Central force

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A central force is one whose magnitude depends only on the scalar distance r of the object from the origin and is directed radially outward from the origin. Since the force depends only on the distance from the chosen origin, the field is spherically symmetric.

This has some important consequences: 1.) The angular momentum of the system is conserved and 2.) the energy of the system is conserved. Since a central force is always parallel to the object's position vector, the torque exerted by a central force on the object is zero and the motion takes place in a plane perpendicular to the angular momentum vector. The statement that energy is conserved in a central force is equivalent to saying that a central force is a conservative field.

[edit] Properties

A central force can always be expressed as the negative gradient of a potential:

\mathbf{F}(\mathbf{r}) = - \mathbf{\nabla} V(|\mathbf{r}|)

As a consequence the curl of a central field is zero:

\nabla \times \mathbf{F}(\mathbf{r}) = 0

An object in a central force field obeys Kepler's second law due to conservation of angular momentum.

[edit] Examples

Gravitational force and Coulomb force are two familiar examples with F(r) being proportional to 1/r2.

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