# Work

In physics, work is a scalar quantity that can be described as the product of a force times the distance through which it acts, and it is called the work of the force. Only the component of a force in the direction of the movement of its point of application does work. The term work was first coined in 1826 by the French mathematician Gaspard-Gustave Coriolis.[1][2]
If a constant force of magnitude F acts on a point that moves s in the direction of the force, then the work W done by this force is calculated W=Fs. In particular, a force of 10 newtons (F=10N) acting along the path of 2 metres (s=2m) will do the work W=(10N)(2m)=20Nm=20J, where joule (J) is the SI unit for work (derived in order to replace the product Nm).
Calculating the work as "force times straight path segment" can only be done in the simple circumstances described above. If the force is changing, if the body is moving along a curved path, possibly rotating and not necessarily rigid, then only the path of the application point of the force is relevant for the work done, and only the component of the force parallel to the application point velocity is doing work (positive work when in the same direction, and negative when in the opposite direction of the velocity). This component of the force can be described by the scalar quantity called scalar tangential component ($\scriptstyle F\cos\theta$, where $\scriptstyle \theta$ is the angle between the force and the velocity). And then the most general definition of work can be formulated as follows:
Work of a force is the line integral of its scalar tangential component times the velocity along the path of its application point.
Simpler (intermediate) formulas for work and the transition to the general definition are described in the text below.
At any instant, the rate of the work done by a force is the scalar product of the force with the velocity vector of the point of application. This scalar product of force and velocity is called instantaneous power, and work is the time integral of instantaneous power along the trajectory of the point of application. Because changes in this trajectory will change the calculation of work, work is said to be path dependent.
The first law of thermodynamics states that when work is done to a system its energy state changes by the same amount. This equates work and energy. In the case of rigid bodies, Newton's laws can be used to derive a similar relationship called the work-energy theorem.

## Units

The SI unit of work is the joule (J), which is defined as the work done by a force of one newton acting over a distance of one metre. This definition is based on Sadi Carnot's 1824 definition of work as "weight lifted through a height", which is based on the fact that early steam engines were principally used to lift buckets of water, through a gravitational height, out of flooded ore mines. The dimensionally equivalent newton-metre (N·m) is sometimes used instead; however, it is also sometimes reserved for torque to distinguish its units from work or energy.
Non-SI units of work include the erg, the foot-pound, the foot-poundal, and the litre-atmosphere.
Heat conduction is not considered to be a form of work, since the energy is transferred into atomic vibration rather than a macroscopic displacement.

## Zero work

A baseball pitcher does positive work on the ball by transferring energy into it.

Work can be zero even when there is a force. The centripetal force in a uniform circular motion, for example, does zero work since the kinetic energy of the moving object doesn't change. This is because the force is always perpendicular to the motion of the object; only the component of a force parallel to the velocity vector of an object can do work on that object. Likewise when a book sits on a table, the table does no work on the book despite exerting a force equivalent to mg upwards, because no energy is transferred into or out of the book.
If the table were moving upward at a constant velocity, then it will be doing work on the book, since the force of the table on the book will be acting through a distance. However, the force of gravity will be doing equal and opposite work on the book, and the net rate of work done on the book will still be zero, as evidenced by the fact that its kinetic energy remains constant throughout the process. A current that generates a magnetic field can also produce a magnetic force where a charged particle exerts a force on a magnetic field, but the magnetic force can do no work because the charge velocity is perpendicular to the magnetic field and in order for a force or an object to perform work, the force has to be in the same direction as the distance that it moves.

