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For a simple one dimensional example of the relationship between force and potential energy, assume that the potential energy of a particle is given by the equation
The realtionship between force and potential energy is
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(1) |
For our 1D example, where the potential energy is dependent only on the position, 
Taking the derivative yields
Therefore, the force on the particle is governed by the equation
![$\displaystyle F = -\frac{A}{x^2} \left [ 1 + B e^{-x/c} \right ] + \frac{A}{x} \left [ -\frac{B}{C}e^{-x/c} \right ]$](http://images.physicslibrary.org/cache/objects/261/l2h/img6.png "$\displaystyle F = -\frac{A}{x^2} \left [ 1 + B e^{-x/c} \right ] + \frac{A}{x} \left [ -\frac{B}{C}e^{-x/c} \right ]$") |
(2) |
If we further assume that the constant C is so much larger than x, the force will simplify. Rearraning to get
Because  ,
 , we get
Further simplifying
 gives the force under this assumption
![$\displaystyle F = -\frac{A}{x^2} \left [ 1 + B \right ]$](http://images.physicslibrary.org/cache/objects/261/l2h/img12.png "$\displaystyle F = -\frac{A}{x^2} \left [ 1 + B \right ]$") |
(3) |
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![[parent]](https://images.physicslibrary.org/images/uparrow.png)
![$\displaystyle U(x) = -\frac{A}{x} \left [ 1 + B e^{-x/c} \right] $](http://images.physicslibrary.org/cache/objects/261/l2h/img1.png "$\displaystyle U(x) = -\frac{A}{x} \left [ 1 + B e^{-x/c} \right] $")
![$\displaystyle \frac{dU}{dx} = \frac{A}{x^2} \left [ 1 + B e^{-x/c} \right ] - \frac{A}{x} \left [ -\frac{B}{C}e^{-x/c} \right ]$](http://images.physicslibrary.org/cache/objects/261/l2h/img5.png "$\displaystyle \frac{dU}{dx} = \frac{A}{x^2} \left [ 1 + B e^{-x/c} \right ] - \frac{A}{x} \left [ -\frac{B}{C}e^{-x/c} \right ]$")
![$\displaystyle F = -\frac{A}{x^2} \left [ 1 + B e^{-x/c} -\frac{Bx}{C}e^{-x/c} \right ] $](http://images.physicslibrary.org/cache/objects/261/l2h/img7.png "$\displaystyle F = -\frac{A}{x^2} \left [ 1 + B e^{-x/c} -\frac{Bx}{C}e^{-x/c} \right ] $")
![$\displaystyle F = -\frac{A}{x^2} \left [ 1 + B - \frac{Bx}{C} \right ] $](http://images.physicslibrary.org/cache/objects/261/l2h/img10.png "$\displaystyle F = -\frac{A}{x^2} \left [ 1 + B - \frac{Bx}{C} \right ] $")