Conservative force

From Wikipedia, the free encyclopedia

A conservative force is a force whose work is path-independent.

The gravitational energy of the object only depends on it's height (0 or h) and not on the path along which it is moved. Reason: the gravitational force is conservative.

1 Informal definition
2 Mathematical description
- 2.1 Nonconservative forces
3 See also

[edit] Informal definition

Informally, a conservative force can be thought of as a force that conserves mechanical energy.

The definition above actually states the following: when moving an object from point A to point B, the total mechanical work done is independent of the path that the object took. This implies that when conservative forces are at work, an object may be moved around and afterwards restored to it's initial position, without any energy being transferred to or from the object. An obvious example is the gravitational force. The amount of work done by this force only depends on the vertical distance the object is moved: it doesn't matter whether the object falls straight down, moved in a spiral or in any weird motion. When returning the object to it's initial height, no work will have been done. Other examples are the electric force, the spring force and the magnetic force (in a time-independent electric field). Friction is an example of a non-conservative force. When an object is moved around but returned to it's initial position, the frictional force will have done work and energy is extracted from the system.

[edit] Mathematical description

A force F is called conservative if it meets any of these (equivalent - proof) conditions:

The curl of F is zero:

$\nabla \times \vec{F} = 0. \,$

The work, W, is zero for any simple closed path:

$W = \oint_C \vec{F} \cdot \mathrm{d}\vec r = 0.\,$

The force can be written as the gradient of a potential, $Φ$ :

$\vec{F} = -\nabla \Phi. \,$

Conservative force fields are curl-less as a direct consequence of Helmholtz decomposition. The term conservative force comes from the fact that when a conservative force exists, it conserves mechanical energy. The most familiar conservative forces are gravity, the electric force, and spring force.

[edit] Nonconservative forces

Nonconservative forces arise due to neglected degrees of freedom. For instance, friction may be treated without resorting to the use of nonconservative forces by treating heat as kinetic energy; however that means every molecule's motion must be considered rather than handling it through statistical methods. For macroscopic systems the nonconservative approximation is far easier to deal with than millions of degrees of freedom. Examples of non-conservative forces are friction and non-elastic material stress.