In physics, potential energy is the

_{p}''.
Potential energy is the energy by virtue of an object's position relative to other objects. Potential energy is often associated with restoring forces such as a spring or the force of

_{''A''} and x_{''B''} to obtain the work over any trajectory between these two points. It is tradition to define this function with a negative sign so that positive work is a reduction in the potential, that is
$$W\; =\backslash int\_\; \backslash mathbf\; \backslash cdot\; d\backslash mathbf\; =\; U(\backslash mathbf\_A)-U(\backslash mathbf\_B)$$
where ''C'' is the trajectory taken from A to B. Because the work done is independent of the path taken, then this expression is true for any trajectory, ''C'', from A to B.
The function ''U''(x) is called the potential energy associated with the applied force. Examples of forces that have potential energies are gravity and spring forces.

_{x}'', is ''x''^{2}/2.
The function
$$U(x)\; =\; \backslash frackx^2,$$
is called the potential energy of a linear spring.
Elastic potential energy is the potential energy of an elastic object (for example a

_{''r''} and e_{''t''} are the radial and tangential unit vectors directed relative to the vector from ''M'' to ''m''. Use this to simplify the formula for work of gravity to,
$$W\; =\; -\backslash int^\_\; \backslash frac\; (r\backslash mathbf\_r)\backslash cdot(\backslash dot\backslash mathbf\_r\; +\; r\backslash dot\backslash mathbf\_t)\backslash ,dt\; =\; -\backslash int^\_\backslash fracr\backslash dotdt\; =\; \backslash frac-\backslash frac.$$
This calculation uses the fact that
$$\backslash fracr^\; =\; -r^\backslash dot\; =\; -\backslash frac.$$

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'' Hawaiian Electric Company''. Accessed: 13 February 2012. Gravitational potential energy is also used to power clocks in which falling weights operate the mechanism. It's also used by

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Ski Lifts Help Open $25 Billion Market for Storing Power

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What is potential energy?

{{DEFAULTSORT:Potential Energy Forms of energy Gravity

energy
In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of ...

held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors.
Common types of potential energy include the gravitational potential energy of an object, the elastic potential energy
Elastic energy is the mechanical potential energy stored in the configuration of a material or physical system as it is subjected to elastic deformation by work performed upon it. Elastic energy occurs when objects are impermanently compressed, s ...

of an extended spring, and the electric potential energy of an electric charge in an electric field
An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field f ...

. The unit for energy in the International System of Units
The International System of Units, known by the international abbreviation SI in all languages and sometimes pleonastically as the SI system, is the modern form of the metric system and the world's most widely used system of measurement. E ...

(SI) is the joule
The joule ( , ; symbol: J) is the unit of energy in the International System of Units (SI). It is equal to the amount of work done when a force of 1 newton displaces a mass through a distance of 1 metre in the direction of the force appli ...

, which has the symbol J.
The term ''potential energy'' was introduced by the 19th-century Scottish engineer and physicist William Rankine, although it has links to Greek philosopher Aristotle
Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatetic school of phi ...

's concept of potentiality. Potential energy is associated with forces that act on a body in a way that the total work done by these forces on the body depends only on the initial and final positions of the body in space. These forces, that are called ''conservative forces'', can be represented at every point in space by vectors expressed as gradients of a certain scalar function called ''potential''.
Since the work of potential forces 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, there is a function known as ''potential'' that can be evaluated at the two positions to determine this work.
Overview

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 theCoulomb force
Coulomb's inverse-square law, or simply Coulomb's law, is an experimental law of physics that quantifies the amount of force between two stationary, electrically charged particles. The electric force between charged bodies at rest is conventio ...

is called electric potential energy; work of the strong nuclear force
The strong interaction or strong force is a fundamental interaction that confines quarks into proton, neutron, and other hadron particles. The strong interaction also binds neutrons and protons to create atomic nuclei, where it is called the ...

or weak nuclear force acting on the baryon charge is called nuclear potential energy; work of intermolecular forces
An intermolecular force (IMF) (or secondary force) is the force that mediates interaction between molecules, including the electromagnetic forces of attraction
or repulsion which act between atoms and other types of neighbouring particles, e.g. a ...