## Mathematical calculation

### Force and displacement

If a force F that is constant with respect to time acts on an object while the object is translationally displaced for a displacement vector d, the work done by the force on the object is the dot product of the vectors F and d:[3]
$W = \bold{F} \cdot \bold{d} = F d \cos\theta$      (1)
where $\textstyle\theta$ is the angle between the force and the displacement vector.
Gravity F=mg does work W=mgh along any descending path

Whereas the magnitude and direction of the force must remain constant, the object's path may have any shape: the work done is independent of the path and is determined only by the total displacement vector $\scriptstyle\bold{d}$. A most common example is the work done by gravity – see diagram. The object descends along a curved path, but the work is calculated from $\scriptstyle d \cos\theta = h$ , which gives the familiar result $\scriptstyle mgh$.
More generally, if the force causes (or affects) rotation of the body, or if the body is not rigid, displacement of the point to which the force is applied (the application point) must be used to calculate the work. This is also true for the case of variable force (below) where, however, magnitude of $\scriptstyle \mathrm{d}\bold{x}$ can equally be interpreted as differential displacement magnitude or differential length of the path of the application point. (Although use of displacement vector most frequently can simplify calculation of work, in some cases simplification is achieved by use of the path length, as in the work of torque calculation below.)
In situations where the force changes over time, equation (1) is not generally applicable. But it is possible to divide the motion into small steps, such that the force is well approximated as being constant for each step, and then to express the overall work as the sum over these steps. This will give an approximate result, which can be improved by further subdivisions into smaller steps (numerical integration). The exact result is obtained as the mathematical limit of this process, leading to the general definition below.
The general definition of mechanical work is given by the following line integral:
$W_C = \int_{C} \bold{F} \cdot \mathrm{d}\bold{x} = \int_{C}\bold{F}\cdot \bold{v}dt,$       (2)
where:
$\textstyle _C$ is the path or curve traversed by the application point of the force;
$\bold F$ is the force vector;
$\bold x$ is the position vector; and
$\bold{v} = d\bold{x}/dt$ is its velocity.
The expression $\delta W = \bold{F} \cdot \mathrm{d}\bold{x}$ is an inexact differential which means that the calculation of $\textstyle{ W_C}$ is path-dependent and cannot be differentiated to give $\bold{F} \cdot \mathrm{d}\bold{x}$.
Equation (2) explains how a non-zero force can do zero work. The simplest case is where the force is always perpendicular to the direction of motion, making the integrand always zero. This is what happens during circular motion. However, even if the integrand sometimes takes nonzero values, it can still integrate to zero if it is sometimes negative and sometimes positive.
The possibility of a nonzero force doing zero work illustrates the difference between work and a related quantity, impulse, which is the integral of force over time. Impulse measures change in a body's momentum, a vector quantity sensitive to direction, whereas work considers only the magnitude of the velocity. For instance, as an object in uniform circular motion traverses half of a revolution, its centripetal force does no work, but it transfers a nonzero impulse.
A force of constant magnitude and perpendicular to the lever arm

### Torque and rotation

Work done by a torque can be calculated in a similar manner, as is easily seen when a force of constant magnitude is applied perpendicularly to a lever arm. After extraction of this constant value, the integral in the equation (2) gives the path length of the application point, i.e. the circular arc $\ s$ , and the work done is $\ W=Fs$ .
However, the arc length can be calculated from the angle of rotation $\varphi\;$ (expressed in radians) as $\ s= r \varphi\;$ , and the ensuing product $\ Fr \;$ is equal to the torque $\tau\;$ applied to the lever arm. Therefore, a constant torque does work as follows:
$W= \tau \varphi\$

## Work and kinetic energy

According to the work-energy theorem, if one or more external forces act upon a rigid object, causing its kinetic energy to change from Ek1 to Ek2, then the work (W) done by the net force is equal to the change in kinetic energy. For translational motion, the theorem can be specified as:[4]
$W = \Delta E_k = E_{k_2} - E_{k_1} = \tfrac12 m (v_2^2 - v_1^2) \,\!$
where m is the mass of the object and v is the object's velocity.
The theorem is particularly simple to prove for a constant force acting in the direction of motion along a straight line. For more complex cases, however, it can be claimed that very concept of work is defined in such a way that the work-energy theorem remains valid.

