# Potential energy

Potential energy | |
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In the case of a bow and arrow, when the archer does work on the bow, drawing the string back, some of the chemical energy of the archer's body is transformed into elastic potential-energy in the bent limbs of the bow. When the string is released, the force between the string and the arrow does work on the arrow. Thus, the potential energy in the bow limbs is transformed into the kinetic energy of the arrow as it takes flight. | |

Common symbols |
PE, U, or V |

SI unit | joule (J) |

Derivations from other quantities |
U = -m · B (magnetic) |

Classical mechanics |
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'"`UNIQ--postMath-00000001-QINU`"' |

Core topics |

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In physics, The term If the work of a force field acting on a body that moves from a start to an end position is determined only by these two positions, and does not depend on the trajectory of the body, then there is a function known as ## Contents- 1 Overview
- 2 Work and potential energy
- 3 Potential energy for near Earth gravity
- 4 Potential energy for a linear spring
- 5 Potential energy for gravitational forces between two bodies
- 6 Potential energy for electrostatic forces between two bodies
- 7 Reference level
- 8 Gravitational potential energy
- 9 Chemical potential energy
- 10 Electric potential energy
- 11 Nuclear potential energy
- 12 Forces, potential and potential energy
- 13 Notes
- 14 References
- 15 External links
## Overview
The more formal definition is that potential energy is the energy difference between the energy of an object in a given position and its energy at a reference position. There are various types of potential energy, each associated with a particular type of force. For example, the work of an elastic force is called elastic potential energy; work of the gravitational force is called gravitational potential energy; work of the Coulomb force is called electric potential energy; work of the strong nuclear force or weak nuclear force acting on the baryon charge is called nuclear potential energy; work of intermolecular forces is called intermolecular potential energy. Chemical potential energy, such as the energy stored in fossil fuels, is the work of the Coulomb force during rearrangement of mutual positions of electrons and nuclei in atoms and molecules. Thermal energy usually has two components: the kinetic energy of random motions of particles and the potential energy of their mutual positions. Forces derivable from a potential are also called conservative forces. The work done by a conservative force is - '"`UNIQ--postMath-00000002-QINU`"'
where '"`UNIQ--postMath-00000003-QINU`"' is the change in the potential energy associated with the force. The negative sign provides the convention that work done against a force field increases potential energy, while work done by the force field decreases potential energy. Common notations for potential energy are ## Work and potential energyPotential energy is closely linked with forces. If the work done by a force on a body that moves from If the work for an applied force is independent of the path, then the work done by the force is evaluated at the start and end of the trajectory of the point of application. This means that there is a function - '"`UNIQ--postMath-00000004-QINU`"'
where The function ## Derivable from a potentialIn this section the relationship between work and potential energy is presented in more detail. The line integral that defines work along curve - '"`UNIQ--postMath-00000005-QINU`"'
In this case, work along the curve is given by - '"`UNIQ--postMath-00000006-QINU`"'
which can be evaluated using the gradient theorem to obtain - '"`UNIQ--postMath-00000007-QINU`"'
This shows that when forces are derivable from a scalar field, the work of those forces along a curve Potential energy - '"`UNIQ--postMath-00000008-QINU`"'
In this case, the application of the del operator to the work function yields, - '"`UNIQ--postMath-00000009-QINU`"'
and the force ## Computing potential energyGiven a force field - '"`UNIQ--postMath-0000000A-QINU`"'
For the force field - '"`UNIQ--postMath-0000000B-QINU`"'
The power applied to a body by a force field is obtained from the gradient of the work, or potential, in the direction of the velocity - '"`UNIQ--postMath-0000000C-QINU`"'
Examples of work that can be computed from potential functions are gravity and spring forces. ## Potential energy for near Earth gravityIn classical physics, gravity exerts a constant downward force F) on the center of mass of a body moving near the surface of the Earth. The work of gravity on a body moving along a trajectory _{z}(t) = (rx(t), y(t), z(t)), such as the track of a roller coaster is calculated using its velocity, =(vv_{x}, v_{y}, v_{z}), to obtain
- '"`UNIQ--postMath-0000000D-QINU`"'
where the integral of the vertical component of velocity is the vertical distance. Notice that the work of gravity depends only on the vertical movement of the curve The function - '"`UNIQ--postMath-0000000E-QINU`"'
