PlanetPhysics/1D Example of the Relation Between Force and Potential Energy

For a simple one dimensional example of the relationship between force and potential [[../CosmologicalConstant/|energy]], assume that the potential energy of a [[../Particle/|particle]] is given by the equation

$U(x)=-\frac{A}{x}[1+B{e}^{-x/c}]$

The relationship between force and potential energy is

$𝐅=-\mathrm{\nabla }U$

For our 1D example, where the potential energy is dependent only on the [[../Position/|position]], $x$

$F=-\frac{dU}{dx}$

Taking the derivative yields

$\frac{dU}{dx}=\frac{A}{x^2}[1+B{e}^{-x/c}]-\frac{A}{x}[-\frac{B}{C}{e}^{-x/c}]$

Therefore, the force on the particle is governed by the equation

$F=-\frac{A}{x^2}[1+B{e}^{-x/c}]+\frac{A}{x}[-\frac{B}{C}{e}^{-x/c}]$

If we further assume that the constant C is so much larger than x, the force will simplify. Rearranging to get

$F=-\frac{A}{x^2}[1+B{e}^{-x/c}-\frac{Bx}{C}{e}^{-x/c}]$

Because $x\ll C$, $e^{-x/c}\Rightarrow e^0=1$, we get

$F=-\frac{A}{x^2}[1+B-\frac{Bx}{C}]$

Further simplifying $\frac{Bx}{C}=0$ gives the force under this assumption

$F=-\frac{A}{x^2}[1+B]$