Potential Energy
A form of energy associated with the presence of conservative interactions such as gravity or a spring.
Motivation for Concept
Conservative interactions like gravity have the ability to "store" kinetic energy. Consider an object thrown up to a high roof. If the object is thrown perfectly, the force of gravity will slow it to a stop just as it reaches the roof. The object will then remain at rest on the roof until disturbed. But, as it falls, gravity will restore all the speed that was removed on the way up. Because we can perfectly recover the kinetic energy removed by gravity, we can consider the total energy to be constant in such a situation if we can associate some energy with the object's height. That energy is the gravitational potential energy.
Mathematical Definition
Finding Potential Energy From Force
The work done by a general force is given by:
\begin
[ W = \int_
\vec
\cdot d\vec
]\end
The Work-Kinetic Energy Theorem tells us that when work is done on a system, the system's kinetic energy will change:
\begin
[ K_
+ W = K_
]\end
Suppose that we consider a system acted upon by a single, conservative force. If we want to define a potential energy U to represent this interaction in such a way that the mechanical energy of the system is conserved, we must take:
\begin
[ U_
- U_
= W^
]\end
With this definition, the work-energy theorem takes the form:
\begin
[ K_
+ (U_
-U_
) = K_
]\end
which is equivalent to the conservation of mechanical energy:
\begin
[ K_
+ U_
= K_
+ U_
]\end
The definition we have arrived at expresses potential energy in terms of force through the application of a path integral:
\begin
[ U_
- U_
= - \int_
\vec
^
\cdot d\vec
]\end
it is important to note, however, that the work done by conservative forces is, by definition, path independent. Thus, the integrals can be done using the most advantageous path, and the value will depend only upon the initial and final positions of the system. We can therefore write:
\begin
[ U_
- U_
= - \int_{\vec
_{i}}^{\vec
_{f}} \vec
^
\cdot d\vec
] \end
Note also that the expression we have found is only useful for computing potential energy differences. The formula's validity does not depend upon the precise value of Uf or Ui, but instead upon the difference. That means that an arbitrary constant can be added to the potential energy without affecting its usefulness. In problems involving potential energy, then, it is customary to specify a zero point for the potential energy (r0) such that:
\begin
[ U(\vec
_
) = 0 ]\end
Finding Force From Potential Energy
Mathematically
Taking the componentwise derivative of the above definition of potential energy with respect to position yields the three expressions:
\begin
[ -\frac
= F^
_
]
[ -\frac
= F^
_
]
[ -\frac
= F^
_
]\end
Thus, given information about the dependence of the potential energy on position, the force acting on the system subject to that potential energy can be determined.
Diagrammatically
A [potential energy curve] is a graphical representation of a system's potential energy as a function of posittion. This can be done for any system, but it is most often drawn for a system confined to move in one dimension (since multidimensional graphs are difficult to draw and interpret). The graph can be useful in furthering both qualitative and quantitative understanding of the system's behavior.
Common Conservative Forces
Near-Earth Gravity
One conservative force which is often encountered in introductory mechanics is near-earth gravity. The customary form of the gravitational potential energy near the earth's surface is:
\begin
[ U
= mgy ]\end
assuming that the y direction is taken to point upward from the earth's surface.
Springs
Springs whose interaction is well described by Hooke's Law are another example of a commonly encountered conservative force. The customary form of the elastic potential energy associated with a spring is:
\begin
[ U
= \frac
kx^
]\end
where one end of the spring is fixed and the other end is constrained to stretch or compress only in the x direction, and the coordinates have been defined such that the free end of the spring provides zero force when it is at the position x = 0.