a force field

Summary

Remarks

Reading path

Further reading

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