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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 importantcustomary 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

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.

h4. Potential Energy CurveCurves

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.  



h2. Common Types

h4. Gravitational Potential Energy Near Earth

Near the earth's surface, if we assume coordinates with the +{_}y_ direction pointing upward, the force of gravity can be written:

{latex}\begin{large}\[ \vec{F} = -mg \hat{y}\]\end{large}{latex}

Since the "natural" ground level varies depending upon the specific situation, it is customary to specify the coordinate system such that:

{latex}\begin{large}\[ U(0) \equiv 0\]\end{large}{latex}

The [gravitational potential energy] at any other height _y_ can then be found by choosing a path for the work integral that is perfectly vertical, such that:

{latex}\begin{large}\[ U(y) = U(0) - \int_{0}^{y} (-mg)\;dy = mgy\]\end{large}{latex}

For an object in vertical freefall (no horizontal motion) the associated potential energy curve would then be:

POTENTIAL ENERGY CURVE

For movement under pure near-earth gravity, then, there is no equilibrium point.  At least one other force, such as a normal force, tension, etc., must be present to produce equilibrium. 

h4. Elastic Potential Energy

Assuming an object attached to a spring that obeys Hooke's Law with the motion confined to the _x_ direction, it is customary to choose the coordinates such that _x_ = 0 when the object is in a position such that the spring is at its natural length.  The force on the object from the spring is then:

{latex}\begin{large}\[ F_{x} = - kx \]\end{large}{latex}

It is also customary to make the assignment:

{latex}\begin{large}\[ U(0) \equiv 0\]\end{large}{latex}

Thus, the potential can be defined:

{latex}\begin{large}\[ U(x) = U(0) - \int_{0}^{x} (-kx)\:dx = \frac{1}{2}kx^{2}\]\end{large}{latex}

For an object moving under the influence of a spring only, the associated potential energy curve would then be:

POTENTIAL ENERGY CURVE

For such a motion, then, there is one stable equilibrium point at _x_ = 0.

h4. Universal Gravitational Potential Energy