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 configurations 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 configuration.
Forces derivable from a potential are also called conservative force
In physics, a conservative force is a force with the property that the total work done in moving a particle between two points is independent of the path taken. Equivalently, if a particle travels in a closed loop, the total work done (the sum o ...

s. The work done by a conservative force is
$$W\; =\; -\backslash Delta\; U$$
where $\backslash Delta\; U$ 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 ''PE'', ''U'', ''V'', and ''Egravity
In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the str ...

. The action of stretching a spring or lifting a mass is performed by an external force that works against the force field of the potential. This work is stored in the force field, which is said to be stored as potential energy. If the external force is removed the force field acts on the body to perform the work as it moves the body back to the initial position, reducing the stretch of the spring or causing a body to fall.
Consider a ball whose mass is and whose height is . The acceleration of free fall is approximately constant, so the weight force of the ball is constant. The product of force and displacement gives the work done, which is equal to the gravitational potential energy, thus
$$U\_g\; =\; mgh$$
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.
Work and potential energy

Potential energy is closely linked with forces. If the work done by a force on a body that moves from ''A'' to ''B'' does not depend on the path between these points (if the work is done by a conservative force), then the work of this force measured from ''A'' assigns a scalar value to every other point in space and defines ascalar potential
In mathematical physics, scalar potential, simply stated, describes the situation where the difference in the potential energies of an object in two different positions depends only on the positions, not upon the path taken by the object in trav ...

field. In this case, the force can be defined as the negative of the vector gradient of the potential field.
If the work for an applied force is independent of the path, then the work done by the force is evaluated from the start to the end of the trajectory of the point of application. This means that there is a function ''U''(x), called a "potential," that can be evaluated at the two points xDerivable from a potential

In this section the relationship between work and potential energy is presented in more detail. The line integral that defines work along curve ''C'' takes a special form if the force F is related to a scalar field U'(x) so that $$\backslash mathbf=\; =\; \backslash left\; (\; \backslash frac,\; \backslash frac,\; \backslash frac\; \backslash right\; ).$$ This means that the units of U' must be this case, work along the curve is given by $$W\; =\; \backslash int\_\; \backslash mathbf\; \backslash cdot\; d\backslash mathbf\; =\; \backslash int\_\; \backslash nabla\; U\text{'}\backslash cdot\; d\backslash mathbf,$$ which can be evaluated using thegradient theorem
The gradient theorem, also known as the fundamental theorem of calculus for line integrals, says that a line integral through a gradient field can be evaluated by evaluating the original scalar field at the endpoints of the curve. The theorem is ...

to obtain
$$W=\; U\text{'}(\backslash mathbf\_B)\; -\; U\text{'}(\backslash mathbf\_A).$$
This shows that when forces are derivable from a scalar field, the work of those forces along a curve ''C'' is computed by evaluating the scalar field at the start point ''A'' and the end point ''B'' of the curve. This means the work integral does not depend on the path between ''A'' and ''B'' and is said to be independent of the path.
Potential energy is traditionally defined as the negative of this scalar field so that work by the force field decreases potential energy, that is
$$W\; =\; U(\backslash mathbf\_A)\; -\; U(\backslash mathbf\_B).$$
In this case, the application of the del operator
Del, or nabla, is an operator used in mathematics (particularly in vector calculus) as a vector differential operator, usually represented by the nabla symbol ∇. When applied to a function defined on a one-dimensional domain, it denote ...

to the work function yields,
$$=\; -\; =\; -\backslash left\; (\; \backslash frac,\; \backslash frac,\; \backslash frac\; \backslash right\; )\; =\; \backslash mathbf,$$
and the force F is said to be "derivable from a potential." This also necessarily implies that F must be a conservative vector field
In vector calculus, a conservative vector field is a vector field that is the gradient of some function. A conservative vector field has the property that its line integral is path independent; the choice of any path between two points does not ...

