yawiki.org entry for 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).

    Potential energy
        Gravitational potential energy
            Uses
            Simplified calculation
            Extended calculation
            Gravitational potential
        Elastic potential energy
        Chemical energy
        Electrical potential energy
        Rest mass energy
        Relation between potential energy and force
        Graphical representation
        See also

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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 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.

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Uses
It helps with science projects on catapults
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 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.

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Simplified calculation
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. 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, U_g, is given by

$E_g = m g h ,$

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.

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Extended calculation
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, U_g, of a system of two masses:

where

$m_1$ and $m_2$ are the masses of the two objects

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

$h_1$ is the reference level (the separation at which potential energy is considered to be zero)

$h_2$ is the actual distance between the objects.

Subject to the caveats mentioned above, the distances

h_1

and

h_2

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,

m_1

and

m_2

are respectively the masses of the earth and the object,

h_1

is the distance from the earth's centre to the earth's surface, and

h_2

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:

$U_g = rac$

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.

U_g 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.

See also two-body problem and gravitational binding energy.

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Gravitational potential

Gravitational potential is the potential energy per unit mass of an object due to its position in a gravitational field. The gravitational potential due to a point mass:

$E(r) = rac$

where:

$G$ is the universal gravitational constant,

$r$ is the distance to the center of mass of the object,

$m$ is the mass of the point object.

In astrodynamics 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 zonal coefficients:

$E(r,phi) = rac left 1 - sum_{n=2}^N J_{n} left (rac{R}{r}
ight)^2 P_n (sin phi)
ight$

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Elastic potential energy
Elastic potential energy in one dimensional is defined as the work done by an elastic force. In the case of Hooke's law, it is equal to:

$U_e = kx^2$

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 (see Main Article: Elastic potential energy).

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 ε_ij:

$f(epsilon_) = lambda left ( sum_^ epsilon_$

ight)^2+2mu sum_^ sum_^ epsilon_^2

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

$sigma_ = left ( rac$

ight)_S

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Chemical energy
Chemical energy is a form of potential energy related to the breaking and forming of chemical bonds. It is stored in food, fuels and batteries, and is released as other forms of energy during chemical reactions. Green plants trap light energy from the sun and convert it to chemical energy during photosynthesis. Other forms of energy, such as electrical energy, are sometimes converted into chemical energy to allow them to be stored for future use.

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Electrical potential energy
The electrical potential energy of an electrically charged object 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 d away from A₂ is given by:

$W=krac d$

where k is Coulomb's constant, equal to

rac 1

.

This equation is obtained by integrating the Coulomb force between the limits of infinity and d.

A related measure called electrical potential is equivalent to electrical potential energy divided by electric charge.

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Rest mass energy
Albert Einstein's famous equation, derived in his special theory of relativity, 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 speed of light in a vacuum. (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 petajoules per kilogram ≈ 21 megaton of TNT per kilogram)

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Relation between potential energy 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 vector gradient of the potential field.

For example, gravity is a conservative force. 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

$U_ = (b - a) + (a - b) = 0 ,$

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

$U_ = (b + c) - (a + c) = b - a ,$

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 Earth 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 electromagnetic forces try to keep the atoms at their optimal distance and so elastic potential is actually electromagnetic potential. Having said that, scientists rarely talk about forces on an atomic 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 differential geometry as a closed form. Because Euclidean space is contractible, its de Rham cohomology 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.

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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.

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