### Derivation for a particle

In rigid body dynamics, a formula equating work and the change in kinetic energy of the system is obtained as a first integral of Newton's second law of motion.
To see this, consider a particle P that follows the trajectory X(t) with a force F acting on it. Newton's second law provides a relationship between the force and the acceleration of the particle as
$\mathbf{F}=m\ddot{\mathbf{X}},$
where m is the mass of the particle.
The scalar product of each side of Newton's law with the velocity vector yields
$\mathbf{F}\cdot\dot{\mathbf{X}} = m\ddot{\mathbf{X}}\cdot\dot{\mathbf{X}},$
which is integrated from the point X(t1) to the point X(t2) to obtain
$\int_{t_1}^{t_2} \mathbf{F}\cdot\dot{\mathbf{X}} dt = m\int_{t_1}^{t_2}\ddot{\mathbf{X}}\cdot\dot{\mathbf{X}}dt.$
The left side of this equation is the work of the force as it acts on the particle along the trajectory from time t1 to time t2. This can also be written as
$W = \int_{t_1}^{t_2} \mathbf{F}\cdot\dot{\mathbf{X}} dt = \int_{\mathbf{X}(t_1)}^{\mathbf{X}(t_2)} \mathbf{F}\cdot d\mathbf{X}.$
This integral is computed along the trajectory X(t) of the particle and is therefore path dependent.
The right side of the first integral of Newton's equations can be simplified using the identity
$\frac{1}{2}\frac{d}{dt}(\dot{\mathbf{X}}\cdot \dot{\mathbf{X}}) = \ddot{\mathbf{X}}\cdot\dot{\mathbf{X}},$
which can be integrated explicitly to obtain the change in kinetic energy,
$\Delta K = m\int_{t_1}^{t_2}\ddot{\mathbf{X}}\cdot\dot{\mathbf{X}}dt = \frac{m}{2}\int_{t_1}^{t_2}\frac{d}{dt}(\dot{\mathbf{X}}\cdot \dot{\mathbf{X}}) dt = \frac{m}{2}\dot{\mathbf{X}}\cdot \dot{\mathbf{X}}(t_2) - \frac{m}{2}\dot{\mathbf{X}}\cdot \dot{\mathbf{X}} (t_1),$
where the kinetic energy of the particle is defined by the scalar quantity,
$K = \frac{m}{2}\dot{\mathbf{X}}\cdot \dot{\mathbf{X}}.$
The result is the work-energy principle for rigid body dynamics,
$W = \Delta K. \!$
This derivation can be generalized to arbitrary rigid body systems.

## Frame of reference

The work done by a force acting on an object depends on the choice of reference frame because displacements and velocities are dependent on the reference frame in which the observations are being made.[5]
The change in kinetic energy also depends on the choice of reference frame because kinetic energy is a function of velocity. However, regardless of the choice of reference frame, the work energy theorem remains valid and the work done on the object is equal to the change in kinetic energy.[6]

# Energy

Lightning is the electric breakdown of air by strong electric fields, which produce a force on charges. When these charges move through a distance, a flow of energy occurs. The electric potential energy in the atmosphere then is transformed into thermal energy, light, and sound, which are other forms of energy.

In physics, energy (Ancient Greek: ἐνέργεια energeia "activity, operation"[1]) is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems.[2][3] Since work is defined as a force acting through a distance (a length of space), energy is always equivalent to the ability to exert pulls or pushes against the basic forces of nature, along a path of a certain length.

