Elastic Potential Energy Calculator
A stretched or compressed spring stores energy that it can release as motion. For an ideal spring obeying Hooke's law, that elastic potential energy equals one half of the spring constant times the displacement squared. This calculator takes the spring constant in newtons per metre and the displacement in metres, returns the stored energy in joules, and also reports the restoring force and the same energy in foot-pounds, so you can analyse springs, bungee systems, archery and trampolines with confidence.
Elastic potential energy formula
Stored energy U = 0.5 * k * x^2 (joules)
Restoring force F = k * x (newtons)
Energy in foot-pounds = U / 1.355818
Energy in calories = U / 4.184
The energy is the area under the linear force-displacement line, which is a triangle of height k*x and base x, giving one half k x squared. Force follows Hooke's law directly. The conversions express the same joules in customary units.
Spring energy context
- Hooke's law (F = k x) holds only within the elastic limit of the material.
- The spring constant k is in newtons per metre; a stiffer spring has a higher k.
- Energy grows with the square of displacement, so doubling stretch quadruples energy.
- One joule equals about 0.7376 foot-pounds; one calorie equals 4.184 joules.
- The restoring force always points back toward the natural length.
Elastic potential energy: frequently asked questions
What is the elastic potential energy formula?
The energy stored in an ideal (Hookean) spring is U = 0.5 * k * x^2, where k is the spring constant in newtons per metre and x is the displacement from the natural length in metres. The result is in joules. This is the area under the force-versus-displacement line for a linear spring.
Why is displacement squared?
Spring force grows linearly with displacement (F = k * x by Hooke's law), so the energy, which is the integral of force over distance, grows with the square of displacement. Stretching a spring twice as far stores four times the energy, not twice.
What is the spring constant?
The spring constant k measures stiffness: the force needed per unit of stretch, in newtons per metre. A stiff spring has a high k. You can find it from a known force and displacement (k = F / x) or from the manufacturer's specification. Enter your spring's value.
What is the restoring force at this displacement?
By Hooke's law the restoring force is F = k * x, in newtons, directed back toward the natural length. This calculator reports that force alongside the stored energy so you can see both the instantaneous force and the total energy held in the spring.
Does this apply beyond a spring's elastic limit?
No. Hooke's law and this energy formula hold only while the material behaves elastically, that is, returns to its original shape. Past the elastic limit the relationship becomes nonlinear and some energy is lost to permanent deformation, so the formula no longer applies.
Official sources
- NASA Glenn Research Center: Beginner's Guide to Aeronautics (energy).
- National Institute of Standards and Technology: SI Units (joule, newton).
Reviewed by the CalculatorHub team, edited by James Graham, 16 June 2026. See our methodology.