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

_{''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
$$\backslash fracr^\; =\; -r^\backslash dot\; =\; -\backslash frac.$$

_{0} is the

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What is potential energy?

{{DEFAULTSORT:Potential Energy Forms of energy Gravity

energy
In , energy is the that must be to a or to perform on the body, or to it. Energy is a ; the law of states that energy can be in form, but not created or destroyed. The unit of measurement in the (SI) of energy is the , which is the ...

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
Gravitational energy or gravitational potential energy is the potential energy
In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or othe ...

of an object that depends on its mass
Mass is the quantity
Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value ...

and its distance from the center of mass
In physics, the center of mass of a distribution of mass
Mass is the physical quantity, quantity of ''matter'' in a physical body. It is also a measure (mathematics), measure of the body's ''inertia'', the resistance to acceleration (change ...

of another object, the elastic potential energy
Elastic energy is the mechanical potential energy
In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors.
Common types of p ...

of an extended spring, and the electric potential energy
Electric potential energy, is a potential energy
In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors.
Common types of ...

of an electric charge
Electric charge is the physical property of matter that causes it to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons respectively). Like c ...

in an electric field
An electric field (sometimes E-field) is the physical field that surrounds electrically-charged particle
In physics
Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' ' ...

. 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 Pleonasm#Acronyms_and_initialisms, pleonastically as the SI system, is the modern form of the metric system and the world's most wi ...

(SI) is the joule
The joule ( ; symbol: J) is a SI derived unit, derived unit of energy in the International System of Units. It is equal to the energy transferred to (or work (physics), work done on) an object when a force of one Newton (unit), newton acts on th ...

, which has the symbol J.
The term ''potential energy'' was introduced by the 19th-century Scottish engineer and physicist William Rankine
William John Macquorn Rankine (; 5 July 1820 – 24 December 1872) was a Scottish mechanical engineer who also contributed to civil engineering, physics and mathematics. He was a founding contributor, with Rudolf Clausius and William Thomson, 1 ...

, although it has links to Greek philosopher Aristotle
Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher
A philosopher is someone who practices philosophy
Philosophy (from , ) is the study of general and fundamental quest ...

'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 law, or Coulomb's inverse-square law, is an experimental law
Law is a system
A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole.
A system ...

is called electric potential energy
Electric potential energy, is a potential energy
In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors.
Common types of ...

; work of the strong nuclear force
In nuclear physics
Nuclear physics is the field of physics that studies atomic nuclei and their constituents and interactions. Other forms of nuclear matter are also studied.
Nuclear physics should not be confused with atomic physics, which ...

or weak nuclear force
In nuclear physics
Nuclear physics is the field of physics that studies atomic nuclei and their constituents and interactions. Other forms of nuclear matter are also studied.
Nuclear physics should not be confused with atomic physics, which ...

acting on the baryon
In particle physics
Particle physics (also known as high energy physics) is a branch of that studies the nature of the particles that constitute and . Although the word ' can refer to various types of very small objects (e.g. , gas partic ...

charge
Charge or charged may refer to:
Arts, entertainment, and media Films
* ''Charge, Zero Emissions/Maximum Speed'', a 2011 documentary
Music
* Charge (David Ford album), ''Charge'' (David Ford album)
* Charge (Machel Montano album), ''Charge'' (Mac ...

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 neighboring particles, e.g. atom
...

is called intermolecular potential energy. Chemical potential energy, such as the energy stored in fossil fuels
A fossil fuel is a hydrocarbon
In organic chemistry, a hydrocarbon is an organic compound
, CH4; is among the simplest organic compounds.
In chemistry, organic compounds are generally any chemical compounds that contain carbon-hydrogen che ...

, 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
A conservative force is a force
In physics
Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (phy ...

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 ''Eforce
In physics, a force is an influence that can change the motion (physics), motion of an Physical object, object. A force can cause an object with mass to change its velocity (e.g. moving from a Newton's first law, state of rest), i.e., to acce ...

s such as a spring
Spring(s) may refer to:
Common uses
* Spring (season), a season of the year
* Spring (device), a mechanical device that stores energy
* Spring (hydrology), a natural source of water
* Spring (mathematics), a geometric surface in the shape of a heli ...

or the force of gravity
Gravity (), or gravitation, is a by which all things with or —including s, s, , and even —are attracted to (or ''gravitate'' toward) one another. , gravity gives to s, and the causes the s of the oceans. The gravitational attracti ...

