Potential
energy

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This article is about a form of energy in physics. For the
statistical method, see Potential energy
statistics.

Potential energy

In the case of a bow and arrow, the
energy is converted from the potential energy in the archer's arm to the
potential energy in the bent limbs of the bow when the string is drawn
back. When the string is released, the potential energy in the bow limbs
is transferred back through the string to become kinetic energy in
the arrow as it takes flight.

Common symbol(s):

PE, U, or V

SI unit:

joule
(J)

Derivations from other quantities:

U = m · g · h (gravitational)

U = ½ · k · x2
(elastic)
U
= C · V2
/ 2 (electric)
U
= -m · B (magnetic)

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In physics,
potential energy is the energy of an object or a system due to the
position of the body or the arrangement of the particles of the system.[1]
The SI unit
for measuring work and energy is the joule (symbol J).

The term potential energy was coined by the 19th century Scottish
engineer and physicist William Rankine,[2][3]
although it has links to Greek philosopher Aristotle's concept of potentiality.
Potential energy is associated with a set of forces that act on a body in a way
that depends only on the body's position in space. This allows the set of forces
to be considered as having a specified vector at every point in space forming
what is known as a vector field of forces, or
a force field. If
the work of forces of this type acting on a body that moves from a start to an
end position is defined only by these two positions and does not depend on the
trajectory of the body between the two, then there is a function known as a
potential that can be evaluated at the two positions to determine this
work. Furthermore, the force field is defined by this potential function, also
called potential energy.

Contents

1 Overview

2 Work and potential
energy

2.1 Potential function for
near earth gravity

2.2 Potential function for
a linear spring

3 Reference
level

4 Gravitational potential
energy

4.1 Local
approximation

4.2 General
formula

4.3 Why choose a convention
where gravitational energy is negative?

4.4 Uses

5 Elastic potential
energy

5.1 Calculation of elastic
potential energy

6 Chemical potential
energy

7 Electric potential
energy

7.1 Electrostatic potential
energy

7.2 Magnetic potential
energy

8 Nuclear potential
energy

9 Relation between potential
energy, potential and force

10 Notes

11 References

12 External
links

Overview[edit]

Potential energy is often associated with restoring forces such as a spring or the force of
gravity. The action of
stretching the spring or lifting the 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.

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.

There are various types of potential energy, each associated with a
particular type of force. More specifically, every conservative force
gives rise to potential energy. 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 the Coulomb
force is called electric potential
energy; work of the strong nuclear
force or weak nuclear force
acting on the baryon charge is called
nuclear potential energy; work of intermolecular
forces is called intermolecular potential energy. Chemical potential energy,
such as the energy stored in fossil fuels, is the work
of the Coulomb force during rearrangement of mutual positions 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
mutual positions.

As a general rule, the work done by a conservative force F will be

where
is the change in the potential energy associated with that particular force.
Common notations for potential energy are U, V, and
Ep.

Work and potential
energy[edit]

The work of a force acting on a moving body yields a difference in potential
energy when the integration of the work is path independent. The scalar product
of a force F and the velocity v of its point of application
defines the power input to a system
at an instant of time. Integration of this power over the trajectory of the
point of application, d=x(t), defines the work input to the system
by the force.

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
x(t1) and x(t2) 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

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.

In this case, the application of the del
operator to the work function yields

and the force F is said to be "derivable from a potential."[4]

Because the potential U defines a force F at every point
x in space, the set of forces is called a force field. 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 body, that
is

Examples of work that can be computed from potential functions are gravity
and spring forces.[5]

Potential function for near earth gravity[edit]

Gravity exerts a constant downward force F=(0, 0,
Fz) 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
s(t) = (x(t), y(t), z(t)), such as the track
of a roller coaster is calculated using its velocity,
v=(vx, vy,
vz), to obtain

where the integral of the vertical component of velocity is the vertical
distance. Notice that the work of gravity depends only on the vertical movement
of the curve s(t).

The function U(s)=mgh is called the potential
energy of a near earth gravity field.

Potential
function for a linear spring[edit]

A horizontal spring exerts a force F=(kx, 0, 0) that is
proportional to its deflection in the x direction. The work of this
spring on a body moving along the space curve s(t) = (x(t),
y(t), z(t)), is calculated using its velocity,
v=(vx, vy,
vz), to obtain

For convenience, consider contact with the spring occurs at t = 0,
then the integral of the product of the distance x and the
x-velocity, xvx, is (1/2)x2.

The function U(x)= 1/2 kx2 is called the
potential energy of a linear spring.

Reference level[edit]

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[edit]

Main articles: Gravitational
potential and Gravitational
energy

Gravitational energy is the potential energy associated with gravitational force,
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.