. The potential ''U'' defines a force F at every point x in space, so the set of forces is called a force field.
Computing potential energy

Given a force field F(x), evaluation of the work integral using thegradient theorem
The gradient theorem, also known as the fundamental theorem of calculus for line integrals, says that a line integral through a gradient field can be evaluated by evaluating the original scalar field at the endpoints of the curve. The theorem is ...

can be used to find the scalar function associated with potential energy. This is done by introducing a parameterized curve from to , and computing,
$$\backslash begin\; \backslash int\_\; \backslash nabla\backslash Phi(\backslash mathbf)\; \backslash cdot\; d\backslash mathbf\; \&=\backslash int\_a^b\; \backslash nabla\backslash Phi(\backslash mathbf(t))\; \backslash cdot\; \backslash mathbf\text{'}(t)\; dt,\; \backslash \backslash \; \&=\backslash int\_a^b\; \backslash frac\backslash Phi(\backslash mathbf(t))dt\; =\backslash Phi(\backslash mathbf(b))-\backslash Phi(\backslash mathbf(a))\; =\backslash Phi\backslash left(\backslash mathbf\_B\backslash right)-\backslash Phi\backslash left(\backslash mathbf\_A\backslash right).\; \backslash end$$
For the force field F, let , then the gradient theorem
The gradient theorem, also known as the fundamental theorem of calculus for line integrals, says that a line integral through a gradient field can be evaluated by evaluating the original scalar field at the endpoints of the curve. The theorem is ...

yields,
$$\backslash begin\; \backslash int\_\; \backslash mathbf\; \backslash cdot\; d\backslash mathbf\; \&=\backslash int\_a^b\; \backslash mathbf\; \backslash cdot\; \backslash mathbf\; \backslash ,\; dt,\; \backslash \backslash \; \&=\; -\backslash int\_a^b\; \backslash frac\; U(\backslash mathbf(t))\; \backslash ,\; dt\; =U(\backslash mathbf\_A)-\; U(\backslash mathbf\_B).\; \backslash end$$
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 v of the point of application, that is
$$P(t)\; =\; -\; \backslash cdot\; \backslash mathbf\; =\; \backslash mathbf\backslash cdot\backslash mathbf.$$
Examples of work that can be computed from potential functions are gravity and spring forces.
Potential energy for near Earth gravity

For small height changes, gravitational potential energy can be computed using $$U\_g\; =\; mgh,$$ where ''m'' is the mass in kg, ''g'' is the local gravitational field (9.8 metres per second squared on earth), ''h'' is the height above a reference level in metres, and ''U'' is the energy in joules. In classical physics, gravity exerts a constant downward force 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 , such as the track of a roller coaster is calculated using its velocity, , to obtain $$W\; =\; \backslash int\_^\; \backslash boldsymbol\; \backslash cdot\; \backslash boldsymbol\; \backslash ,\; dt\; =\; \backslash int\_^\; F\_z\; v\_z\; \backslash ,\; dt\; =\; F\_z\backslash Delta\; z.$$ where the integral of the vertical component of velocity is the vertical distance. The work of gravity depends only on the vertical movement of the curve .Potential energy for a linear spring

A horizontal spring exerts a force that is proportional to its deformation in the axial or ''x'' direction. The work of this spring on a body moving along the space curve , is calculated using its velocity, , to obtain $$W\; =\; \backslash int\_0^t\backslash mathbf\backslash cdot\backslash mathbf\backslash ,dt\; =\; -\backslash int\_0^t\; kx\; v\_x\; \backslash ,\; dt\; =-\backslash int\_0^t\; k\; x\; \backslash fracdt\; =\; \backslash int\_^\; k\; x\; \backslash ,\; dx\; =\; \backslash frac\; kx^2$$ For convenience, consider contact with the spring occurs at , then the integral of the product of the distance ''x'' and the ''x''-velocity, ''xvbow
Bow often refers to:
* Bow and arrow, a weapon
* Bowing, bending the upper body as a social gesture
* An ornamental knot made of ribbon
Bow may also refer to:
* Bow (watercraft), the foremost part of a ship or boat
* Bow (position), the rower ...

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
In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of a ...

between the atoms and molecules that constitute the object. If the stretch is released, the energy is transformed into kinetic energy
In physics, the kinetic energy of an object is the energy that it possesses due to its motion.
It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acc ...