. 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 energy, potential energies of an object in two different positions depends only on the positions, not upon the path taken by th ...

field. In this case, the force can be defined as the negative of the 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 at the start and 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. Theline integral
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

that defines work along curve ''C'' takes a special form if the force F is related to a scalar field Φ(x) so that
$$\backslash mathbf=\; =\; \backslash left\; (\; \backslash frac,\; \backslash frac,\; \backslash frac\; \backslash right\; ).$$
In this case, work along the curve is given by
$$W\; =\; \backslash int\_\; \backslash mathbf\; \backslash cdot\; d\backslash mathbf\; =\; \backslash int\_\; \backslash nabla\; \backslash Phi\backslash cdot\; d\backslash mathbf,$$
which can be evaluated using the gradient theorem
The gradient theorem, also known as the fundamental theorem of calculus for line integrals, says that a line integral through a Conservative vector field, gradient field can be evaluated by evaluating the original scalar field at the endpoints of ...

to obtain
$$W=\; \backslash Phi(\backslash mathbf\_B)\; -\; \backslash Phi(\backslash mathbf\_A).$$
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
Vector calculus, or vector analysis, is concerned with derivative, differentiation and integral, integration of vector fields, primarily in 3-dimensional E ...

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 (mathematics), function. Conservative vector fields have the property that the line integral is path independent; the choice of any path betwee ...

. 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 Conservative vector field, gradient field can be evaluated by evaluating the original scalar field at the endpoints of ...

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 Conservative vector field, gradient field can be evaluated by evaluating the original scalar field at the endpoints of ...

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:
Boats
* Bow (ship), the foremost part
* Bow (rowing), the foremost part of a boat
Knot ...

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
Electromagnetism is a branch of physics
Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related ...

between the atoms and molecules that constitute the object. If the stretch is released, the energy is transformed into kinetic energy
In physics
Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular ...

.
Potential energy for gravitational forces between two bodies

The gravitational potential function, also known asgravitational potential energy
Gravitational energy or gravitational potential energy is the potential energy
In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or othe ...

, 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 . 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
between two point charge
A point particle (ideal particle or point-like particle, often spelled pointlike particle) is an idealization of particle
In the Outline of physical science, physical sciences, a particle (or corpuscule in older te ...

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

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

.
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
420px, S represents the light source, while r represents the measured points. The lines represent the flux emanating from the sources and fluxes.
The total number of flux lines depends on the strength of the light source and is constant with in ...

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
Gravity (), or gravitation, is a natural phenomenon
Types of natural phenomena include:
Weather, fog, thunder, tornadoes; biological processes, decomposition, germination
seedlings, three days after germination.
Germination is th ...

, 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
Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular ...

. 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
The standard acceleration due to gravity (or standard acceleration of free fall), sometimes abbreviated as standard gravity, usually denoted by or , is the nominal gravitational acceleration
In physics, gravitational acceleration is the accele ...

"). In this case, a simple expression for gravitational potential energy can be derived using the equation for work
Work may refer to:
* Work (human activity), intentional activity people perform to support themselves, others, or the community
** Manual labour, physical work done by humans
** House work, housework, or homemaking
* Work (physics), the product of ...

, 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
Mass is the quantity
Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value ...

, gravity
Gravity (), or gravitation, is a by which all things with or —including s, s, , and even —are attracted to (or ''gravitate'' toward) one another. , gravity gives to s, and the causes the s of the oceans. The gravitational attracti ...

, and altitude
Altitude or height (also sometimes known as depth) is a distance measurement, usually in the vertical or "up" direction, between a reference and a point or object. The exact definition and reference datum varies according to the context (e.g. ...

, 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 kilogram
The kilogram (also kilogramme) is the base unit of mass
Mass is the physical quantity, quantity of ''matter'' in a physical body. It is also a measure (mathematics), measure of the body's ''inertia'', the resistance to acceleration (change ...

s, in m/smetre
The metre ( Commonwealth spelling) or meter (American spelling
Despite the various English dialects spoken from country to country and within different regions of the same country, there are only slight regional variations in English o ...

s then will be calculated in joule
The joule ( ; symbol: J) is a SI derived unit, derived unit of energy in the International System of Units. It is equal to the energy transferred to (or work (physics), work done on) an object when a force of one Newton (unit), newton acts on th ...

