Potential Energy is energy associated with the configuration or position of a system relative to a chosen reference state, particularly when conservative forces are involved. It is not necessarily an intrinsic property of an isolated object because it often belongs to an interacting system. Gravitational potential energy, for example, is properly associated with the Earth-object system rather than simply with the object alone.
Potential energy is defined so that changes in potential energy correspond to work performed by or against a conservative force. For a conservative force, the change in potential energy is related to force through the negative gradient of the potential, commonly written F = −∇U. The SI unit is the joule.
Near Earth's surface, gravitational potential energy is commonly approximated by U = mgh relative to an arbitrarily selected zero-height reference. Here m is mass, g is gravitational acceleration and h is vertical height relative to the chosen reference. The absolute numerical value of potential energy depends on the reference convention, but differences in potential energy have physical significance. Raising an object increases its gravitational potential energy relative to a lower reference because external work is required against gravity. As the object falls, gravitational potential energy can decrease while kinetic energy increases, provided that other energy transfers such as air resistance are negligible. For large changes in distance from Earth, the approximation mgh becomes less accurate because gravitational acceleration varies with altitude.
The more general Newtonian gravitational potential energy for two point masses is U = −GMm/r when the zero of potential energy is chosen at infinite separation. The negative value indicates that a gravitationally bound configuration has lower potential energy than the reference state at infinite separation. As two masses move farther apart, gravitational potential energy increases towards zero. As they move closer together, potential energy decreases and gravitational forces can convert that decrease into kinetic energy. This mathematical form is important in orbital mechanics, celestial dynamics and escape-velocity calculations. It also demonstrates why potential energy is not restricted to objects positioned above Earth's surface; gravitational potential energy applies throughout gravitational systems.
Elastic potential energy is another important form. An ideal spring that obeys Hooke's law stores elastic potential energy according to U = ½kx², where k is the spring constant and x is displacement from the equilibrium position. The energy increases with the square of displacement, so doubling the extension or compression quadruples the stored energy within the ideal model. Real materials can deviate from Hooke's law, especially at large deformations, and may experience plastic deformation, hysteresis or permanent structural changes. Elastic potential energy can be released to produce motion, as in springs, bows and certain mechanical energy-storage systems. The relevant potential function depends on the physical force law governing the system.
Potential energy is especially useful because conservative-force problems can be analysed through total mechanical energy. If only conservative forces act, the sum of kinetic and potential energy remains constant, so a decrease in potential energy corresponds to an equal increase in kinetic energy and vice versa. This approach can simplify problems involving falling objects, oscillators and orbital motion. When non-conservative forces such as friction act, mechanical energy is not generally conserved, but total energy remains conserved if the resulting thermal and other energy changes are included. Potential energy therefore provides a convenient accounting mechanism rather than a claim that a separate physical substance is stored at a particular location.
Potential-energy functions are closely connected to equilibrium and stability. An equilibrium position occurs where the net conservative force is zero, corresponding in one-dimensional systems to a point where the derivative of potential energy with respect to position is zero. A local minimum of potential energy generally corresponds to stable equilibrium because small displacements tend to generate restoring forces. A local maximum can correspond to unstable equilibrium because small displacements can move the system farther away from the equilibrium configuration. This relationship is widely used in mechanics, molecular physics and field theory. The shape of a potential-energy curve can therefore reveal the possible motion of a system without directly solving the force equation at every point.
Potential energy has many specialised forms beyond gravity and springs. Electrostatic potential energy depends on electric charges and their separation, while chemical potential energy is associated with molecular configurations and chemical composition. In nuclear physics, changes in nuclear binding energy can be interpreted through differences in the energy of nuclear configurations. In thermodynamics and statistical mechanics, chemical potential provides a different but related concept describing how thermodynamic energy changes with particle number under specified conditions. These uses demonstrate why the word potential must be interpreted according to the physical interaction and mathematical framework involved. Not every energy associated with configuration is represented by a simple mechanical U = mgh expression.
Several misconceptions concern the arbitrary zero of potential energy and the idea that potential energy belongs exclusively to an object. The zero can be selected conveniently because only differences in potential energy affect ordinary force and energy calculations. A book on a table can be assigned zero gravitational potential energy if the tabletop is selected as the reference, even though it could be assigned a different value relative to the floor or Earth's centre.
Potential energy also requires appropriate system boundaries: gravitational potential energy is conventionally associated with the interacting masses. Finally, potential energy should not be confused with stored energy in every circumstance, because the mathematical concept applies to conservative interactions and configurations even when describing the system in more abstract physical terms.