.
Potential energy for gravitational forces between two bodies

The gravitational potential function, also known as gravitational potential energy, is: $$U=-\backslash frac,$$ The negative sign follows the convention that work is gained from a loss of potential energy.Derivation

The gravitational force between two bodies of mass ''M'' and ''m'' separated by a distance ''r'' is given by Newton's law $$\backslash mathbf=-\backslash frac\backslash mathbf,$$ where $\backslash mathbf$ is a vector of length 1 pointing from ''M'' to ''m'' and ''G'' is the gravitational constant. Let the mass ''m'' move at the velocity then the work of gravity on this mass as it moves from position to is given by $$W\; =\; -\backslash int^\_\; \backslash frac\; \backslash mathbf\backslash cdot\; d\backslash mathbf\; =\; -\backslash int^\_\; \backslash frac\; \backslash mathbf\backslash cdot\backslash mathbf\; \backslash ,\; dt.$$ The position and velocity of the mass ''m'' are given by $$\backslash mathbf\; =\; r\backslash mathbf\_r,\; \backslash qquad\backslash mathbf=\backslash dot\backslash mathbf\_r\; +\; r\backslash dot\backslash mathbf\_t,$$ where ePotential energy for electrostatic forces between two bodies

The electrostatic force exerted by a charge ''Q'' on another charge ''q'' separated by a distance ''r'' is given byCoulomb's Law
Coulomb's inverse-square law, or simply Coulomb's law, is an experimental law of physics that quantifies the amount of force between two stationary, electrically charged particles. The electric force between charged bodies at rest is convention ...

$$\backslash mathbf=\backslash frac\backslash frac\backslash mathbf,$$
where $\backslash mathbf$ is a vector of length 1 pointing from ''Q'' to ''q'' and ''ε''vacuum permittivity
Vacuum permittivity, commonly denoted (pronounced "epsilon nought" or "epsilon zero"), is the value of the absolute dielectric permittivity of classical vacuum. It may also be referred to as the permittivity of free space, the electric const ...

. This may also be written using the Coulomb constant
The Coulomb constant, the electric force constant, or the electrostatic constant (denoted , or ) is a proportionality constant in electrostatics equations. In SI base units it is equal to .Derived from ''k''e = 1/(4''πε''0) – It was nam ...

.
The work ''W'' required to move ''q'' from ''A'' to any point ''B'' in the electrostatic force field is given by the potential function
$$U(r)\; =\; \backslash frac\backslash frac.$$
Reference level

The 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 ofinverse-square law
In science, an inverse-square law is any scientific law stating that a specified physical quantity is inversely proportional to the square of the distance from the source of that physical quantity. The fundamental cause for this can be understo ...

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 ''relative'' positions of its components only, so the reference state can also be expressed in terms of relative positions.
Gravitational potential energy

Gravitational energy is the potential energy associated withgravitational force
In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the strong ...

, 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
In physics, the kinetic energy of an object is the energy that it possesses due to its motion.
It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acc ...

. 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. "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 approximation

The 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 (" standard gravity"). In this case, a simple expression for gravitational potential energy can be derived using the equation for work, and the equation $$W\_F\; =\; -\backslash Delta\; U\_F.$$ The amount of gravitational potential energy held 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 ). The upward force required while moving at a constant velocity is equal to the weight, , of an object, so the work done in lifting it through a height is the product . Thus, when accounting only for mass,gravity
In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the str ...

, and altitude, the equation is:
$$U\; =\; mgh$$
where is the potential energy of the object relative to its being on the Earth's surface, is the mass of the object, is the acceleration due to gravity, and ''h'' is the altitude of the object. If is expressed in kilograms, in m/sjoule
The joule ( , ; symbol: J) is the unit of energy in the International System of Units (SI). It is equal to the amount of work done when a force of 1 newton displaces a mass through a distance of 1 metre in the direction of the force appli ...

s.
Hence, the potential difference is
$$\backslash Delta\; U\; =\; mg\; \backslash Delta\; h.$$
General formula

However, over large variations in distance, the approximation that is constant is no longer valid, and we have to usecalculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arith ...

