Central force

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A central force is one whose magnitude is a function of only the scalar distance r of the object from the origin and is directed radially, either outward from the origin or inward towards it. Since the force magnitude (but not direction) depends only on the distance from the chosen origin, the field is spherically symmetric.

This has some important consequences:

• The angular momentum of the system is conserved and
• The energy of the system is conserved, though it may shift between potential and kinetic energy, depending on the nature of the system. 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, which can be shown mathematically by showing that curl(F) = 0.
• A body moving solely under the influence of a central force experiences no torque.

[edit] Properties

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

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

[edit] Examples

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

An object in a central force field proportional to 1/r² obeys Kepler's second law due to conservation of angular momentum.