Central force

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A central force acting on an object is one whose magnitude depends only on the scalar distance r of the object from the origin and whose direction is along the position vector [ \mathbf ] from the origin to the object.

This has some important consequences. The energy of the object is conserved in time. Because of this last property, a central force field is an example of a conservative field. 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. As a result, the angular momentum of an object in a central force field is constant, and the motion of the object takes place in a plane perpendicular to the angular momentum.

Properties

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

[ \mathbf(\mathbf) = - \mathbf V(|\mathbf|) ]

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

[ \nabla \times \mathbf(\mathbf) = 0 ]

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

Examples

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

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