and the general mathematical definition of work to determine gravitational potential energy. For the computation of the potential energy, we can integrate the gravitational force, whose magnitude is given by Newton's law of gravitation, with respect to the distance between the two bodies. Using that definition, the gravitational potential energy of a system of masses and at a distance using gravitational constant is
$$U\; =\; -G\; \backslash frac\; +\; K,$$
where is an arbitrary constant dependent on the choice of datum from which potential is measured. Choosing the convention that (i.e. in relation to a point at infinity) makes calculations simpler, albeit at the cost of making negative; for why this is physically reasonable, see below.
Given this formula for , the total potential energy of a system of bodies is found by summing, for all $\backslash frac$ pairs of two bodies, the potential energy of the system of those two bodies.
Considering the system of bodies as the combined set of small particles the bodies consist of, and applying the previous on the particle level we get the negative gravitational binding energy
The gravitational binding energy of a system is the minimum energy which must be added to it in order for the system to cease being in a gravitationally bound state. A gravitationally bound system has a lower (''i.e.'', more negative) gravitati ...

. This potential energy is more strongly negative than the total potential energy of the system of bodies as such since it also includes the negative gravitational binding energy of each body. The potential energy of the system of bodies as such is the negative of the energy needed to separate the bodies from each other to infinity, while the gravitational binding energy is the energy needed to separate all particles from each other to infinity.
$$U\; =\; -\; m\; \backslash left(G\; \backslash frac+\; G\; \backslash frac\backslash right)$$
therefore,
$$U\; =\; -\; m\; \backslash sum\; G\; \backslash frac\; ,$$
Negative gravitational energy

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 ''r'' over another, there seem to be only two reasonable choices for the distance at which becomes zero: $r\; =\; 0$ and $r\; =\; \backslash infty$. The choice of $U\; =\; 0$ at infinity may seem peculiar, and the consequence that gravitational energy is always negative may seem counterintuitive, but this choice allows gravitational potential energy values to be finite, albeit negative. Thesingularity
Singularity or singular point may refer to:
Science, technology, and mathematics Mathematics
* Mathematical singularity, a point at which a given mathematical object is not defined or not "well-behaved", for example infinite or not differentiab ...

at $r\; =\; 0$ in the formula for gravitational potential energy means that the only other apparently reasonable alternative choice of convention, with $U\; =\; 0$ for $r\; =\; 0$, would result in potential energy being positive, but infinitely large for all nonzero values of , and would make calculations involving sums or differences of potential energies beyond what is possible with the real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one- dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Ever ...

system. Since physicists abhor infinities in their calculations, and is always non-zero in practice, the choice of $U\; =\; 0$ at infinity is by far the more preferable choice, even if the idea of negative energy in a gravity well
The Hill sphere of an astronomical body is the region in which it dominates the attraction of satellites. To be retained by a planet, a moon must have an orbit that lies within the planet's Hill sphere. That moon would, in turn, have a Hill s ...

appears to be peculiar at first.
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.
Uses

Gravitational potential energy has a number of practical uses, notably the generation ofpumped-storage hydroelectricity
Pumped-storage hydroelectricity (PSH), or pumped hydroelectric energy storage (PHES), is a type of hydroelectric energy storage used by electric power systems for load balancing. The method stores energy in the form of gravitational potenti ...

. 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.Jacob, ThierrPumped 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
The University of Colorado (CU) is a system of public universities in Colorado. It consists of four institutions: University of Colorado Boulder, University of Colorado Colorado Springs, University of Colorado Denver, and the University ...

'', December 2007. Accessed: 12 February 2012.Yang, Chi-JenPumped Hydroelectric Storage

''

Duke University
Duke University is a private research university in Durham, North Carolina. Founded by Methodists and Quakers in the present-day city of Trinity in 1838, the school moved to Durham in 1892. In 1924, tobacco and electric power industrialist Jame ...