is called the potential energy of a near earth gravity field. ## Potential energy for a linear springA horizontal spring exerts a force - '"`UNIQ--postMath-0000000F-QINU`"'
For convenience, consider contact with the spring occurs at x^{2}/2.
The function - '"`UNIQ--postMath-00000010-QINU`"'
is called the potential energy of a linear spring. Elastic potential energy is the potential energy of an elastic object (for example a bow or a catapult) that is deformed under tension or compression (or stressed in formal terminology). It arises as a consequence of a force that tries to restore the object to its original shape, which is most often the electromagnetic force between the atoms and molecules that constitute the object. If the stretch is released, the energy is transformed into kinetic energy. ## Potential energy for gravitational forces between two bodiesGravitational potential energy between two bodies in space is obtained from the force exerted by a mass - '"`UNIQ--postMath-00000011-QINU`"'
where This can also be expressed as - '"`UNIQ--postMath-00000012-QINU`"'
where '"`UNIQ--postMath-00000013-QINU`"' is a vector of length 1 pointing from Let the mass - '"`UNIQ--postMath-00000014-QINU`"'
Notice that the position and velocity of the mass - '"`UNIQ--postMath-00000015-QINU`"'
where - '"`UNIQ--postMath-00000016-QINU`"'
This calculation uses the fact that - '"`UNIQ--postMath-00000017-QINU`"'
The function - '"`UNIQ--postMath-00000018-QINU`"'
is the gravitational potential function, also known as gravitational potential energy. The negative sign follows the convention that work is gained from a loss of potential energy. ## Potential energy for electrostatic forces between two bodiesThe electrostatic force exerted by a charge - '"`UNIQ--postMath-00000019-QINU`"'
where The work - '"`UNIQ--postMath-0000001A-QINU`"'
## Reference levelThe potential energy is a function of the state a system is in, and is defined relative to that for a particular state. This reference state is not always a real state, it may also be a limit, such as with the distances between all bodies tending to infinity, provided that the energy involved in tending to that limit is finite, such as in the case of inverse-square law forces. Any arbitrary reference state could be used, therefore it can be chosen based on convenience. Typically the potential energy of a system depends on the ## Gravitational potential energyGravitational energy is the potential energy associated with gravitational force, as work is required to elevate objects against Earth's gravity. The potential energy due to elevated positions is called gravitational potential energy, and is evidenced by water in an elevated reservoir or kept behind a dam. If an object falls from one point to another point inside a gravitational field, the force of gravity will do positive work on the object, and the gravitational potential energy will decrease by the same amount. Consider a book placed on top of a table. As the book is raised from the floor, to the table, some external force works against the gravitational force. If the book falls back to the floor, the "falling" energy the book receives is provided by the gravitational force. Thus, if the book falls off the table, this potential energy goes to accelerate the mass of the book and is converted into kinetic energy. When the book hits the floor this kinetic energy is converted into heat, deformation and sound by the impact. The factors that affect an object's gravitational potential energy are its height relative to some reference point, its mass, and the strength of the gravitational field it is in. Thus, a book lying on a table has less gravitational potential energy than the same book on top of a taller cupboard, and less gravitational potential energy than a heavier book lying on the same table. An object at a certain height above the Moon's surface has less gravitational potential energy than at the same height above the Earth's surface because the Moon's gravity is weaker. Note that "height" in the common sense of the term cannot be used for gravitational potential energy calculations when gravity is not assumed to be a constant. The following sections provide more detail. ## Local approximationThe strength of a gravitational field varies with location. However, when the change of distance is small in relation to the distances from the center of the source of the gravitational field, this variation in field strength is negligible and we can assume that the force of gravity on a particular object is constant. Near the surface of the Earth, for example, we assume that the acceleration due to gravity is a constant - '"`UNIQ--postMath-0000001B-QINU`"'
The amount of gravitational potential energy possessed by an elevated object is equal to the work done against gravity in lifting it. The work done equals the force required to move it upward multiplied with the vertical distance it is moved (remember - '"`UNIQ--postMath-0000001C-QINU`"'
