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## PowerPedia:Potential energy

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Potential energy is Energy that is "captured" in an object, with the potential to be released. There are various different types of potential energy. Many of these – such as gravitational, elastic, or electrical potential energy – arise from the relative positions or configurations of objects. The potential energy may then be defined as the work that must be done against a particular force – in these examples, gravitational, electrical or elastic force – so as to achieve that configuration. Chemical potential energy is slightly different, at least in its macroscopic manifestation: it is the energy that is available for release from chemical reactions (for example, by burning a fuel).

#### Electrical potential energy

The electrical potential energy of an Electric charge object is defined as the There was an error working with the wiki: Code[1] 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 d away from A2 is given by:

:W=k\frac {q_1q_2} d

where k is There was an error working with the wiki: Code[8], equal to \frac 1 {4\pi\epsilon_0}.

This equation is obtained by integrating the There was an error working with the wiki: Code[9] between the limits of infinity and d. A related measure called There was an error working with the wiki: Code[10] is equivalent to electrical potential energy divided by electric charge.

#### Chemical potential energy

Chemical potential energy is a form of potential energy related to the breaking and forming of Chemical bonds.

#### Gravitational potential energy

Gravitational potential energy is the energy that would be released if an object in a gravitational field (such as the earth's gravitational field) were allowed to fall from its current position to a given reference level (such as the surface of the earth). Equivalently, it is the energy required to raise the object from the reference level to the given height. For example, a book lying on a table has greater gravitational potential energy than the same book on the floor, but less than if it were on top of a tall cupboard. To raise the book from the floor to the table, work must be done, and energy supplied. (If the book is lifted by a person then this is provided by the chemical energy obtained from that person's food and then stored in the chemicals of the body.) Assuming perfect efficiency (no energy losses), the energy supplied to lift the book is exactly the same as the increase in the book's gravitational potential energy. The book's potential energy can be realised (released) by knocking it off the table. As the book falls, its potential energy is converted to 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: the mass of the object, the distance that it is raised, and the gravitational field strength. For example, raising the same object to the same height on the Moon would require less energy than on earth because the force of gravity on the Moon's surface is weaker. See the formulas below for more details.

##### Uses

Gravitational potential energy has a number practical uses, notably the generation of hydroelectricity. For example in Dinorwig, Wales there are two lakes, one higher than the other. At times when surplus electricity is not required (and so is cheap), water is pumped up to the higher lake, converting the electrical energy to gravitational potential energy. At times of peak demand for electricity, the water flows back down through turbines, converting the potential energy 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.) Assuming that the opposing gravitational force is constant, the work done in raising an object is equal to the force applied multiplied by the distance through which the object is raised.

##### Mathematics

The gravitational force that must be overcome is equal to the object's mass multiplied by the acceleration due to gravity, so the object's gravitational potential energy, Ug, is given by

: U_g = m g h \,

where

: m\, is the mass of the object

: g\, is the acceleration due to gravity (approximately 9.8 m/s2 at the earth's surface)

: h\, is the height to which the object is raised, relative to a given reference level (such as the earth's surface).

When applying this equation it is essential to use consistent units. Most scientific work is now done in SI units, in which case mass is measured in kilograms (kg), acceleration in metres per second squared (m/s2), and distance (here height) in metres (m). The resulting energy is expressed in joules (kg m²/s2). The equation shows that gravitational potential energy is proportional to both mass and height. For example, raising two similar objects, or raising the same object twice as far, doubles the potential energy.

The "mgh" formula works well provided that the acceleration due to gravity, g, is very nearly constant over the distance h. On or close to the surface of the earth this assumption is reasonable, but over the much larger distances applying, for example, to spacecraft and astronomical bodies, it is not. To calculate potential energy with varying g it is necessary to sum all the individual increments of potential energy as the masses are separated, taking account of the varying value of g as we go. In the limit, as the increments become "infinitely small", the sum becomes an integral.

To simplify the evaluation of the integral we can make the assumption that the gravitational forces act as if the objects' masses were concentrated at their respective centres of mass. This assumption is mathematically exactly correct for a spherically symmetrical object (such as, to a reasonable approximation, a planet). It is not generally correct in other cases, though if the dimensions of an object are very small compared to the distance of separation then it is reasonable to consider it as a point mass and ignore the details of its shape. With this simplifying assumption, integrating force over distance leads to the following general expression for the gravitational potential energy, Ug, of a system of two masses:

{|

|-

|style="width:15pt"|

|U_g\,

|= \int_{h_1}^{h_2} {G m_1 m_2 \over r^2} dr

|-

|

|

|= G m_1 m_2 \left ( \frac{1}{h_1} - \frac{1}{h_2} \right )

|}

where

:m1 and m2 are the masses of the two objects

:G is the gravitational constant (not to be confused with the g used earlier)

:h1 is the reference level (the separation at which potential energy is considered to be zero)

:h2 is the actual distance between the objects.