The total energy contained in an object is identified with its mass, and energy (like mass), cannot be created
or destroyed. When matter (ordinary material particles) is changed into energy (such as energy of motion, or into radiation), the mass of the system does not change through the transformation process. However, there may be mechanistic limits as to how much of the matter in an object may be changed into other types of energy and thus into work, on other systems. Energy, like mass, is a scalar physical quantity. In the International System of Units (SI), energy is measured in joules, but in many fields other units, such as kilowatt-hours and kilocalories, are customary. All of these units translate to units of work, which is always defined in terms of forces and the distances that the forces act through.
A system can transfer energy to another system by simply transferring matter to it (since matter is equivalent to energy, in accordance with its mass). However, when energy is transferred by means other than matter-transfer, the transfer produces changes in the second system, as a result of work done on it. This work manifests itself as the effect of force(s) applied through distances within the target system. For example, a system can emit energy to another by transferring (radiating) electromagnetic energy, but this creates forces upon the particles that absorb the radiation. Similarly, a system may transfer energy to another by physically impacting it, but that case the energy of motion in an object, called kinetic energy, results in forces acting over distances (new energy) to appear in another object that is struck. Transfer of thermal energy by heat occurs by both of these mechanisms: heat can be transferred by electromagnetic radiation, or by physical contact in which direct particle-particle impacts transfer kinetic energy.
Energy may be stored in systems without being present as matter, or as kinetic or electromagnetic energy. Stored energy is created whenever a particle has been moved through a field it interacts with (requiring a force to do so), but the energy to accomplish this is stored as a new position of the particles in the field—a configuration that must be "held" or fixed by a different type of force (otherwise, the new configuration would resolve itself by the field pushing or pulling the particle back toward its previous position). This type of energy "stored" by force-fields and particles that have been forced into a new physical configuration in the field by doing work on them by another system, is referred to as potential energy. A simple example of potential energy is the work needed to lift an object in a gravity field, up to a support. Each of the basic forces of nature is associated with a different type of potential energy, and all types of potential energy (like all other types of energy) appears as system mass, whenever present. For example, a compressed spring will be slightly more massive than before it was compressed. Likewise, whenever energy is transferred between systems by any mechanism, an associated mass is transferred with it.
Any form of energy may be transformed into another form. For example, all types of potential energy are converted into kinetic energy when the objects are given freedom to move to different position (as for example, when an object falls off a support). When energy is in a form other than thermal energy, it may be transformed with good or even perfect efficiency, to any other type of energy, including electricity or production of new particles of matter. With thermal energy, however, there are often limits to the efficiency of the conversion to other forms of energy, as described by the second law of thermodynamics.
In all such energy transformation processes, the total energy remains the same, and a transfer of energy from one system to another, results in a loss to compensate for any gain. This principle, the conservation of energy, was first postulated in the early 19th century, and applies to any isolated system. According to Noether's theorem, the conservation of energy is a consequence of the fact that the laws of physics do not change over time.[4]
Although the total energy of a system does not change with time, its value may depend on the frame of reference. For example, a seated passenger in a moving airplane has zero kinetic energy relative to the airplane, but non-zero kinetic energy (and higher total energy) relative to the Earth.

# Conservative force

 A conservative force is a force with the property that the work done in moving a particle between two points is independent of the path taken.[1] Equivalently, if a particle travels in a closed loop, the net work done (the sum of the force acting along the path multiplied by the distance travelled) by a conservative force is zero.[2] It is possible to define a numerical value of potential at every point in space for a conservative force. When an object moves from one location to another, the force changes the potential energy of the object by an amount that does not depend on the path taken. Gravity is an example of a conservative force, while friction is an example of a non-conservative force.

## Informal definition

Informally, a conservative force can be thought of as a force that conserves mechanical energy. Suppose a particle starts at point A, and there is a constant force F acting on it. Then the particle is moved around by other forces, and eventually ends up at A again. Though the particle may still be moving, at that instant when it passes point A again, it has traveled a closed path. If the net work done by F at this point is 0, then F passes the closed path test. Any force that passes the closed path test for all possible closed paths is classified as a conservative force.
The gravitational force, spring force, magnetic force (according to some definitions, see below) and electric force (at least in a time-independent magnetic field, see Faraday's law of induction for details) are examples of conservative forces, while friction and air drag are classical examples of non-conservative forces.
For non-conservative forces, the mechanical energy that is lost (not conserved) has to go somewhere else, by conservation of energy. Usually the energy is turned into heat, for example the heat generated by friction. In addition to heat, friction also often produces some sound energy. The water drag on a moving boat converts the boat's mechanical energy into not only heat and sound energy, but also wave energy at the edges of its wake. These and other energy losses are irreversible because of the second law of thermodynamics.