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 mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations ...

and the general mathematical definition of work to determine gravitational potential energy. For the computation of the potential energy, we can integrate
Integration may refer to:
Biology
*Modular integration, where different parts in a module have a tendency to vary together
*Multisensory integration
*Path integration
* Pre-integration complex, viral genetic material used to insert a viral genome ...

the gravitational force, whose magnitude is given by Newton's law of gravitation
Newton's law of universal gravitation is usually stated as that every particle
In the Outline of physical science, physical sciences, a particle (or corpuscule in older texts) is a small wikt:local, localized physical body, object to which can ...

, 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 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 differentiabl ...

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
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...

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
Gravity (), or gravitation, is a list of natural phenomena, natural phenomenon by which all things with mass or energy—including planets, stars, galaxy, galaxies, and even light—are brought toward (or ''gravitate'' toward) one another. ...

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
In physical cosmology
Physical cosmology is a branch of cosmology concerned with the study of cosmological models. A cosmological model, or simply cosmology, provides a description of the largest-scale structures and dynamics of the univ ...

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 hydroelectricity, hydroelectric energy storage used by electric power systems for load balancing (electrical power), load balancing. The method s ...

. For example, in Dinorwig
Dinorwig sometimes spelled Dinorwic (; ), is a small village located high above Llyn Padarn, near Llanberis, in North Wales. It is thought that it was part of the territory of the Ordovices tribe, and that 'Dinorwig' means "Fort of the Ordovice ...

, 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
A university ( la, universitas, 'a whole') is an institution of higher (or tertiary) education and research which awards academic degrees in various academic discipl ...

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

''

Duke University
Duke University is a Private university, private research university in Durham, North Carolina. Founded by Methodists and Quakers in the present-day town of Trinity, North Carolina, Trinity in 1838, the school moved to Durham in 1892. In 1924, t ...

''. 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
400px, Simple crane
A counterweight is a weight that, by applying an opposite force, provides balance and stability of a mechanical system
A machine is any physical system with ordered structural and functional properties. It may represent ...

s for lifting up an elevator
U-Bahn
Rapid transit in Germany consists of four U-Bahn systems and fourteen S-Bahn systems. The U-Bahn or Untergrundbahn (''underground railway'') are conventional rapid transit systems that run mostly underground, while the S-Bahn or ...

, crane, or sash window
A sash window or hung sash window is made of one or more movable panels, or "sashes". The individual sashes are traditionally paned windows, but can now contain an individual sheet (or sheets, in the case of double glazing
Insulating glass (IG ...

.
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
In physics
Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular ...

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
''The Economist'' is an international weekly newspaper
A weekly newspaper is a general-news or current affairsCurrent affairs may refer to:
Media
* Current Affairs (magazine), ''Current Affairs'' (magazine), a bimonthly magazine of cult ...

'', 3 March 2012Downing, LouiseSki Lifts Help Open $25 Billion Market for Storing Power

Bloomberg News
Bloomberg News (originally Bloomberg Business News) is an international news agency
A news agency is an organization that gathers news reports and sells them to subscribing news organizations, such as newspapers, magazines and All-news rad ...

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 atom
An atom is the smallest unit of ordinary matter
In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyda ...

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 substance
A chemical substance is a form of matter
In classical physics and general chemistry, matter is any substance that has mass and t ...

. 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 Solar irradiance, 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 ...

to chemical energy through the process known as photosynthesis
Photosynthesis is a process used by plants and other organisms to convert
Conversion or convert may refer to:
Arts, entertainment, and media
* Conversion (Doctor Who audio), "Conversion" (''Doctor Who'' audio), an episode of the audio drama ' ...

, and electrical energy can be converted to chemical energy through electrochemical
Electrochemistry is the branch of physical chemistry
Physical chemistry is the study of macroscopic
The macroscopic scale is the length scale on which objects or phenomena are large enough to be visible with the naked eye, without magnifying ...

reactions.
The similar term chemical potential
In thermodynamics
Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, radiation, and physical properties of matter. The behavior of these quantities is governed ...

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 itselectric charge
Electric charge is the physical property of matter that causes it to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons respectively). Like c ...

