Elastic Potential Energy Calculator

How to use this calculator

Select a spring preset from the dropdown to automatically load a typical spring constant, or enter your own values. Then choose the calculation mode and click Calculate.

Mode 0 (Calculate Energy): Enter the spring constant k (N/m) and the extension or compression x (m). The calculator returns U = ½kx² in Joules.

Mode 1 (Calculate Spring Constant): Enter the stored energy U (J) and the extension x (m). The calculator returns k = 2U/x².

Mode 2 (Calculate Extension): Enter the stored energy U (J) and the spring constant k (N/m). The calculator returns x = √(2U/k).

All modes also report the spring force F = kx (the force the spring exerts at that extension). The Energy vs Extension chart shows the parabolic energy curve (U = ½kx²), and the Force vs Extension chart shows the linear Hooke’s law relationship (F = kx).

Hooke’s Law: history and meaning

The foundation of elastic potential energy is Hooke’s Law, named after the English scientist Robert Hooke. In 1678, Hooke published his discovery in an anagram (a common way to establish priority while keeping the result secret): “ceiiinosssttuv,” which unscrambles to the Latin “ut tensio, sic vis” meaning “as the extension, so the force.”

The modern form of Hooke’s Law is:

Where F is the restoring force (in Newtons), k is the spring constant (in N/m, also written as kg/s²), and x is the displacement from the natural length (in meters). The restoring force always acts opposite to the displacement, pulling the spring back toward its natural length.

The law states that the force is proportional to the displacement. This holds true for small deformations of almost any elastic material, whether it is a metal spring, a rubber band, a wooden plank bending, or even an atomic bond. It breaks down when deformation becomes large enough that the material’s microstructure is permanently altered (the elastic limit or yield point).

Hooke’s Law applies to springs, beams, rubber, biological tissues, and even bonds between atoms in a crystal. It is one of the most universal laws in mechanics.

Deriving U = ½kx² from the work integral

The elastic potential energy formula is derived by calculating the work done against the spring force as the spring is deformed.

When a spring with constant k is compressed or stretched from x = 0 to x = X, the force required at each position is F(x) = kx (equal and opposite to the restoring force). The work done against this force is:

This is the area under the F-x graph (a triangle with base X and height kX), which equals ½ × base × height = ½ × X × kX = ½kX².

The geometric interpretation: since the force varies linearly from 0 to kX over the extension X, the average force is kX/2, and work = average force × distance = (kX/2) × X = ½kX².

The quadratic dependence on x means that doubling the extension stores four times the energy, not twice. A spring compressed twice as far stores four times as much energy. This has important implications for safety in spring-loaded systems and for energy storage efficiency.

The spring constant and what it tells you

The spring constant k (also called the stiffness coefficient) quantifies how stiff or soft a spring is. A high k means a stiff spring (requires large force for small extension). A low k means a soft spring (large extension from small force).

Units of k: N/m (Newtons per meter). This can also be written as kg/s².

Typical spring constants across different applications:

Application	k (N/m)
Watch spring	~0.1
Rubber band	~20
Bungee cord	~100
Trampoline spring	~2,000
Door return spring	~500 - 2,000
Car suspension	~20,000 - 50,000
Stiff industrial spring	100,000 - 1,000,000

The spring constant depends on the material (Young’s modulus E), the wire diameter, the number of coils, and the coil diameter. For a helical spring made of wire with diameter d, coil radius R, n coils, and shear modulus G:

This formula shows that k increases steeply with wire diameter (to the 4th power) and decreases with coil radius cubed.

Springs in series and parallel

When multiple springs are combined, their effective spring constant follows rules similar to resistors in electrical circuits.

Springs in parallel (both ends connected, forces add):

Parallel springs are stiffer. A car with two identical springs per wheel has double the spring constant of a single spring.

Springs in series (end-to-end, extensions add):

Series springs are softer. A bungee cord made of two identical sections in series is half as stiff as a single section.

