Potential energy is
energy
stored within a physical system as a result of the position or
configuration of the different parts of that system. It is called
potential energy because it has the potential to be
converted into other forms of energy, such as
kinetic energy, and to do
work in the process. The
SI unit of measure for energy
(including potential energy) and work is the
joule (symbol J).
The term "potential energy" was coined by the 19th century Scottish
engineer and physicist
William
Rankine.
Overview
Potential energy is energy that is stored within a system. It
exists when there is a
force that tends to
pull an object back towards some lower energy position. This force
is often called a
restoring force.
For example, when a spring is stretched to the left, it exerts a
force to the right so as to return to its original, unstretched
position. Similarly, when a mass is lifted up, the force of
gravity will act so as to bring it back
down. The initial action of stretching the spring or lifting the
mass both require energy to perform. The energy that went into
lifting up the mass is stored in its position in the gravitational
field, while similarly, the energy it took to deform the spring is
stored in the metal. According to the principle of
conservation of energy, energy cannot
be created or destroyed; hence this energy cannot disappear.
Instead, it is stored as potential energy. If the spring is
released or the mass is dropped, this stored energy will be
converted into
kinetic energy by the
restoring force, which is
elasticity in the case of the spring,
and
gravity in the case of the mass. Think
of a roller coaster. When the coaster climbs a hill it has
potential energy. At the very top of the hill is its maximum
potential energy. When the car speeds down the hill potential
energy turns into kinetic. Kinetic energy is greatest at the
bottom.
The more formal definition is that potential energy is the energy
difference between the energy of 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
elastic force is called elastic
potential energy; work of
gravitational force is called gravitational
potential energy, work of the
Coulomb
force is called
electric
potential energy; work of
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 motion of particles
and potential energy of their mutual positions.
As a general rule, the work done by a conservative force
F
will be
- \,W = -\Delta U
where \Delta U is the change in the potential energy associated
with that particular force. The most common notations for potential
energy are
PE and
U.
Reference level
The potential energy is a function of the state a system is in,
defined relative to an arbitrary reference energy. This energy can
be chosen for convenience, and/or such that for a particular state
the potential energy is
zero. Typically the
reference is chosen such that the potential energy depends on the
relative positions of its components only.
In the case of
inverse-square law
forces, a common choice is to define the potential energy as
tending to zero when the distances between all bodies tend to
infinity.
Gravitational potential energy
Gravitational energy is the potential energy associated with
gravitational force. If an
object falls from one point to another 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.
For example, 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 called 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. (This follows from
Newton's law of gravitation
because the mass of the moon is much smaller than that of the
Earth.) It is important to 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.
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 ("
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.. 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.
However, if the force of gravity varies too much for this
approximation to be valid, then we have to use the general,
integral definition of work to determine gravitational potential
energy. Now taking the arbitrary reference point where
U =
0 to be when the two objects are infinite distance apart:
The (now negative) gravitational potential energy of a system of
masses
m_{1} and
m_{2} at a
distance
R using
gravitational constant G
is
- U = -G \frac{m_1 m_2}{R}.
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 from to .
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.
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.
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
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 (often termed under the word
stress by physicists). 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, it is transformed into
mechanical energy.
Calculation of elastic potential energy
In the case of a spring of natural length
l and
modulus of elasticity λ under
an extension of
x, elastic potential energy can be
calculated using the formula:
- E = \frac{\lambda x^2}{2l}
This formula is obtained from the integral of
Hooke's Law:
- U_e = -\int\vec{F}\cdot d\vec{x}=-\int {-k x}\, dx = \frac {1}
{2} k x^2
The equation is often used in calculations of positions of
mechanical equilibrium.
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_{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 λ and μ 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
For a material of Young's modulus,
Y (same as modulus of
elasticity
λ), cross sectional area,
A_{0}, initial length,
l_{0},
which is stretched by a length, \Delta l:
- U_e = \int {\frac{Y A_0 \Delta l} {l_0}}\, dl = \frac {Y A_0
{\Delta l}^2} {2 l_0}
- where U_{e} is the elastic potential
energy.
The elastic potential energy per unit volume is given by:
- \frac{U_e} {A_0 l_0} = \frac {Y {\Delta l}^2} {2 l_0^2} = \frac
{1} {2} Y {\varepsilon}^2
- where \varepsilon = \frac {\Delta l} {l_0} is the strain in the
material.
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 by chemists to indicate the potential of a
substance to undergo a chemical reaction.
Electrical 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).
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
A
_{1} and A
_{2} with electrical charges
q_{1} and
q_{2}. The work
W required to move A
_{1} from an infinite distance
to a distance
r away from A
_{2} is given by:
- W=\frac{1}{4\pi\varepsilon_0}\frac{q_1q_2}{r},
where
ε_{0} 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.
Electrodynamic potential energy
In case a charged object or its constituent charged particles are
not at rest, it generates a
magnetic
field giving rise to yet another form of potential energy,
often termed as
magnetic potential energy. This
kind of potential energy is a result of the phenomenon
magnetism, whereby an object that is magnetic has
the potential to move other similar objects. Magnetic objects are
said to have some
magnetic
moment. Magnetic fields and their effects are best
studied under
electrodynamics.
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 light, 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
vector gradient of the potential field.
For example, gravity is a
conservative force. The associated
potential is the
gravitational
potential, often denoted by \Phi or V, corresponding to the
energy of a unit mass as a function of position. As the potential
energy of a particle of mass
m at a distance
R
from a body of mass
M is
- U = -\frac{G M m}{R},
the gravitational potential at a distance
R from a body of
mass
M is given by
- \Phi = -\frac{G M}{R}.
The work done against gravity by moving a unit mass from point A
with U = a to point B with U = b is (b - a) and the work done going
back the other way is (a - b) so that the total work done going
from A to B 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
\Phi 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 exact, for example, 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.
Notes
- Hyperphysics - Gravitational Potential
Energy
References