The Elastic Potential Energy Calculator finds the energy stored in a stretched or compressed spring, using the spring constant and the displacement from its unstressed length. Enter k and Δx, and the tool returns the stored energy in joules with the substituted arithmetic shown.
Elastic potential energy is the reason a stretched rubber band snaps back, a compressed spring pushes a toy car forward, and a bow releases an arrow. Unlike gravitational potential energy, it does not depend on height at all, only on how far the elastic object has been deformed from its natural length.
Calculate elastic potential energy
A spring, or any Hookean elastic element, stores energy proportional to the square of its displacement from equilibrium.
U = ½kΔx²| Symbol | Quantity | SI unit |
|---|---|---|
| U | elastic potential energy | joules (J) |
| k | spring constant | newtons per metre (N/m) |
| Δx | extension or compression | metres (m) |
Worked example: a spring with k = 200 N/m is stretched by 0.1 m.
1. Square the displacement. (0.1)² = 0.01.
2. Multiply by the spring constant. 200 × 0.01 = 2.
3. Multiply by one-half. ½ × 2 = 1 J.
See how displacement drives the energy
Because displacement is squared in the formula, doubling the stretch quadruples the stored energy rather than simply doubling it. Doubling the spring constant, by contrast, doubles the energy directly, since k appears only to the first power. Worked comparison: double the stretch to 0.2 m, keeping k = 200 N/m.
U = ½ × 200 × (0.2)² = ½ × 200 × 0.04 = 4 J, four times the original 1 J, confirming the squared relationship.
Halving k to 100 N/m at the original 0.1 m stretch instead gives U = ½ × 100 × 0.01 = 0.5 J, exactly half of the original value.
Connect elastic energy to Hooke's law
The formula for elastic potential energy comes directly from Hooke's law, F = −kx, which describes the restoring force a spring exerts when displaced.
Elastic potential energy is the work done to displace the spring by that amount, which works out to the area under a force-versus-displacement graph, a triangle with base Δx and height kΔx, giving an area of ½ × Δx × kΔx = ½kΔx².
The restoring force itself, at the same 0.1 m stretch used above, is F = kx = 200 × 0.1 = 20 N. Force and energy answer different questions about the same spring: force describes the push or pull at a given instant, while energy describes the total work stored up to that point.
Know the limits of Hooke's law
The formula ½kΔx² only holds while the spring obeys Hooke's law, meaning force stays proportional to displacement. Beyond the elastic limit of the material, that proportionality breaks down, and the true stored energy departs from the simple quadratic formula. Springs used well within their rated range in everyday problems generally satisfy this assumption.
Handle springs in series and parallel
When multiple springs work together, their combined stiffness is not simply the sum of individual constants unless they are arranged in parallel. Springs in parallel add their constants directly (k_total = k1 + k2).
Springs in series combine like resistors, with the reciprocal of the total equal to the sum of the reciprocals (1/k_total = 1/k1 + 1/k2).
Always compute the effective spring constant for the actual arrangement before applying the energy formula, rather than using a single coil's rating when several coils share the load.
Work through a compression example
A spring with k = 150 N/m is compressed by 0.2 m. 1. Square the displacement. (0.2)² = 0.04. 2. Multiply by the spring constant. 150 × 0.04 = 6. 3. Multiply by one-half. ½ × 6 = 3 J.
The result is positive despite the spring being compressed rather than stretched, confirming that the squared displacement term treats compression and extension identically.
Frequently asked questions
What is the formula for elastic potential energy?
U = ½kΔx², where k is the spring constant in newtons per metre and Δx is the displacement from the spring's natural length in metres. For k = 200 N/m and Δx = 0.1 m, U = 1 J.
Does it matter if the spring is stretched or compressed?
No. Because displacement is squared in the formula, a compression of a given magnitude stores the same energy as an extension of the same magnitude; the sign of Δx cancels out.
How does doubling the stretch affect stored energy?
It quadruples the stored energy, since displacement is squared in the formula. Doubling the spring constant instead only doubles the energy, since k appears to the first power.
What is the difference between elastic potential energy and spring force?
Elastic potential energy (U = ½kΔx²) describes the total work stored in the spring. Spring force (F = kx, from Hooke's law) describes the instantaneous restoring force at a given displacement. They use related but different formulas.
Does this formula work for any amount of stretch?
Only within the elastic limit of the material, where force stays proportional to displacement as Hooke's law describes. Beyond that limit, the spring deforms permanently and the simple quadratic formula no longer applies.
How do you find the spring constant for springs in series?
Add the reciprocals of each individual spring constant, then take the reciprocal of that sum: 1/k_total = 1/k1 + 1/k2. This gives a lower combined stiffness than either spring alone.
Can elastic potential energy convert to kinetic energy?
Yes. When a stretched or compressed spring is released, its stored elastic potential energy converts to kinetic energy of whatever it pushes or pulls, following conservation of energy in an idealized frictionless system.
Summary
Elastic potential energy stored in a spring follows U = ½kΔx², giving 1 J for a 200 N/m spring stretched 0.1 m, and 4 J if that same spring is stretched to 0.2 m, since displacement is squared in the formula.
The relationship comes directly from Hooke's law and holds only within the spring's elastic limit. Springs in series and parallel combine to different effective constants, which must be calculated before the energy formula is applied.