Subject to the caveats mentioned above, the distances h1 and h2 are measured between the objects' centres of mass.

For example, in the case of a small object above the surface of the earth, with reference level at the surface, m1 and m2 are respectively the masses of the earth and the object, h1 is the distance from the earth's centre to the earth's surface, and h2 is the distance from the earth's centre to the object. If we try to calculate an "absolute" potential energy by setting the reference level at zero then the formula "blows up" with division by zero. In other words, we can only actually use this formula to measure the difference in potential energy between one non-zero separation and another. In practice it is often convenient to take the reference level at infinite separation (i.e. h_1 = \infty), in which case the formula becomes:

:U_g = \frac{-G m_1 m_2}{r}

where r is now the distance between the centres of mass of the two objects (again noting the earlier caveats). For a small object above the surface of the earth, r is the distance from the object to the earth's centre (and similarly for other spherical bodies). Using this convention, potential energy is zero when r is infinitely large, and negative at any finite r. However, the difference in potential energy at different values of r – the quantity we are actually interested in – takes the expected sign.

Ug as calculated above measures the potential energy of the whole system. This can be visualised as if two bodies in space were released from rest and allowed to come together under the force of gravity. The sum of the kinetic energy gained by the two objects is exactly equal to the decrease in the potential energy of the system. The ratio of the objects' individual kinetic energy gains is equal to the reciprocal of the ratio of their masses. So, in the case of a relatively light object falling towards a very massive object (such as the earth), the contribution from the massive object is insignificant. In some sense, therefore, we can say that almost all the potential energy of the system is embodied in the light object, and almost none in the very massive object.

##### Related phenomenon

Related concepts include classical mechanics the There was an error working with the wiki: Code[11] and There was an error working with the wiki: Code[12]. The There was an error working with the wiki: Code[11] is to determine the motion of two point particles that interact only with each other. Common examples include the Moon orbiting the Earth, a planet orbiting a star, two stars orbiting each other (a binary star), and a classical electron orbiting an atomic nucleus. As described below, Newton's laws of motion allow us to reduce the two-body problem to an equivalent one-body problem, i.e., to solving for the motion of one particle in an external potential. Since one-body problems can usually be solved exactly, the corresponding two-body problem can also be solved. By contrast, the three-body problem (and, more generally, the n-body problem for n\geq 3) cannot be solved, except in special cases. Any classical system of two particles is, by definition, a two-body problem. In many cases, however, one particle is significantly heavier than the other, e.g., the Earth and the Sun. In such cases, the heavier particle is approximately the center of mass, and the reduced mass is approximately the lighter mass. Hence, the heavier mass may be treated roughly as a fixed center of force, and the motion of the lighter mass may be solved for directly by one-body methods. In other cases, however, the masses of the two bodies are roughly equal, so that neither of them can be approximated as being at rest. Astronomical examples include:

a binary star, e.g. Alpha Centauri (approx. the same mass)

a double planet, e.g. Pluto with its moon Charon (mass ratio 0.147)

The gravitational binding energy of an object consisting of loose material, held together by gravity alone, is the amount of energy required to pull all of the material apart, to infinity. It is also the amount of energy that is liberated (usually in the form of heat) during the accretion of such an object from material falling from infinity. The gravitational binding energy of a system is equal to the negative of the gravitational potential energy. For a system consisting of a celestial body and a satellite, the gravitational binding energy will have a larger absolute value than the potential energy of the satellite with respect to the celestial body, because for the latter quantity, only the separation of the two components is taken into account, keeping each intact.

##### Gravitational potential

Gravitational potential is the potential energy per unit mass of an object due to its position in a There was an error working with the wiki: Code[14]. The gravitational potential due to a There was an error working with the wiki: Code[15]:

:U(r) = \frac{-Gm}{r} \

where:

:G \ is the There was an error working with the wiki: Code[16],

:r \ is the distance to the center of mass of the object,

:m \ is the mass of the point object.

In There was an error working with the wiki: Code[17] the gravitational potential function has to account for the non-spherical and non-homogeneous nature of typical sources of gravitational potential. In this case a gravitational potential may depend on polar \phi\!\, and azimuth \lambda\!\, direction of vector r\!\,.