''. Accessed: 12 February 2012.Energy Storage'' Hawaiian Electric Company''. Accessed: 13 February 2012. Gravitational potential energy is also used to power clocks in which falling weights operate the mechanism. It's also used by

counterweight
A counterweight is a weight that, by applying an opposite force, provides balance and stability of a mechanical system. The purpose of a counterweight is to make lifting the load faster and more efficient, which saves energy and causes less wea ...

s for lifting up an elevator, crane, or sash window.
Roller coasters
A roller coaster, or rollercoaster, is a type of amusement ride that employs a form of elevated railroad track designed with tight turns, steep slopes, and sometimes inversions. Passengers ride along the track in open cars, and the rides are ...

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
In physics, the kinetic energy of an object is the energy that it possesses due to its motion.
It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acc ...

obtained from the 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).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 2012Downing, Louise

Ski Lifts Help Open $25 Billion Market for Storing Power

Bloomberg News
Bloomberg News (originally Bloomberg Business News) is an international news agency headquartered in New York City and a division of Bloomberg L.P. Content produced by Bloomberg News is disseminated through Bloomberg Terminals, Bloomberg Tele ...

online, 6 September 2012
Chemical potential energy

Chemical potential energy is a form of potential energy related to the structural arrangement of atoms or molecules. This arrangement may be the result ofchemical bond
A chemical bond is a lasting attraction between atoms or ions that enables the formation of molecules and crystals. The bond may result from the electrostatic force between oppositely charged ions as in ionic bonds, or through the sharing of e ...

s within a molecule or otherwise. Chemical energy of a chemical substance can be transformed to other forms of energy by a chemical reaction
A chemical reaction is a process that leads to the chemical transformation of one set of chemical substances to another. Classically, chemical reactions encompass changes that only involve the positions of electrons in the forming and breaki ...

. 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
Solar energy is radiant light and heat from the Sun that is harnessed using a range of technologies such as solar power to generate electricity, solar thermal energy (including solar water heating), and solar architecture. It is an e ...

to chemical energy through the process known as photosynthesis, and electrical energy can be converted to chemical energy through electrochemical
Electrochemistry is the branch of physical chemistry concerned with the relationship between electrical potential difference, as a measurable and quantitative phenomenon, and identifiable chemical change, with the potential difference as an outc ...

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 energy

An 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 energy

Electrostatic potential energy between two bodies in space is obtained from the force exerted by a charge ''Q'' on another charge ''q'' which is given by $$\backslash mathbf\_\; =\; -\backslash frac\; \backslash frac\; \backslash mathbf,$$ where $\backslash mathbf$ is a vector of length 1 pointing from ''Q'' to ''q'' and ''ε''vacuum permittivity
Vacuum permittivity, commonly denoted (pronounced "epsilon nought" or "epsilon zero"), is the value of the absolute dielectric permittivity of classical vacuum. It may also be referred to as the permittivity of free space, the electric const ...

. This may also be written using the Coulomb constant
The Coulomb constant, the electric force constant, or the electrostatic constant (denoted , or ) is a proportionality constant in electrostatics equations. In SI base units it is equal to .Derived from ''k''e = 1/(4''πε''0) – It was nam ...

.
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 ''W'' required to move ''q'' from ''A'' to any point ''B'' in the electrostatic force field is given by
$$\backslash Delta\; U\_()=-\backslash int\_^\; \backslash mathbf\; \backslash cdot\; d\backslash mathbf$$
typically given in J for Joules. A related quantity called '' electric potential'' (commonly denoted with a ''V'' for voltage) is equal to the electric potential energy per unit charge.
Magnetic potential energy

The energy of a magnetic moment $\backslash boldsymbol$ in an externally produced magnetic B-field has potential energy $$U=-\backslash boldsymbol\backslash cdot\backslash mathbf.$$ Themagnetization
In classical electromagnetism, magnetization is the vector field that expresses the density of permanent or induced magnetic dipole moments in a magnetic material. Movement within this field is described by direction and is either Axial or Dia ...

