In nature there are four different types of forces. The gravitational forces, the electromagnetic forces, the nuclear strong and nuclear weak forces. The first two type of forces are long range forces, in what it concern the distance over they are effective. In the following will will refer only to gravitational force, the study of electromagnetic forces being the same.
As we know the gravitational force can be written as
`F = G*(m*M)/R^2` and we can associate a force field to it
`E = GM/R^2`
(the force field is equal to the force that acts on the unity mass `m=1` kg).
Now, every force field can do work on an exterior body. The work of the gravitational force is just
`W =int_0^(+oo) F*dR =-G(mM)/R =-m*(GM/R) =-m*U` (1)
Experimentally it has been found that the work of the gravitational field doe not depend on the path taken. (The value of the integral above does not depend on the path between starting point `R=0` and last point `R=+oo` ). This means that the filed is conservative. In these conditions one can define a potential `U` of the field `E` (see the above relation) such that
`E= -grad U` (2)
Therefore a potential is defined for a conservative field using the above relation(2). As can be seen from (1) a high potential means that the field can do work on a mass moving it to a lower potential.
The mathematical study of potentials is known as potential theory; it is the study of harmonic functions on manifolds. This mathematical formulation arises from the fact that, in physics, the scalar potential is irrotational, and thus has a vanishing Laplacian — the very definition of a harmonic function.
In physics, 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 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).
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 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.
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.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 ΔU is the change in the potential energy associated with that particular force. Common notations for potential energy are U, Ep, and PE.
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.