Elastic Potential Energy⚠ unverified
Physics / Mechanics · Compute the elastic potential energy stored in a linear spring
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| spring_constant | springconstant | N/m | 1.0 | Spring stiffness |
| displacement | displacement | m | 1.0 | Displacement from the spring's natural (unstretched) length |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | PEspring | J | Stored elastic potential energy, in joules (J) |
The science & history
Understanding the Parameters
- $k$ — stiffer spring stores more energy at the same deflection.
-
$x$ — energy scales as $x^{2}$; sign of $x$ does not matter ($x^{2}$). Measured from the unstressed length for the ideal Hooke model.
-
$PE$ — recoverable as kinetic energy if the spring is released (ideal, no damping).
Derivation (Approaching a Proof)
Hooke’s law: restoring force $F = -k x$. Work to move slowly from $0$ to $x$ against the spring is
$$W = \int_0^{x} k x'\,dx' = \frac{1}{2} k x^{2}.$$
That work is stored as elastic potential energy. Dual of magnetic/electric storage forms ($\tfrac12 L I^{2}$, $\tfrac12 C V^{2}$) and translational Kinetic Energy ($\tfrac12 m v^{2}$).
History
Hooke’s law (17th century) plus energy methods of the 19th century give the standard spring PE used in every vibration and mechanism model.
Related Concepts: Kinetic Energy, Potential Energy, Work Constant Force, Rotational Kinetic Energy
Notes: Registry calculator elastic-potential-energy (unverified). Linear Hookean spring only.