AcademyPotential Energy and Conservation
Academy
Force from Potential Energy
Level 1 - Physics topic page in Potential Energy and Conservation.
Principle
A conservative force points in the direction where potential energy decreases fastest.
Notation
\(F_x\)
x-component of force
\(\mathrm{N}\)
\(U(x)\)
potential energy as a function of position
\(\mathrm{J}\)
\(\vec{F}\)
force vector
\(\mathrm{N}\)
\(\nabla U\)
gradient of potential energy
\(\mathrm{J\,m^{-1}}\)
Rules
One dimension
\[F_x=-\frac{dU}{dx}\]
Three dimensions
\[\vec{F}=-\nabla U\]
Equilibrium
\[\frac{dU}{dx}=0\]
Stable equilibrium
\[\frac{d^2U}{dx^2}>0\]
Method
Differentiate potential
\[\frac{dU}{dx}\]
Apply minus sign
\[F_x=-\frac{dU}{dx}\]
Classify equilibrium
\[U''(x)>0\ \text{stable},\quad U''(x)<0\ \text{unstable}\]
Examples
Question
For
\[U(x)=\frac{1}{2}kx^2\]
find \(F_x\).Answer
\[F_x=-\frac{dU}{dx}=-kx\]
This recovers Hooke's law for a spring centered at \[x=0\]
Checks
- The force is minus the slope of \(U(x)\).
- A positive slope means force points negative.
- Equilibrium occurs where the slope is zero.
- Minima of \(U\) are stable equilibria.