in a field is
$$U\; =\; -\backslash frac\backslash int\; \backslash mathbf\backslash cdot\backslash mathbf\; \backslash ,\; dV,$$
where the integral can be over all space or, equivalently, where is nonzero.
Magnetic potential energy is the form of energy related not only to the distance between magnetic materials, but also to the orientation, or alignment, of those materials within the field. For example, the needle of a compass has the lowest magnetic potential energy when it is aligned with the north and south poles of the Earth's magnetic field. If the needle is moved by an outside force, torque is exerted on the magnetic dipole of the needle by the Earth's magnetic field, causing it to move back into alignment. The magnetic potential energy of the needle is highest when its field is in the same direction as the Earth's magnetic field. Two magnets will have potential energy in relation to each other and the distance between them, but this also depends on their orientation. If the opposite poles are held apart, the potential energy will be higher the further they are apart and lower the closer they are. Conversely, like poles will have the highest potential energy when forced together, and the lowest when they spring apart.
Nuclear potential energy

Nuclear potential energy is the potential energy of the particles inside an atomic nucleus. The nuclear particles are bound together by thestrong nuclear force
The strong interaction or strong force is a fundamental interaction that confines quarks into proton, neutron, and other hadron particles. The strong interaction also binds neutrons and protons to create atomic nuclei, where it is called the ...

. 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 can have less mass than if they were individually free, in which case this mass difference can be 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 and potential energy

Potential energy is closely linked with forces. If the work done by a force on a body that moves from ''A'' to ''B'' does not depend on the path between these points, then the work of this force measured from ''A'' assigns a scalar value to every other point in space and defines ascalar potential
In mathematical physics, scalar potential, simply stated, describes the situation where the difference in the potential energies of an object in two different positions depends only on the positions, not upon the path taken by the object in trav ...

field. In this case, the force can be defined as the negative of the vector gradient of the potential field.
For example, gravity is a conservative force
In physics, a conservative force is a force with the property that the total work done in moving a particle between two points is independent of the path taken. Equivalently, if a particle travels in a closed loop, the total work done (the sum o ...

. The associated potential is the gravitational potential
In classical mechanics, the gravitational potential at a location is equal to the work (energy transferred) per unit mass that would be needed to move an object to that location from a fixed reference location. It is analogous to the electric ...

, often denoted by $\backslash phi$ or $V$, corresponding to the energy per unit mass as a function of position. The gravitational potential energy of two particles of mass ''M'' and ''m'' separated by a distance ''r'' is
$$U\; =\; -\backslash frac,$$
The gravitational potential ( specific energy) of the two bodies is
$$\backslash phi\; =\; -\backslash left(\; \backslash frac\; +\; \backslash frac\; \backslash right)=\; -\backslash frac\; =\; -\backslash frac\; =\; \backslash frac.$$
where $\backslash mu$ is the reduced mass
In physics, the reduced mass is the "effective" inertial mass appearing in the two-body problem of Newtonian mechanics. It is a quantity which allows the two-body problem to be solved as if it were a one-body problem. Note, however, that the mas ...

.
The work done against gravity by moving an infinitesimal mass from point A with $U\; =\; a$ to point B with $U\; =\; b$ is $(b\; -\; a)$ and the work done going back the other way is $(a\; -\; b)$ so that the total work done in moving from A to B and returning to A is
$$U\_\; =\; (b\; -\; a)\; +\; (a\; -\; b)\; =\; 0.$$
If the potential is redefined at A to be $a\; +\; c$ and the potential at B to be $b\; +\; c$, where $c$ is a constant (i.e. $c$ 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
$$U\_\; =\; (b\; +\; c)\; -\; (a\; +\; c)\; =\; b\; -\; a$$
as before.
In practical terms, this means that one can set the zero of $U$ and $\backslash phi$ anywhere one likes. One may set it to be zero at the surface of the Earth
Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large volumes of water can be found throughout the Solar System, only Earth sustains liquid surface water. About 71% of Earth's surface ...

, 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
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...

is contractible
In mathematics, a topological space ''X'' is contractible if the identity map on ''X'' is null-homotopic, i.e. if it is homotopic to some constant map. Intuitively, a contractible space is one that can be continuously shrunk to a point within tha ...

, its de Rham cohomology
In mathematics, de Rham cohomology (named after Georges de Rham) is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adap ...

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

References

* *External links

What is potential energy?

{{DEFAULTSORT:Potential Energy Forms of energy Gravity