## Path independence

The work done by the gravitational force on an object depends only on its change in height because the gravitational force is conservative.

A direct consequence of the closed path test is that the work done by a conservative force on a particle moving between any two points does not depend on the path taken by the particle. Also the work done by a conservative force is equal to the negative of change in potential energy during that process. For a proof of that, let's imagine two paths 1 and 2, both going from point A to point B. The variation of energy for the particle, taking path 1 from A to B and then path 2 backwards from B to A, is 0; thus, the work is the same in path 1 and 2, i.e., the work is independent of the path followed, as long as it goes from A to B.
For example, if a child slides down a frictionless slide, the work done by the gravitational force on the child from the top of the slide to the bottom will be the same no matter what the shape of the slide; it can be straight or it can be a spiral. The amount of work done only depends on the vertical displacement of the child.

## Mathematical description

A force field F, defined everywhere in space (or within a simply-connected volume of space), is called a conservative force or conservative vector field if it meets any of these three equivalent conditions:
1. The curl of F is zero:
$\nabla \times \vec{F} = 0. \,$
2. There is zero net work (W) done by the force when moving a particle through a trajectory that starts and ends in the same place:
$W \equiv \oint_C \vec{F} \cdot \mathrm{d}\vec r = 0.\,$
3. The force can be written as the gradient of a potential, Φ:
$\vec{F} = -\nabla \Phi. \,$
The term conservative force comes from the fact that when a conservative force exists, it conserves mechanical energy. The most familiar conservative forces are gravity, the electric force (in a time-independent magnetic field, see Faraday's law), and spring force.
Many forces (particularly those that depend on velocity) are not force fields. In these cases, the above three conditions are not mathematically equivalent. For example, the magnetic force satisfies condition 2 (since the work done by a magnetic field on a charged particle is always zero), but does not satisfy condition 3, and condition 1 is not even defined (the force is not a vector field, so one cannot evaluate its curl). Accordingly, some authors classify the magnetic force as conservative,[3] while others do not.[4] The magnetic force is an unusual case; most velocity-dependent forces, such as friction, do not satisfy any of the three conditions, and therefore are unambiguously nonconservative.

## Nonconservative forces

Nonconservative forces can only arise in classical physics due to neglected degrees of freedom. For instance, friction may be treated without resorting to the use of nonconservative forces by considering the motion of individual molecules; however that means every molecule's motion must be considered rather than handling it through statistical methods. For macroscopic systems the nonconservative approximation is far easier to deal with than millions of degrees of freedom. Examples of nonconservative forces are friction and non-elastic material stress.

# Potential energy

In physics, potential energy is the energy stored in a body or in a system due to its position in a force field or due to its configuration.[1] The SI unit of measure for energy and work is the Joule (symbol J). The term "potential energy" was coined by the 19th century Scottish engineer and physicist William Rankine.[2][3]

## 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 of inverse-square law forces. Any arbitrary reference state could be used, therefore it can be chosen based on convenience.
Typically the potential energy of a system depends on the 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 with gravitational force. 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.
Gravitational force keeps the planets in orbit around the Sun.
A trebuchet uses the gravitational potential energy of the counterweight to throw projectiles over long distances.
Consider a book placed on top of a table. When 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 same work will be done by the gravitational force. Thus, if the book falls off the table, this potential energy goes to accelerate the mass of the book and is converted into kinetic energy. When the book hits the floor this kinetic energy is converted into heat and sound by the impact.
The factors that affect an object's gravitational potential energy are its height relative to some reference point, its mass, and the strength of the gravitational field it is in. Thus, a book lying on a table has less gravitational potential energy than the same book on top of a taller cupboard, and less gravitational potential energy than a heavier book lying on the same table. An object at a certain height above the Moon's surface has less gravitational potential energy than at the same height above the Earth's surface because the Moon's gravity is weaker. Note that "height" in the common sense of the term cannot be used for gravitational potential energy calculations when gravity is not assumed to be a constant. The following sections provide more detail.

