Elastic Potential Energy Calculator

Enter a spring's stiffness and displacement to get the stored elastic potential energy (E = ½kx²), the Hooke's Law restoring force, and the release velocity for a given mass.

Quick Facts

Energy formula
E = ½kx²
Doubling the displacement quadruples the stored energy.
Hooke's Law
F = kx
The restoring force grows linearly with displacement, opposite in direction.
Unit conversion
1 J = 0.7376 ft·lb
Joules are the SI unit for energy; 1 J = 1 N·m.

Your Results

Calculated
Elastic Potential Energy
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E = ½kx², in joules
Energy (ft·lb)
-
Same energy in foot-pounds
Restoring Force
-
F = kx (Hooke's Law), at this displacement
Release Velocity
-
v = √(2E/m), if all energy converts to kinetic energy

Ready

Enter a spring constant and displacement, then press Calculate.

Formula and Method for Elastic Potential Energy

Elastic potential energy is the energy stored in a spring or other elastic object when it is stretched or compressed away from its natural, unstressed length. For an ideal spring that obeys Hooke's Law, the restoring force at displacement x is F = kx, where k is the spring constant. Because that force grows linearly from 0 up to kx as the spring is deformed, the work done — and therefore the energy stored — is the average force (½kx) times the distance x, giving the standard result E = ½kx², where k is in newtons per meter, x is in meters, and E is in joules.

How the calculation works

Enter the spring constant k and choose its unit (N/m, N/cm, N/mm, lbf/in, or lbf/ft), then enter the displacement x from the spring's natural length and choose its unit. The calculator converts both to SI units (N/m and m) and applies E = ½kx² to get the stored energy in joules, then converts that same energy to foot-pounds. It also reports the instantaneous restoring force F = kx in newtons and pounds-force. If you provide an object mass, it computes the release velocity v = √(2E/m) — the speed the object would reach if all the stored elastic energy converted into kinetic energy with no losses.

Common mistakes

  • Using total length instead of displacement: x is the change from the spring's natural, unstretched length — not the spring's overall length while deformed.
  • Forgetting the ½: the restoring force at displacement x is kx, but the stored energy is ½kx² because the force ramps up from zero as the spring deforms.
  • Mixing spring constant units: a spring rated at 10 lbf/in is not the same as 10 N/m — convert to one consistent unit system before comparing springs.
  • Exceeding the elastic limit: E = ½kx² only holds within the spring's linear-elastic range; beyond that, the spring deforms permanently and the formula overstates recoverable energy.

Real-world applications

  • Vehicle suspension and shock-absorber springs store and release energy to smooth out road impacts.
  • Archery bows, catapults, and mousetraps convert stored elastic energy into the kinetic energy of a launched object.
  • Mechanical clocks, wind-up toys, and some fitness equipment use coiled or extension springs as compact energy storage.
  • Structural and mechanical engineers use the elastic-limit check to size springs and flexible members safely.

Frequently Asked Questions

What is the formula for elastic potential energy?
Elastic potential energy is E = ½kx², where k is the spring constant in newtons per meter and x is the displacement from the spring's natural (unstretched) length in meters. The result is in joules.
How is elastic potential energy different from Hooke's Law force?
Hooke's Law, F = kx, gives the instantaneous restoring force at a displacement x. Elastic potential energy is the work needed to reach that displacement, found by integrating force over distance from 0 to x, which is why the energy formula carries a factor of one-half: E = ½kx².
Does compressing a spring store more energy than stretching it the same distance?
No. Because the energy formula depends on x² and not on the sign of x, a spring compressed 10 cm stores the same elastic potential energy as one stretched 10 cm, as long as both stay within the spring's elastic limit.
What happens if I exceed the spring's elastic limit?
Hooke's Law and E = ½kx² only hold while the spring deforms elastically. Beyond its elastic limit the spring yields permanently, the constant k no longer applies, and this calculator's result will overstate the recoverable energy — check the manufacturer's maximum travel spec first.

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