Potential energy

In physics, potential energy is the energy of an object or system due to the body's position relative to other objects, or the configuration of its particles. The energy is equal to the work done against any restoring forces, such as gravity or those in a spring.
The term potential energy was introduced by the 19th-century Scottish engineer and physicist William Rankine, although it has links to the ancient Greek philosopher Aristotle's concept of potentiality.
Common types of potential energy include gravitational potential energy, the elastic potential energy of a deformed spring, and the electric potential energy of an electric charge and an electric field. The unit for energy in the International System of Units is the joule.
Potential energy is associated with forces that act on a body in a way that the total work done by these forces on the body depends only on the initial and final positions of the body in space. These forces, whose total work is path independent, are called conservative forces. If the force acting on a body varies over space, then one has a force field; such a field is described by vectors at every point in space, which is, in turn, called a vector field. A conservative vector field can be simply expressed as the gradient of a certain scalar function, called a scalar potential. The potential energy is related to, and can be obtained from, this potential function.

Overview

There are various types of potential energy, each associated with a particular type of force. 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 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 configurations 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 configuration.
Forces derivable from a potential are also called conservative forces. The work done by a conservative force is
where is the change in the potential energy associated with the force. The negative sign provides the convention that work done against a force field increases potential energy, while work done by the force field decreases potential energy. Common notations for potential energy are PE, U, V, and E_p.
Potential energy is the energy by virtue of an object's position relative to other objects. Potential energy is often associated with restoring forces such as a spring or the force of gravity. The action of stretching a spring or lifting a 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.
Consider a ball whose mass is dropped from height. The acceleration of free fall is approximately constant, so the weight force of the ball is constant. The product of force and displacement gives the work done, which is equal to the gravitational potential energy, thus
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.

History

From around 1840 scientists sought to define and understand energy and work.
The term "potential energy" was coined by William Rankine a Scottish engineer and physicist in 1853 as part of a specific effort to develop terminology. He chose the term as part of the pair "actual" vs "potential" going back to work by Aristotle. In his 1867 discussion of the same topic Rankine describes potential energy as 'energy of configuration' in contrast to actual energy as 'energy of activity'. Also in 1867, William Thomson introduced "kinetic energy" as the opposite of "potential energy", asserting that all actual energy took the form of ². Once this hypothesis became widely accepted, the term "actual energy" gradually faded.

Work and potential energy

Potential energy is closely linked with forces. If the work done by a force on a body that moves from A to B does not depend on the path between these points, then the work of this force measured from A assigns a scalar value to every other point in space and defines a scalar potential field. In this case, the force can be defined as the negative of the vector gradient of the potential field.
If the work for an applied force is independent of the path, then the work done by the force is evaluated from the start to the end of the trajectory of the point of application. This means that there is a function U, called a "potential", that can be evaluated at the two points x_A and x_B 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
where C is the trajectory taken from A to B. Because the work done is independent of the path taken, then this expression is true for any trajectory, C, from A to B.
The function U is called the potential energy associated with the applied force. Examples of forces that have potential energies are gravity and spring forces.

Derivable from a potential

In this section the relationship between work and potential energy is presented in more detail. The line integral that defines work along curve C takes a special form if the force F is related to a scalar field U′ so that
This means that the units of U′ must be this case, work along the curve is given by
which can be evaluated using the gradient theorem to obtain
This shows that when forces are derivable from a scalar field, the work of those forces along a curve C is computed by evaluating the scalar field at the start point A and the end point B of the curve. This means the work integral does not depend on the path between A and B and is said to be independent of the path.
Potential energy is traditionally defined as the negative of this scalar field so that work by the force field decreases potential energy, that is
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". This also necessarily implies that F must be a conservative vector field. The potential U defines a force F at every point x in space, so the set of forces is called a force field.

Computing potential energy

Given a force field F, evaluation of the work integral using the gradient theorem can be used to find the scalar function associated with potential energy. This is done by introducing a parameterized curve from to, and computing,
For the force field F, let, then the gradient theorem yields,
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 point of application, that is
Examples of work that can be computed from potential functions are gravity and spring forces.

Potential energy for near-Earth gravity

For small height changes, gravitational potential energy can be computed using
where m is the mass in kilograms, g is the local gravitational field, h is the height above a reference level in metres, and U is the energy in joules.
In classical physics, gravity exerts a constant downward force 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, such as the track of a roller coaster is calculated using its velocity,, to obtain
where the integral of the vertical component of velocity is the vertical distance. The work of gravity depends only on the vertical movement of the curve.

Potential energy for a linear spring

A horizontal spring exerts a force that is proportional to its deformation in the axial or x-direction. The work of this spring on a body moving along the space curve, is calculated using its velocity,, to obtain
For convenience, consider contact with the spring occurs at, then the integral of the product of the distance x and the x-velocity, xv_x, is ²/2.
The function
is called the potential energy of a linear spring.
Elastic potential energy is the potential energy of an elastic object that is deformed under tension or compression. 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.

Potential energy for gravitational forces between two bodies

The gravitational potential function, also known as gravitational potential energy, is:
The negative sign follows the convention that work is gained from a loss of potential energy.

Derivation

The gravitational force between two bodies of mass M and m separated by a distance r is given by Newton's law of universal gravitation
where is a vector of length 1 pointing from M to m and G is the gravitational constant.
Let the mass m move at the velocity then the work of gravity on this mass as it moves from position to is given by
The position and velocity of the mass m are given by
where e_r and e_t are the radial and tangential unit vectors directed relative to the vector from M to m. Use this to simplify the formula for work of gravity to,
This calculation uses the fact that