AcademyPotential Energy and Conservation
Academy
Elastic Potential Energy
Level 1 - Physics topic page in Potential Energy and Conservation.
Principle
Elastic potential energy is stored when a spring or elastic element is displaced from equilibrium.
Notation
\(U_s\)
elastic potential energy
\(\mathrm{J}\)
\(k\)
spring constant
\(\mathrm{N\,m^{-1}}\)
\(x\)
extension or compression from equilibrium
\(\mathrm{m}\)
\(F_s\)
spring force
\(\mathrm{N}\)
Rules
Spring force
\[F_s=-kx\]
Elastic potential energy
\[U_s=\frac{1}{2}kx^2\]
Work by spring
\[W_s=-\Delta U_s\]
Energy exchange
\[\frac{1}{2}kx_i^2+K_i=\frac{1}{2}kx_f^2+K_f\]
Method
Measure from equilibrium
\[x=0\ \text{at the spring's natural length}\]
Store energy
\[U_s=\frac{1}{2}kx^2\]
Convert energy
\[\Delta K=-\Delta U_s\]
Examples
Question
A spring with constant \(k\) is compressed by \(x\). Find the launch speed of a mass \(m\) on a frictionless surface.
Answer
\[\frac{1}{2}kx^2=\frac{1}{2}mv^2\]
so \[v=x\sqrt{\frac{k}{m}}\]
Checks
- Elastic energy is never negative for an ideal spring.
- The sign of \(x\) affects force direction, not stored energy sign.
- Spring energy grows quadratically with displacement.
- Hooke's-law spring formulas apply only within the elastic range.