# What is the conservative force

1. Jul 24, 2014

### Greg Bernhardt

Definition/Summary

A force is conservative (the following definitions are all equivalent):

if it complies with the work-energy theorem: work done equals change in mechanical energy

if the work done is path-independent

if the work done on a closed path is zero: $\oint_C \mathbf{F} \cdot d\mathbf{s} =0$

if the overall gain or loss of mechanical energy is path-independent

if the overall gain or loss of mechanical energy on a closed path is zero

if the force is a field with a potential (in which case it can be written as minus the gradient of the potential: $\mathbf{F}\ =\ -\mathbf{\nabla}\Phi$, and so $\mathbf{\nabla}\times\mathbf{F}\ =\ \mathbf{\nabla}\times \mathbf{\nabla}\Phi\ =\ 0$)

if the force is a field whose curl is zero: $\mathbf{\nabla}\times\mathbf{F}\ =\ 0$

Equations

$$\oint_C \vec F d \vec s =0$$

Extended explanation

A conservative force is a force such that $$\oint_C \vec F d \vec s =0$$.
Examples of conservative forces : Gravitational force, static friction force and elastic forces.

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