Potential energy

Potential energy is energy which depends on mutual positions of bodies. The energy is defined as a work against a specific force such as gravity, an elastic force of a spring in a clockwork motor, electric force, etc (and is usually named after that specific force).

Examples

A book on a table has greater gravitational potential energy than the same book on the floor. However the same book has less gravitational potential energy than if it were even higher. In raising the book from the floor to the table, work was done by someone which is now stored as potential energy. (This energy was provided by the chemical energy stored in that person's food, and then in the chemicals of their body). The presence of this potential energy could be demonstrated by sliding the book off the table. The book would gain kinetic energy from its velocity until it reached the floor. The kinetic energy would then be converted into heat and sound by the impact.

In another example in Wales at Dinorwig there are two lakes, one higher than the other. At times when surplus electricity is not required (and so is cheap), water is pumped to the higher lake. At times of peak demand for electricity, the water flows through turbines and generates electricity once more (see also pumped storage). (The process is not completely efficient and much of the original energy from the surplus electricity is wasted by friction). In this example the potential energy is stored by doing work against the force of gravity.

The factors that affect the amount of gravitational potential energy that is created are: the mass of the object, the distance that it is raised and the gravitational field strength. 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 less.

Calculation

The work done in raising an object is the force applied multiplied by the distance that it was raised. Thus raising two similar objects or raising the object twice as far would produce twice as much potential energy. The gravitational force that must be overcome is the mass multiplied by the acceleration of gravity. The gravitational potential energy of a body is:

$m g h$ where

m is the mass of the object,
g the acceleration due to gravity and
h the height above a chosen reference level

(typical units would be kilograms for m, metres per second squared for g, and metres for h).

Types

Gravitational potential energy

The gravitational potential energy

U_g

of an object is equal to the work done by a constant gravitational force

F=mg

on the object to move it from a given position to some reference position by a distance h between them, and is equal to

U_g = m g h \,

where

m

is the mass of the object, and

g

the acceleration due to gravity.

In relation to 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} {G m_1 m_2 \over r^2} dr

where

h_0

is the radius of the sphere,

m_2

is the mass of the sphere, and

G is the gravitational constant.

If the structure of an object is spherically symmetrical, e.g. the Earth, then the potential is the same as that of a point object having the same mass. Also, the concept of center of mass generalizes this idea to an object of any shape and density.

As a result, the gravitational potential energy of an object relative to some other object is:

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

where

U_g

is the potential energy of the object in question defined to equal 0 when r=∞,

m_1

and

m_2

are the masses of each object, and

r is the distance between the centers of mass of the two objects.

Note that the potential energy of both objects is the same, so the potential energy of the whole system is 2× $U_g$ (as defined above). Note also that the convention for describing potential energy is to define 0 potential energy at r=∞. This is done for mathematical simplicity in some cases. The consequence of this is that the potential energy described using this convention is always negative. However, this does not imply that differences in potential are always negative.

When an object is lifted, the object and the Earth are moved apart such that each is moved a distance inversely proportional to its mass. In both moves, the force is of the same magnitude, so the energy involved in moving the Earth is much smaller. Similarly when the object is dropped the velocities are inversely proportional to the masses, so the kinetic energy also. See also two-body problem and gravitational binding energy.

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:

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

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:

U(r,\phi) = \frac{GM}{r} \left - \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 work done by an elastic force. In the case of Hooke's law, 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 (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_{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

Chemical energy

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

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)

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=\frac {kq_1q_2} d

where k is Coulomb's constant, equal to

\frac 1 {4\pi\epsilon_0}

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.

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_{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

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.

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.

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.

References

Energy | Introductory physics | Gravity | Astrodynamics

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Examples

Calculation

Types

Gravitational potential energy

Gravitational potential

Elastic potential energy

Chemical energy

Rest mass energy

Electrical potential energy

Relation between potential energy and force

Graphical representation

See also

References