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Potential energy {{#invoke:main|main}} Instead of a force, often the mathematically related concept of a potential energy field can be used for convenience. For instance, the gravitational force acting upon an object can be seen as the action of the gravitational field that is present at the object's location. Restating mathematically the definition of energy (via the definition of work), a potential scalar field <math>\scriptstyle{U(\vec{r})}</math> is defined as that field whose gradient is equal and opposite to the force produced at every point:

<math>\vec{F}=-\vec{\nabla} U.</math>

Forces can be classified as conservative or nonconservative. Conservative forces are equivalent to the gradient of a potential while nonconservative forces are not.<ref name=FeynmanVol1 /><ref name=Kleppner />

Conservative forces

{{#invoke:main|main}} A conservative force that acts on a closed system has an associated mechanical work that allows energy to convert only between kinetic or potential forms. This means that for a closed system, the net mechanical energy is conserved whenever a conservative force acts on the system. The force, therefore, is related directly to the difference in potential energy between two different locations in space,<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> and can be considered to be an artifact of the potential field in the same way that the direction and amount of a flow of water can be considered to be an artifact of the contour map of the elevation of an area.<ref name=FeynmanVol1 /><ref name=Kleppner />

Conservative forces include gravity, the electromagnetic force, and the spring force. Each of these forces has models that are dependent on a position often given as a radial vector <math>\scriptstyle \vec{r}</math> emanating from spherically symmetric potentials.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> Examples of this follow:

For gravity:

<math>\vec{F} = - \frac{G m_1 m_2 \vec{r}}{r^3}</math>

where <math>G</math> is the gravitational constant, and <math>m_n</math> is the mass of object n.

For electrostatic forces:

<math>\vec{F} = \frac{q_{1} q_{2} \vec{r}}{4 \pi \epsilon_{0} r^3}</math>

where <math>\epsilon_{0}</math> is electric permittivity of free space, and <math>q_n</math> is the electric charge of object n.

For spring forces:

<math>\vec{F} = - k \vec{r}</math>

where <math>k</math> is the spring constant.<ref name=FeynmanVol1 /><ref name=Kleppner />

Nonconservative forces

For certain physical scenarios, it is impossible to model forces as being due to gradient of potentials. This is often due to macrophysical considerations that yield forces as arising from a macroscopic statistical average of microstates. For example, friction is caused by the gradients of numerous electrostatic potentials between the atoms, but manifests as a force model that is independent of any macroscale position vector. Nonconservative forces other than friction include other contact forces, tension, compression, and drag. However, for any sufficiently detailed description, all these forces are the results of conservative ones since each of these macroscopic forces are the net results of the gradients of microscopic potentials.<ref name=FeynmanVol1 /><ref name=Kleppner />

The connection between macroscopic nonconservative forces and microscopic conservative forces is described by detailed treatment with statistical mechanics. In macroscopic closed systems, nonconservative forces act to change the internal energies of the system, and are often associated with the transfer of heat. According to the Second law of thermodynamics, nonconservative forces necessarily result in energy transformations within closed systems from ordered to more random conditions as entropy increases.<ref name=FeynmanVol1 /><ref name=Kleppner />


Force sections
Intro  Development of the concept  Pre-Newtonian concepts  Newtonian mechanics  Special theory of relativity  Descriptions  Fundamental forces  Non-fundamental forces  Rotations and torque  Kinematic integrals  Potential energy  Units of measurement  Force measurement  See also  Notes  References  Further reading  External links  

Potential energy
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