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. 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. 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[edit]

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 = 9.8
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

The amount of gravitational potential energy possessed 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 W = Fd). The upward force required while moving at a
constant velocity is equal to the weight, mg, of an object, so the work
done in lifting it through a height h is the product mgh. Thus,
when accounting only for mass, gravity, and altitude, the
equation is:[6]

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.[7]
If m is expressed in kilograms, g in m/s2
and h in metres then U will be
calculated in joules.

Hence, the potential difference is

General formula[edit]

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

,

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
pairs of two bodies, the potential energy of the system of those two bodies.

Gravitational potential summation

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.

therefore,

,

Why choose a convention where gravitational energy is
negative?[edit]

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:
and .
The choice of
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

in the formula for gravitational potential energy means that the only other
apparently reasonable alternative choice of convention, with
for ,
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
at infinity is by far the more preferable choice, even if the idea of negative
energy in a gravity well 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[edit]

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.[citation
needed] (The process is not completely efficient and some
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.

It's also used by counterweights for lifting
up an Elevator, crane, or Sash
window.

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 obtained
from 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.

Elastic potential
energy[edit]

Springs are used for
storing elastic potential
energy

Archery is one of humankind's
oldest applications of elastic potential energy.

Main article: 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[edit]

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:

an ideal spring will follow Hooke's Law:

The work done (and therefore the stored potential energy) will then be:

The units are in joules (J).

The equation is often used in calculations of positions of mechanical
equilibrium. This equation can also be stated as:

Chemical potential
energy[edit]

Main article: Chemical 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[edit]

Main article: 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[edit]

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:

where ε0 is the vacuum permittivity.
This may also be written in a simpler form, resembling better the natural
parallelism with Newton's gravitation equation, by using the electrostatic
constant (Coulomb's
constant), defined as ke = 1 ⁄ 4πε0.

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[edit]

The energy of a magnetic moment m in an externally produced magnetic B-field B has potential energy[8]

The magnetization M in a field is

where the integral can be over all space or, equivalently, where M is nonzero.[9]
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 it is perpendicular to 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 the highest when they are near the
edge of their attraction, and the lowest when they pull together. Conversely,
like poles will have the highest potential energy when forced together, and the
lowest when they spring apart.[10][11]

Nuclear potential
energy[edit]

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[edit]

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
or ,
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

The gravitational potential (specific energy)
of the two bodies is

where
is the reduced mass.

The work done against gravity by moving an infinitesimal mass from
point A with
to point B with
is
and the work done going back the other way is
so that the total work done in moving from A to B and returning to A is

If the potential is redefined at A to be
and the potential at B to be ,
where
is a constant (i.e.
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

as before.

In practical terms, this means that one can set the zero of
and
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).

A conservative force can be expressed in the language of differential
geometry as a closed form. As
Euclidean space is contractible, its de Rham cohomology
vanishes, so every closed form is also an exact form, 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[edit]

^
McCall, Robert P.
(2010). "Energy, Work and Metabolism". Physics of the Human
Body. JHU Press. p. 74. ISBN 978-0-8018-9455-8.

^
William John Macquorn Rankine (1853) "On the
general law of the transformation of energy," Proceedings of the
Philosophical Society of Glasgow, vol. 3, no. 5, pages 276-280; reprinted
in: (1) Philosophical Magazine, series 4, vol. 5, no. 30, pages 106-117 (February 1853); and (2) W. J. Millar,
ed., Miscellaneous Scientific Papers: by W. J. Macquorn Rankine, ...
(London, England: Charles Griffin and Co., 1881), part II, pages 203-208.

^
Smith, Crosbie (1998).
The Science of Energy - a Cultural History of Energy Physics in Victorian
Britain. The University of Chicago Press. ISBN 0-226-76420-6.

^
J. R. Taylor, Classical Mechanics, University Science Books,
2005.

^
B. Paul, Kinematics and Dynamics of Planar Machinery,
Prentice-Hall, 1979.

^
Feynman, Richard P.
(2011). "Work and potential energy". The Feynman Lectures on
Physics, Vol. I. Basic Books. p. 13. ISBN 978-0-465-02493-3.

^
Hyperphysics - Gravitational Potential Energy

^
Aharoni, Amikam (1996).
Introduction to the theory of ferromagnetism (Repr. ed.). Oxford:
Clarendon Pr. ISBN 0-19-851791-2.

^
Jackson,
John David (1975). Classical electrodynamics (2d ed.). New York:
Wiley. ISBN 0-471-43132-X.

^
James D. Livingston, Rising Force: The Magic of
Magnetic Levitation - President and Fellows of Harvard College 2011, p.
152

^
Narinder Kumar, Comprehensive Physics XII,
Laxmi Publications 2004, p. 713

References[edit]

Serway, Raymond A.; Jewett, John W. (2010).
Physics for Scientists and Engineers (8th ed.). Brooks/Cole cengage. ISBN 1-4390-4844-4.

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

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