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# Potential energy

Potential energy or potential is energy stored in an object or substance. In most scientific circles, potential energy is symbolized by $U$.

Gravitational potential energy This energy is stored as a result of the elevated position of an object such as a rock on top of a hill or water behind a dam. It is stated as

$U_g = m g h,$
where $m$ is the mass of the object in kilograms, $g$ the acceleration due to gravity in m/s2 and $h$ the height in meters above a chosen reference level.

In relation with spacecraft and astronomy g is not constant and the formula becomes an integral. In the case of a sphere of uniform mass (such as a planet), with h measured above the surface, the integral takes the form:

$U_g = \int_{h_0}^{h + h_0} {GmM \over r^2} dr$

Where $h_0$ is the radius of the sphere, M is the mass of the sphere, and G is the gravitational constant.

If h is instead taken to be the distance from the center of the sphere, then outside the sphere the potential energy relative to that at the center has two terms:

$U_g = \int_{h_0}^h {GmM \over r^2} dr + \int_0^{h_0} {GmM \over h_0^2} {r \over h_0} dr$,

which evaluates to:

$U_g = GmM [{1 \over h_0} - {1 \over h}] + {1 \over 2} {GmM \over h_0} = GmM [{3 \over 2h_0} - {1 \over h}]$

[We may also want to link to an explanation of that second term (the gravitational forces created by hollow spherical shells)]

A frequently adopted convention is that an object infinitely far away from an attracting source has zero potential energy. Relative to this, an object at a finite distance r from a source of gravitation has negative potential energy. If the source is approximated as a point mass, the potential energy simplifies to:

$U_g = - {GmM \over r}$

This energy is stored as the result of a deformed solid such as a stretched spring. It is stated as

$U_e = {k x^2 \over 2},$
where $k$ is the spring constant, expressed in N/m, and $x$ is the displacement from the relaxed position, expressed in meters.

Chemical potential energy This energy is stored in a molecular substance such as a hydrocarbon, which may be released by a chemical reaction (see oxidation)

Electrical potential energy The electrical potential energy per unit charge is called electrical potential. It is expressed in volt. The fact that a potential is always relative to a reference point is often made explicit by using the term potential difference. The term voltage is also common.

Relation between potential energy and force Potential energy is closely linked with forces. If the work done going around a loop 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. 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

$U_{A \to B \to A} = (b - a) + (a - b) = 0$
The nice thing about potential energy is that you can add any number to all points in space and it doesn't affect the physics. 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 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.

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