Elastic Potential Energy Calculator

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Use this calculator to find spring energy, spring constant, displacement, or equivalent stiffness for two springs with clear units.

Pick what you want to calculate. The calculator will hide inputs you do not need for that choice.
How far the spring is stretched or compressed from its relaxed length. Use the size (magnitude). Example: 0.10 m is 10 cm.
Unit used for the displacement input x.
How stiff the spring is. Bigger k means stiffer. Must be greater than 0.
Unit used for the spring constant k.
Energy stored in the spring. Use a non-negative value (0 or more) for this Hooke's law model.
Unit used to display the elastic potential energy result.
Parallel makes the system stiffer (k adds). Series makes it softer (reciprocal adds).
Spring constant for the first spring, in the selected k unit. Must be greater than 0.
Spring constant for the second spring, in the selected k unit. Must be greater than 0.
Advanced options
Display
How to format numbers. Auto switches to scientific notation only when values are extremely large or small.
Used mainly for scientific notation or very small/large values. Typical: 3 to 6.
Energy density (materials)
If turned on, enter stress and strain to compute u = 0.5 * stress * strain. This is for materials (not just a single spring).
Stress is force per area. Example: 1,000,000 Pa is 1 MPa. Must be 0 or greater.
Unit for stress in the energy density formula.
Strain is fractional change in length: delta_length / original_length. Unitless. Must be 0 or greater.
Note: u = 0.5 * stress * strain gives J/m^3 when stress is in Pa.
Calculating...
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How to use our Elastic Potential Energy Calculator

  1. Choose "What do you want to solve for?" and pick Energy (U), Spring constant (k), Displacement (x), or Combine springs (k_eq).
  2. Enter the needed values using the labels shown, such as "Displacement magnitude |x| (selected length unit)" and "Spring constant k (selected k unit)".
  3. Pick matching units in "Displacement unit", "Spring constant unit", and "Output energy unit" before you calculate.
  4. If you choose "Spring constant (k)" or "Displacement (x)", enter "Elastic potential energy U (selected energy unit)" in the energy unit you selected.
  5. If you choose "Combine springs (k_eq)", select "Spring arrangement (for combine springs mode)" and enter both "Spring 1 constant k1 (selected k unit)" and "Spring 2 constant k2 (selected k unit)".
  6. Open "Advanced options" only if you need a different "Number format", custom "Significant figures (when needed)", or "Also compute energy density u (J per cubic meter)".
  7. Click "Calculate" to see the result cards for energy, force, equivalent spring constant, or warnings.
  8. Sanity-check the result: a bigger displacement should raise energy a lot because energy depends on displacement squared, and a stiffer spring should give a bigger force at the same displacement.

Definitions

Elastic potential energy U: Energy stored in a spring when it is stretched or compressed. In the Hooke's-law range, that stored energy can be recovered as work when the spring is released.[1]

Spring constant k: A measure of stiffness. A larger k means the spring is harder to stretch or compress.

Displacement magnitude |x|: How far the spring moves from its relaxed position. This calculator uses the size only, so stretch and compression give the same energy.

Spring force magnitude F at displacement |x|: The size of the spring's push or pull at that displacement, based on Hooke's law.[2]

Equivalent spring constant k_eq: The single stiffness that would act like two springs together in series or parallel.

Elastic energy density u (if enabled): Energy stored per unit volume in a material, measured in J/m^3.

Common mistakes and quick fixes

Mistake: Entering a negative value in "Displacement magnitude |x| (selected length unit)" even though the label asks for magnitude.
Fix: Enter a non-negative size only, such as 0.10, and let "Spring force magnitude F at displacement |x|" stay as a size rather than a direction.

Mistake: Using "Spring constant k (selected k unit)" with the wrong "Spring constant unit".
Fix: Make sure the number and unit match. For example, if your value is in N/cm, change "Spring constant unit" to N/cm before calculating.

Mistake: Trying to solve from "Elastic potential energy U (selected energy unit)" while leaving "Output energy unit" on a different unit than the value you typed.
Fix: Set "Output energy unit" to the same unit as the energy input first, then calculate.

Mistake: Entering 0 in "Displacement magnitude |x| (selected length unit)" when solving for "Spring constant (k)" with a positive "Elastic potential energy U (selected energy unit)".
Fix: Use a displacement greater than 0, or if displacement is truly 0, then the energy must also be 0 in this model.

Mistake: In combine mode, filling "Spring 1 constant k1 (selected k unit)" and "Spring 2 constant k2 (selected k unit)" but forgetting to choose the correct "Spring arrangement (for combine springs mode)".
Fix: Pick Parallel if the springs share the same displacement, or Series if they are end-to-end and share the same force.

Mistake: Turning on "Also compute energy density u (J per cubic meter)" and then entering negative "Stress (selected stress unit)" or negative "Strain (unitless)".
Fix: Use non-negative magnitudes for both inputs in this simplified material-energy option.

Limitations & Key Assumptions / Boundary Conditions

  • This calculator uses the ideal linear spring model, so results are most reliable only while Hooke's law is a good approximation.
  • Elastic potential energy is treated as non-negative and uses displacement magnitude, so it does not show direction of motion or direction of force.
  • Spring constants must be greater than 0. Zero or negative stiffness is blocked as a non-physical input for this model.
  • When solving for spring constant, a positive energy with zero displacement is impossible in this model and produces an error.
  • Combined-springs mode is limited to two springs and uses standard series or parallel formulas only.
  • The optional energy density result uses u = 0.5 times stress times strain for simple linear elastic, uniaxial behavior. Real materials can differ under nonlinear, multiaxial, or plastic deformation.
  • Real springs can become nonlinear at large deformation or near the elastic limit, so lab or manufacturer data may differ from this calculator's result.

Methodology

Core formulas

This calculator converts all needed inputs to SI units first, does the physics in SI, then converts the displayed energy to your chosen output unit.

U = (1/2) * k * x^2

F = k * |x|

k = 2U / x^2

|x| = sqrt(2U / k)

These formulas come from the standard Hooke's-law spring model, where spring energy depends on the square of displacement.[2]

Combined springs

For two springs acting together, the calculator first finds an equivalent spring constant, then uses that value to compute force and energy at the entered displacement.

k_eq = k1 + k2

1 / k_eq = 1 / k1 + 1 / k2

U = (1/2) * k_eq * x^2

Parallel makes the system stiffer, while series makes it softer.

Energy density option

If the material option is enabled, the calculator also computes elastic energy density from stress and strain magnitudes for a simple linear elastic case.

u = (1/2) * stress * strain

Because 1 pascal is 1 newton per square meter, this result is in J/m^3.

Worked mini-example

Suppose k = 200 N/m and |x| = 0.10 m. Then the stored spring energy is 1 J and the force magnitude is 20 N.

U = (1/2) * 200 * (0.10)^2 = 1

F = 200 * 0.10 = 20

This means the spring stores 1 joule of energy at that displacement, and the pull or push size there is 20 newtons.

Assumptions behind the result

The result assumes a spring that follows Hooke's law over the displacement used.[2][2] Real springs can become nonlinear, especially at larger deflections, so the formula may stop matching measured behavior outside the elastic range.[2]

Sources

  1. 7.4 Conservative Forces and Potential Energy - College Physics 2e - Openstax
  2. Masses and Springs - Periodic Motion | Hooke's Law | Conservation of Energy - PhET Interactive Simulations - Colorado
  3. Potential Energy: Elastic Formula - Softschools
  4. [2510.16960] The nonlinearity of helical springs: An energy-based approach - Arxiv