What is Potential Energy?

Potential energy is often associated with restoring forces such as a spring or the force of gravity. The action of stretching the spring or lifting the mass of an object is performed by an external force that works against the force field of the potential. This work is stored in the force field 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 its initial position, reducing the stretch of the spring or causing the body to fall. 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.

Potential Energy in a Bow and Arrow

In the case of a bow and arrow, the energy is converted from the potential energy in the archer's arm to the potential energy in the bent limbs of the bow when the string is drawn back. When the string is released, the potential energy in the bow limbs is transferred back through the string to become kinetic energy in the arrow as it takes flight.

If the work for an applied force is independent of the path, then the work done by the force is evaluated at the start and end of the trajectory of the point of application. This means that there is a function U(x), called a "potential," that can be evaluated at the two points x(t = t₁) and x(t₂) 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 represented as a reduction in the potential:

$\begin{aligned} W &= \int_C \bf{F} \cdot \rm{d}\bf{x} = \int_{\bf{x}(t_1)}^{\bf{x}(t_2)} \bf{F} \cdot \rm{d}\bf{x} \\& = U(\bf{x}(t_1))-U(\bf{x}(t_2)) = -\Delta U. \end{aligned}$

Examples of Potential Energy

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 done by the gravitational force is called gravitational potential energy; and work done by the Coulomb force is called electric potential energy.