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{excerpt}A form of energy associated with the presence of [conservative|conservative force] interactions such as gravity or a spring.{excerpt}

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h2. 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].

h2. Definition

h4. Finding Potential Energy From Force

The [work] done by a general force is given by:

{latex}\begin{large}\[ W = \int_{\rm path} \vec{F}\cdot d\vec{r}\]\end{large}{latex}

The [work energy theorem] tells us that when work is done on a system, the system's kinetic energy will change:

{latex}\begin{large}\[ K_{i} + W = K_{f}\]\end{large}{latex}

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:

{latex}\begin{large}\[ U_{i} - U_{f} =  W^{\rm cons} \]\end{large}{latex}

With this definition, the work-energy theorem takes the form:

{latex}\begin{large}\[ K_{i} + (U_{i}-U_{f}) = K_{f} \]\end{large}{latex}

which is equivalent to the conservation of mechanical energy:

{latex}\begin{large}\[ K_{i} + U_{i} = K_{f} + U_{f}\]\end{large}{latex}

The definition we have arrived at expresses potential energy in terms of force through the application of a path integral:

{latex}\begin{large}\[ U_{f} - U_{i} = - \int_{\rm path} \vec{F}^{\:\rm cons}\cdot d\vec{r}\]\end{large}{latex}

it is important to note, however, that the work done by [conservative forces|conservative force] 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:

{latex}\begin{large}\[ U_{f} - U_{i} = - \int_{\vec{r}_{i}}^{\vec{r}_{f}} \vec{F}^{\:\rm cons}\cdot d\vec{r} \] \end{large}{latex}

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 _U_~f~ or _U_~i~, 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 important to specify a zero point for the potential energy (_r_~0~) such that:

{latex}\begin{large}\[ U(\vec{r}_{0}) = 0 \]\end{large}{latex}

h4. Finding Force From Potential Energy



h2. Common Types

h4. Gravitational Potential Energy

h4. Elastic Potential Energy