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 as "epsilon nought" or "epsilon zero") is the value of the absolute dielectric permittivity of classical vacuum. Alternatively may be referred to as the permittivity of free space, the electr ...

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

.
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
Electric potential energy, is a potential energy (measured in joules) that results from conservative force, conservative Coulomb forces and is associated with the configuration of a particular set of point electric charge, charges within a defin ...

is the energy of an electrically charged particle (at rest) in an electric field. It is defined as the work
Work may refer to:
* Work (human activity), intentional activity people perform to support themselves, others, or the community
** Manual labour, physical work done by humans
** House work, housework, or homemaking
* Work (physics), the product of ...

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
The electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as the amount of work
Work may refer to:
* Work (human activity), intentional activity people perform to support the ...

'' (commonly denoted with a ''V'' for voltage) is equal to the electric potential energy per unit charge.
Magnetic potential energy

The energy of amagnetic moment
The magnetic moment is the magnetic strength and orientation of a or other object that produces a . Examples of objects that have magnetic moments include: loops of (such as s), permanent magnets, s (such as s), various s, and many astronomical ...

$\backslash boldsymbol$ in an externally produced has potential energy
$$U=-\backslash boldsymbol\backslash cdot\backslash mathbf.$$
The magnetization
In classical electromagnetism
Classical electromagnetism or classical electrodynamics is a branch of theoretical physics
Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and s ...

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 theparticles
In the Outline of physical science, physical sciences, a particle (or corpuscule in older texts) is a small wikt:local, localized physical body, object to which can be ascribed several physical property, physical or chemical , chemical properties ...

inside an atomic nucleus
The atomic nucleus is the small, dense region consisting of s and s at the center of an , discovered in 1911 by based on the 1909 . After the discovery of the neutron in 1932, models for a nucleus composed of protons and neutrons were quickl ...

. The nuclear particles are bound together by the strong nuclear force
In nuclear physics
Nuclear physics is the field of physics that studies atomic nuclei and their constituents and interactions. Other forms of nuclear matter are also studied.
Nuclear physics should not be confused with atomic physics, which ...

. Weak nuclear force
In nuclear physics
Nuclear physics is the field of physics that studies atomic nuclei and their constituents and interactions. Other forms of nuclear matter are also studied.
Nuclear physics should not be confused with atomic physics, which ...

s provide the potential energy for certain kinds of radioactive decay, such as beta decay
In , beta decay (''β''-decay) is a type of in which a (fast energetic or ) is emitted from an , transforming the original to an of that nuclide. For example, beta decay of a transforms it into a by the emission of an electron accompanie ...

.
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
The Sun is the star
A star is an astronomical object consisting of a luminous spheroid of plasma (physics), plasma held together by its own gravity. The List of nearest stars and brown dwarfs, nearest star to Earth is the Sun. Many othe ...

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
In physics, and in particular as measured by radiometry, radiant energy is the energy of electromagnetic radiation, electromagnetic and gravitational radiation. As energy, its SI unit is the joule (J). The quantity of radiant energy may be calculat ...

, 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 energy, potential energies of an object in two different positions depends only on the positions, not upon the path taken by th ...

field. In this case, the force can be defined as the negative of the of the potential field.
For example, gravity is a conservative force
A conservative force is a force
In physics
Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (phy ...

. The associated potential is the gravitational potential
In classical mechanics, the gravitational potential at a location is equal to the work
Work may refer to:
* Work (human activity), intentional activity people perform to support themselves, others, or the community
** Manual labour, physical wo ...

, 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
Specific energy or massic energy is energy
In physics
Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, it ...

) 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 massIn physics
Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior through Spac ...

.
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 harbour and support life. 29.2% of Earth's surface is land consisting of continents and islands. The remaining 70.8% is Water distribution on Earth, covered wi ...

, 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
Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds, using the techniques of differential calculus, integral calculus, linear algebra a ...

as a closed form. As Euclidean space
Euclidean space is the fundamental space of classical geometry. Originally, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension (mathematics), dimens ...

is contractible
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

, its de Rham cohomology vanishes, so every closed form is also an exact formIn mathematics, especially vector calculus and differential topology, a closed form is a differential form ''α'' whose exterior derivative is zero (), and an exact form is a differential form, ''α'', that is the exterior derivative of another diffe ...

, 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