### Local 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 g = 9.81 m/s2 ("standard gravity"). In this case, a simple expression for gravitational potential energy can be derived using the W = Fd equation for work, and the equation
$W_F = -\Delta U_F.\!$
When accounting only for mass, gravity, and altitude, the equation is:
$U = mgh\!$
where U is the potential energy of the object relative to its being on the Earth's surface, m is the mass of the object, g is the acceleration due to gravity, and h is the altitude of the object.[4] If m is expressed in kilograms, g in meters per second squared and h in meters then U will be calculated in joules.
Hence, the potential difference is
$\,\Delta U = mg \Delta h.\$

### General formula

Gravitational field potential energy is determined using Newton's law of universal gravitation. :$U = m_1\phi_2 = m_1( \frac{-GM_2}{r})$
However, over large variations in distance, the approximation that g is constant is no longer valid, and we have to use calculus 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 r between the two bodies. Using that definition, the gravitational potential energy of a system of masses m1 and M2 at a distance r using gravitational constant G is
$U = -G \frac{m_1 M_2}{r}\ + K$,
where K is the constant of integration. Choosing the convention that K=0 makes calculations simpler, albeit at the cost of making U negative; for why this is physically reasonable, see below.
Given this formula for U, the total potential energy of a system of n bodies is found by summing, for all $\frac{n ( n - 1 )}{2}$ pairs of two bodies, the potential energy of the system of those two bodies.
Gravitational potential summation $U = - m (G \frac{ M_1}{r_1}+ G \frac{ M_2}{r_2})$
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. 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.
$U = - m (G \frac{ M_1}{r_1}+ G \frac{ M_2}{r_2})$
therefore,
$U = - m \sum G \frac{ M}{r}$,

### Why choose a convention where gravitational energy is negative?

Gravitational potential is a scalar potential energy per unit mass at each point in space associated with the force fields. Notice at r tends to infinity, ϕ tends to 0 :$\phi = -( \frac{GM}{r})$.
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 U becomes zero: r = 0 and $r=\infty$. 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.
The singularity 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 r, and would make calculations involving sums or differences of potential energies beyond what is possible with the real number system. Since physicists abhor infinities in their calculations, and r 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 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 of hydroelectricity. For example in Dinorwig, Wales, there are two lakes, one at a higher elevation than the other. At times when surplus electricity is not required (and so is comparatively cheap), water is pumped up to the higher lake, thus converting the electrical energy (running the pump) to gravitational potential energy. At times of peak demand for electricity, the water flows back down through electrical generator turbines, converting the potential energy into kinetic energy and then back into electricity. (The process is not completely efficient and much of the original energy from the surplus electricity is in fact lost to friction.) See also pumped storage.
Gravitational potential energy is also used to power clocks in which falling weights operate the mechanism.

## Elastic potential energy

Springs are used for storing elastic potential energy
Elastic potential energy is the potential energy of an elastic object (for example a bow or a catapult) that is deformed under tension or compression (or stressed in formal terminology). It arises as a consequence of a force that tries to restore the object to its original shape, which is most often the electromagnetic force between the atoms and molecules that constitute the object. If the stretch is released, the energy is transformed into kinetic energy.

### Calculation of elastic potential energy

The elastic potential energy stored in a stretched spring can be calculated by finding the work necessary to stretch the spring a distance x from its un-stretched length:
$U_e = -\int\vec{F}\cdot d\vec{x}$
an ideal spring will follow Hooke's Law:
${F = -k x}\,$
The work done (and therefore the stored potential energy) will then be:
$U_e = -\int\vec{F}\cdot d\vec{x}=-\int {-k x}\, dx = \frac {1} {2} k x^2.$
The units are in Joules.
The equation is often used in calculations of positions of mechanical equilibrium. More involved calculations can be found at elastic potential energy.

## 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 of chemical bonds within a molecule or otherwise. Chemical energy of a chemical substance can be transformed to other forms of energy by a chemical reaction. As an example, when a fuel is burned the chemical energy is converted to heat, same is the case with digestion of food metabolized in a biological organism. Green plants transform solar energy to chemical energy through the process known as photosynthesis, and electrical energy can be converted to chemical energy through electrochemical reactions.
The similar term chemical potential is used to indicate the potential of a substance to undergo a change of configuration, be it in the form of a chemical reaction, spatial transport, particle exchange with a reservoir, etc.