The most widely used form of the gravitational potential function depends on \phi\!\, (latitude) and potential coefficients, Jn, called the There was an error working with the wiki: Code[18]:

: U(r,\phi) = \frac{GM}{r} \left [1 - \sum_{n=2}^N J_{n} \left (\frac{R}{r} \right)^2 P_n (\sin \phi) \right ]

#### Elastic potential energy

Elastic potential energy in one dimensional is defined as the Mechanical work done by an There was an error working with the wiki: Code[19]. In the case of There was an error working with the wiki: Code[20], it is equal to:

: U_e = {1\over2}kx^2

where k is the spring constant (a measure of the stiffness of the spring), expressed in N/m, and x is the displacement from the equilibrium position, expressed in metres (which is part of There was an error working with the wiki: Code[21]).

In the general case, elastic energy is given by the Helmholtz potential per unit of volume f as a function of the strain tensor components ?ij:

: f(\epsilon_{ij}) = \lambda \left ( \sum_{i=1}^{3} \epsilon_{ii}\right)^2+2\mu \sum_{i=1}^{3} \sum_{j=1}^{3} \epsilon_{ij}^2

Where ? and ? are the Lamé elastical coefficients. The connection between stress tensor components and strain tensor components is:

: \sigma_{ij} = \left ( \frac{\partial f}{\partial \epsilon_{ij}} \right)_S

#### Rest mass energy

There was an error working with the wiki: Code[22]'s famous equation, derived in his There was an error working with the wiki: Code[23], can be written:

:E_0 = m c^2 \,

where E0 is the rest Mass energy, m is the rest mass of the body, and c is the There was an error working with the wiki: Code[24] in a There was an error working with the wiki: Code[25]. (The subscript zero is used here to distinguish this form of energy from the others that follow. In most other contexts, the equation is written with no subscript.)

The rest mass energy is the amount of energy inherent in the mass when it is at rest. This equation quantifies the equivalence of mass and energy: A small amount of mass is equivalent to a very large amount of energy. (i.e., 90 There was an error working with the wiki: Code[2] per kilogram)

#### Relation between potential energy and force

Potential energy is closely linked with There was an error working with the wiki: Code[3]. 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 There was an error working with the wiki: Code[4] of the potential field. For example, gravity is a There was an error working with the wiki: Code[26]. The work done by a unit mass going from point A with U = a to point B with U = b by gravity is (b - a) and the work done going back the other way is (a - b) so that the total work done from

: U_{A \to B \to A} = (b - a) + (a - b) = 0 \,

If we redefine the potential at A to be a + c and the potential at B to be b + c [where c can be any number, positive or negative, but it must be the same number for all points] then the work done going from

: U_{A \to B} = (b + c) - (a + c) = b - a \,

as before.

In practical terms, this means that you can set the zero of U anywhere you like. You might set it to be zero at the surface of the There was an error working with the wiki: Code[27] or you might find it more convenient to set it zero at infinity. 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 you take does affect the amount of work done, and it makes no sense at all 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. Powerful There was an error working with the wiki: Code[5] scale. Everything is phrased in terms of energy rather than force. You can think of potential energy as being derived from force or you can think of force as being derived from potential energy. A conservative force can be expressed in the language of There was an error working with the wiki: Code[6]. Because Euclidean space is There was an error working with the wiki: Code[7], its There was an error working with the wiki: Code[28] vanishes, so every closed form is exact, i.e., is the gradient of a scalar field. This gives a mathematical justification of the fact that all conservative forces are gradients of a potential field.

#### Graphical representation

A graph of a 1D or 2D potential function with the function value scale increasing upward is useful to visualize the potential field: a ball rolling to the lowest part corresponds to an object such as a mass or charge being attracted. When using this type of analogy, a mass, being an area of attraction, is often called a gravitational well, or potential well.

#### Related concepts

energy

binding energy

kinetic energy

potential difference

#### References and external articles

Serway, Raymond A. Jewett, John W. (2004). Physics for Scientists and Engineers (6th ed.). Brooks/Cole. ISBN 0-534-40842-7.

Tipler, Paul (2004). Physics for Scientists and Engineers: Mechanics, Oscillations and Waves, Thermodynamics (5th ed.). W. H. Freeman. ISBN 0-7167-0809-4.

Chandrasekhar, S. 1939, An Introduction to the Study of Stellar Structure (Chicago: U. of Chicago reprinted in New York: Dover), section 9, eqs. 90-92, p. 51 (Dover edition)

Lang, K. R. 1980, Astrophysical Formulae (Berlin: Springer Verlag), p. 272

There was an error working with the wiki: Code[1], Wikipedia: The Free Encyclopedia. Wikimedia Foundation.