where Hence, the potential difference is - '"`UNIQ--postMath-0000001D-QINU`"'
## General formulaHowever, over large variations in distance, the approximation that - '"`UNIQ--postMath-0000001E-QINU`"',
where Given this formula for - '"`UNIQ--postMath-00000021-QINU`"'
therefore, - '"`UNIQ--postMath-00000022-QINU`"',
## Why choose a convention where gravitational energy is negative?As with all potential energies, only differences in gravitational potential energy matter for most physical purposes, and the choice of zero point is arbitrary. Given that there is no reasonable criterion for preferring one particular finite The singularity at '"`UNIQ--postMath-00000026-QINU`"' in the formula for gravitational potential energy means that the only other apparently reasonable alternative choice of convention, with '"`UNIQ--postMath-00000027-QINU`"' for '"`UNIQ--postMath-00000028-QINU`"', would result in potential energy being positive, but infinitely large for all nonzero values of The negative value for gravitational energy also has deeper implications that make it seem more reasonable in cosmological calculations where the total energy of the universe can meaningfully be considered; see inflation theory for more on this. ## UsesGravitational potential energy has a number of practical uses, notably the generation of pumped-storage hydroelectricity. For example, in Dinorwig, Wales, there are two lakes, one at a higher elevation than the other. At times when surplus electricity is not required (and so is comparatively cheap), water is pumped up to the higher lake, thus converting the electrical energy (running the pump) to gravitational potential energy. At times of peak demand for electricity, the water flows back down through electrical generator turbines, converting the potential energy into kinetic energy and then back into electricity. The process is not completely efficient and some of the original energy from the surplus electricity is in fact lost to friction. Roller coasters are an entertaining way to utilize potential energy - chains are used to move a car up an incline (building up gravitational potential energy), to then have that energy converted into kinetic energy as it falls. Another practical use is utilizing gravitational potential energy to descend (perhaps coast) downhill in transportation such as the descent of an automobile, truck, railroad train, bicycle, airplane, or fluid in a pipeline. In some cases the kinetic energy obtained from potential energy of descent may be used to start ascending the next grade such as what happens when a road is undulating and has frequent dips. The commercialization of stored energy (in the form of rail cars raised to higher elevations) that is then converted to electrical energy when needed by an electrical grid, is being undertaken in the United States in a system called Advanced Rail Energy Storage (ARES). *Further information: Gravitational potential energy storage*
## Chemical potential energyChemical potential energy is a form of potential energy related to the structural arrangement of atoms or molecules. This arrangement may be the result of chemical bonds within a molecule or otherwise. Chemical energy of a chemical substance can be transformed to other forms of energy by a chemical reaction. As an example, when a fuel is burned the chemical energy is converted to heat, same is the case with digestion of food metabolized in a biological organism. Green plants transform solar energy to chemical energy through the process known as photosynthesis, and electrical energy can be converted to chemical energy through electrochemical reactions. The similar term chemical potential is used to indicate the potential of a substance to undergo a change of configuration, be it in the form of a chemical reaction, spatial transport, particle exchange with a reservoir, etc. ## Electric potential energyAn object can have potential energy by virtue of its electric charge and several forces related to their presence. There are two main types of this kind of potential energy: electrostatic potential energy, electrodynamic potential energy (also sometimes called magnetic potential energy). ## Electrostatic potential energyElectrostatic potential energy between two bodies in space is obtained from the force exerted by a charge - '"`UNIQ--postMath-0000002A-QINU`"'
where If the electric charge of an object can be assumed to be at rest, then it has potential energy due to its position relative to other charged objects. The electrostatic potential energy is the energy of an electrically charged particle (at rest) in an electric field. It is defined as the work that must be done to move it from an infinite distance away to its present location, adjusted for non-electrical forces on the object. This energy will generally be non-zero if there is another electrically charged object nearby. The work - '"`UNIQ--postMath-0000002B-QINU`"'
A related quantity called ## Magnetic potential energyThe energy of a magnetic moment - '"`UNIQ--postMath-0000002C-QINU`"'