## Electric potential 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).
Plasma formed inside a gas filled sphere

### Electrostatic potential energy

In case the electric charge of an object can be assumed to be at rest, 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, in the absence of any non-electrical forces on the object. This energy is non-zero if there is another electrically charged object nearby.
The simplest example is the case of two point-like objects A1 and A2 with electrical charges q1 and q2. The work W required to move A1 from an infinite distance to a distance r away from A2 is given by:
$W=\frac{1}{4\pi\varepsilon_0}\frac{q_1q_2}{r},$
where ε0 is the electric constant.
This equation is obtained by integrating the Coulomb force between the limits of infinity and r.
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 m in an externally-produced magnetic B-field B has potential energy[5]
$U=-\mathbf{m}\cdot\mathbf{B}.$
The magnetic potential energy of a magnetization M in a field is
$U = -\frac{1}{2}\int \mathbf{M}\cdot\mathbf{B} dV,$
where the integral can be over all space or just the part where M is nonzero.[6]

## Nuclear potential energy

Nuclear potential energy is the potential energy of the particles inside an atomic nucleus. The nuclear particles are bound together by the strong nuclear force. Weak nuclear forces provide the potential energy for certain kinds of radioactive decay, such as beta decay.
Nuclear particles like protons and neutrons are not destroyed in fission and fusion processes, but collections of them have less mass than if they were individually free, and this mass difference is liberated as heat and radiation in nuclear reactions (the heat and radiation have the missing mass, but it often escapes from the system, where it is not measured). The energy from the Sun is an example of this form of energy conversion. In the Sun, the process of hydrogen fusion converts about 4 million tonnes of solar matter per second into electromagnetic energy, which is radiated into space.

## Relation between potential energy, potential and force

Potential energy is closely linked with forces. If the work done moving along a path which starts and ends in the same location is zero, then the force is said to be conservative and it is possible to define a numerical value of potential associated with every point in space. A force field can be re-obtained by taking the negative of the vector gradient of the potential field.
For example, gravity is a conservative force. The associated potential is the gravitational potential, often denoted by ϕ 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
$U = -\frac{G M m}{r},$
The gravitational potential (specific energy) of the two bodies is
$\phi = -\left( \frac{GM}{r} + \frac{Gm}{r} \right)= -\frac{G(M+m)}{r} = -\frac{GMm}{\mu r} = \frac{U}{\mu}.$
where μ is the reduced mass.
The work done against gravity by moving an infinitesimal mass from point A with U = a to point B with U = b is (ba) and the work done going back the other way is (ab) so that the total work done in moving from A to B and returning to A is
$U_{A \to B \to A} = (b - a) + (a - b) = 0. \,$
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
$U_{A \to B} = (b + c) - (a + c) = b - a \,$
as before.
In practical terms, this means that one can set the zero of U and ϕ anywhere one likes. One may set it to be zero at the surface of the Earth, or may find it more convenient to set zero at infinity (as in the expressions given earlier in this section).
A thing to note about conservative forces is that the work done going from A to B does not depend on the route taken. If it did then it would be pointless to define a potential at each point in space. An example of a non-conservative force is friction. With friction, the route taken does affect the amount of work done, and it makes little sense to define a potential associated with friction.
All the examples above are actually force field stored energy (sometimes in disguise). For example in elastic potential energy, stretching an elastic material forces the atoms very slightly further apart. The equilibrium between electromagnetic forces and Pauli repulsion of electrons (they are fermions obeying Fermi statistics) is slightly violated resulting in a small returning force. Scientists rarely discuss forces on an atomic scale. Often interactions are described in terms of energy rather than force. One may think of potential energy as being derived from force or think of force as being derived from potential energy (though the latter approach requires a definition of energy that is independent from force which does not currently exist).