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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}
 
 
h4. Motivation for Concept


[Conservative interactions|conservative force] like [gravity|gravity (near-earth)] 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.


h4. Mathematical Definition

h6. 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-Kinetic 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 customary 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

h6. Mathematically

Taking the componentwise derivative of the above definition of potential energy with respect to position yields the three expressions:

{latex}\begin{large}\[ -\frac{\partial U}{\partial x} = F^{\rm cons}_{x} \]
\[ -\frac{\partial U}{\partial y} = F^{\rm cons}_{y} \]
\[ -\frac{\partial U}{\partial z} = F^{\rm cons}_{z} \]\end{large}{latex}

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.

h6. Diagrammatically

A *potential energy curve* is a graphical representation of a system's potential energy as a function of position.  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.  

h4. Common Conservative Forces

h6. Near-Earth Gravity

One [conservative force] which is often encountered in introductory mechanics is near-earth [gravity|gravity (near-earth)].  The customary form of the [gravitational potential energy|gravitation (universal)#negpe] near the earth's surface is:

{latex}\begin{large}\[ U(y) = mgy \]\end{large}{latex}

assuming that the _y_ direction is taken to point upward from the earth's surface.

h6. Springs

Springs whose interaction is well described by [Hooke's Law|Hooke's Law for elastic interactions] are another example of a commonly encountered [conservative force].  The customary form of the [elastic potential energy|Hooke's Law for elastic interactions#epe] associated with a spring is:

{latex}\begin{large}\[ U(x) = \frac{1}{2}kx^{2} \]\end{large}{latex}

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.