The magnetization - '"`UNIQ--postMath-0000002D-QINU`"'
where the integral can be over all space or, equivalently, where ## Nuclear potential energyNuclear potential energy is the potential energy of the particles inside an atomic nucleus. The nuclear particles are bound together by the strong nuclear force. Weak nuclear forces provide the potential energy for certain kinds of radioactive decay, such as beta decay. Nuclear particles like protons and neutrons are not destroyed in fission and fusion processes, but collections of them have less mass than if they were individually free, and this mass difference is liberated as heat and radiation in nuclear reactions (the heat and radiation have the missing mass, but it often escapes from the system, where it is not measured). The energy from the Sun is an example of this form of energy conversion. In the Sun, the process of hydrogen fusion converts about 4 million tonnes of solar matter per second into electromagnetic energy, which is radiated into space. ## Forces, potential and potential energyPotential energy is closely linked with forces. If the work done by a force on a body that moves from For example, gravity is a conservative force. The associated potential is the gravitational potential, often denoted by '"`UNIQ--postMath-0000002E-QINU`"' or '"`UNIQ--postMath-0000002F-QINU`"', corresponding to the energy per unit mass as a function of position. The gravitational potential energy of two particles of mass - '"`UNIQ--postMath-00000030-QINU`"'
The gravitational potential (specific energy) of the two bodies is - '"`UNIQ--postMath-00000031-QINU`"'
where '"`UNIQ--postMath-00000032-QINU`"' is the reduced mass. The work done against gravity by moving an infinitesimal mass from point A with '"`UNIQ--postMath-00000033-QINU`"' to point B with '"`UNIQ--postMath-00000034-QINU`"' is '"`UNIQ--postMath-00000035-QINU`"' and the work done going back the other way is '"`UNIQ--postMath-00000036-QINU`"' so that the total work done in moving from A to B and returning to A is - '"`UNIQ--postMath-00000037-QINU`"'
If the potential is redefined at A to be '"`UNIQ--postMath-00000038-QINU`"' and the potential at B to be '"`UNIQ--postMath-00000039-QINU`"', where '"`UNIQ--postMath-0000003A-QINU`"' is a constant (i.e. '"`UNIQ--postMath-0000003B-QINU`"' can be any number, positive or negative, but it must be the same at A as it is at B) then the work done going from A to B is - '"`UNIQ--postMath-0000003C-QINU`"'
as before. In practical terms, this means that one can set the zero of '"`UNIQ--postMath-0000003D-QINU`"' and '"`UNIQ--postMath-0000003E-QINU`"' anywhere one likes. One may set it to be zero at the surface of the Earth, or may find it more convenient to set zero at infinity (as in the expressions given earlier in this section). A conservative force can be expressed in the language of differential geometry as a closed form. As Euclidean space is contractible, its de Rham cohomology vanishes, so every closed form is also an exact form, and can be expressed as the gradient of a scalar field. This gives a mathematical justification of the fact that all conservative forces are gradients of a potential field. ## Notes- ↑
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**Lua error in Module:Citation/CS1 at line 746: Argument map not defined for this variable.** - ↑ William John Macquorn Rankine (1853) "On the general law of the transformation of energy,"
*Proceedings of the Philosophical Society of Glasgow*, vol. 3, no. 5, pages 276-280; reprinted in:**(1)***Philosophical Magazine*, series 4, vol. 5, no. 30, pages 106-117 (February 1853); and**(2)**W. J. Millar, ed.,*Miscellaneous Scientific Papers: by W. J. Macquorn Rankine*, ... (London, England: Charles Griffin and Co., 1881), part II, pages 203-208. - ↑
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**Lua error in Module:Citation/CS1 at line 746: Argument map not defined for this variable.** - ↑ Jacob, Thierry.Pumped storage in Switzerland - an outlook beyond 2000
*Stucky*. Accessed: 13 February 2012. - ↑ Levine, Jonah G. Pumped Hydroelectric Energy Storage and Spatial Diversity of Wind Resources as Methods of Improving Utilization of Renewable Energy Sources page 6,
*University of Colorado*, December 2007. Accessed: 12 February 2012. - ↑ Yang, Chi-Jen. Pumped Hydroelectric Storage
*Duke University*. Accessed: 12 February 2012. - ↑ Energy Storage
*Hawaiian Electric Company*. Accessed: 13 February 2012. - ↑ Packing Some Power: Energy Technology: Better ways of storing energy are needed if electricity systems are to become cleaner and more efficient,
*The Economist*, 3 March 2012 - ↑ Downing, Louise. Ski Lifts Help Open $25 Billion Market for Storing Power, Bloomberg News online, 6 September 2012
- ↑ Kernan, Aedan. Storing Energy on Rail Tracks, Leonardo-Energy.org website, 30 October 2013
- ↑
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## References-
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## External links |
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