Conservative force

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In physics, a conservative force is a force with the property that the total work done by the force in moving a particle between two points is independent of the path taken.^[1] Equivalently, if a particle travels in a closed loop, the total work done (the sum of the force acting along the path multiplied by the displacement) by a conservative force is zero.^[2]

A conservative force depends only on the position of the object. If a force is conservative, it is possible to assign a numerical value for the potential at any point and conversely, when an object moves from one location to another, the force changes the potential energy of the object by an amount that does not depend on the path taken, contributing to the mechanical energy and the overall conservation of energy. If the force is not conservative, then defining a scalar potential is not possible, because taking different paths would lead to conflicting potential differences between the start and end points.

Gravitational force is an example of a conservative force, while frictional force is an example of a non-conservative force.

Other examples of conservative forces are: force in elastic spring, electrostatic force between two electric charges, and magnetic force between two magnetic poles. The last two forces are called central forces as they act along the line joining the centres of two charged/magnetized bodies. A central force is conservative if and only if it is spherically symmetric.^[3]

For conservative forces,

${\displaystyle \mathbf {F_{c}} =-{\frac {\textit {dU}}{d\mathbf {s} }}}$

where ${\displaystyle F_{c}}$ is the conservative force, ${\displaystyle U}$ is the potential energy, and ${\displaystyle s}$ is the position.^[4]