Potential energy: 2011

Monday 4 July 2011

Overview

Potential energy exists when a force acts upon an object that tends to restore it to a lower energy configuration. 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 action of stretching the spring or lifting the mass requires 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 stretch the spring is stored in the metal. According to the law 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 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

$\,W = -\Delta U$

where

Δ U

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

Gravitational potential energy

Main articles: Gravitational potential and Gravitational energy

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

The 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. 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 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.

General formula

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

$U = -G \frac{m_1 m_2}{R}\ + K$ ,

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 $\frac{n ( n - 1 )}{2}$ 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.

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 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

φ

V

, 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

$U = -\frac{G M m}{r},$

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

$\phi = -\left( \frac{GM}{r} + \frac{Gm}{r} \right)= -\frac{G(M+m)}{r} = -\frac{GMm}{\mu r} = \frac{U}{\mu}.$

where

μ

is the reduced mass.
The work done against gravity by moving an infinitesimal mass from point A with

U = a

to point B with

U = b

(b - a)

and the work done going back the other way is

(a - b)

so that the total work done in moving from A to B and returning to A 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

φ

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).

Chemical potential energy

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

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

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:

$W=\frac{1}{4\pi\varepsilon_0}\frac{q_1q_2}{r},$

where ε₀ 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.

Elastic potential energy

Springs are used for storing 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

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:

$U_e = -\int\vec{F}\cdot d\vec{x}$

an ideal spring will follow Hooke's Law:

${F = -k x}\,$

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

$U_e = -\int\vec{F}\cdot d\vec{x}=-\int {-k x}\, dx = \frac {1} {2} k x^2.$

The units are in Joules.
The equation is often used in calculations of positions of mechanical equilibrium. More involved calculations can be found at elastic potential energy.