Elastic PE and simple harmonic motion

When a mass on a spring is displaced from equilibrium and released, it undergoes simple harmonic motion (SHM). The total mechanical energy is conserved, alternating between elastic potential energy and kinetic energy.

At maximum extension x = A (amplitude): all energy is elastic PE.

At equilibrium x = 0: all energy is kinetic.

Therefore: v_max = A × √(k/m)

At any position x between 0 and A:

The angular frequency of oscillation:

And the period:

A heavier mass oscillates more slowly. A stiffer spring oscillates more quickly. This relationship is used in reverse to measure spring constants by measuring the period of oscillation.

Bungee jumping physics

Bungee jumping is a real-world application of elastic potential energy that integrates gravitational PE, kinetic energy, and elastic PE.

At the top (jump point), the jumper has gravitational PE = mgh (where h is measured from the lowest point) and zero kinetic energy. At the lowest point, all energy has converted to elastic PE in the cord (plus any remaining gravitational PE from the cord’s natural length offset).

Let L be the natural length of the cord, k the spring constant, and h_total the total fall distance. At the lowest point, the cord extension is x = h_total - L, and the jumper’s height above ground defines their final gravitational PE.

Energy conservation at the lowest point (taking the lowest point as zero height):

Solving this quadratic equation for h_total gives the maximum fall depth. Safety in bungee design requires this h_total to be less than the height above the ground.

The spring constant of bungee cords is typically 100-200 N/m for recreational cords. For a 70 kg jumper on a 10 m cord with k = 150 N/m, the maximum extension is about 13-15 m, giving a total fall of 23-25 m.

Elastic PE in engineering applications

Elastic potential energy stored in springs is used across a wide range of engineering systems.

Automotive suspension: Springs (coil or leaf) absorb road impacts and store energy that is then released gradually, reducing the transmission of shocks to the vehicle body. Shock absorbers convert this energy to heat rather than returning it, preventing continuous bouncing.

Mechanical watches: The mainspring of a mechanical watch stores elastic PE when wound. As it unwinds, it releases this energy to drive the gear train and escapement. A fully wound watch spring stores roughly 0.01-0.1 J, enough to run for 24-48 hours.

Seismic isolators: Buildings in earthquake zones are sometimes mounted on large elastomeric bearings or spring systems. These store elastic PE during ground motion and return it slowly, reducing the peak forces transmitted to the structure.

Valve springs: In internal combustion engines, valve springs close the intake and exhaust valves. They must store enough elastic PE to close the valve quickly against the inertia of the valve assembly, and must maintain their spring constant over millions of cycles.

Precision scales and balances: Calibration weights are measured by how much they extend a known spring. The spring constant defines the scale’s sensitivity. High-precision scales use leaf springs or flexural elements with very well-characterized k values.

Elastic potential energy beyond Hooke’s Law

The simple formula U = ½kx² assumes linear elastic behavior (Hooke’s Law holds). In real materials, the force-extension relationship is only linear within a limited range. Beyond the proportionality limit, the material still behaves elastically (returns to its original shape when force is removed) but the spring constant k is no longer constant.

For rubber, the force-extension curve is highly non-linear. Rubber is soft at small extensions, becomes stiffer at intermediate extensions, then softens again near failure. This non-linear response is described by hyperelastic material models (such as the Mooney-Rivlin or Neo-Hookean models). Bungee cords, which are rubber-based, must be analyzed with these models rather than the simple ½kx² formula.

For metals, the proportionality limit is followed by the yield strength. Between the proportionality limit and the yield point, the material is still elastic but the stress-strain curve is non-linear. Beyond the yield point, the material deforms plastically (permanently). Springs are designed to always operate below the yield strength. If a spring is over-extended past its yield point, its natural length increases and its spring constant changes permanently.

The elastic limit and ultimate tensile strength define the boundaries of safe spring operation. Modern spring design software uses finite element analysis (FEA) to calculate the full stress distribution in a coil, ensuring no part of the spring exceeds the material’s yield strength.

Energy density of springs

A useful way to compare spring-based energy storage with other storage technologies is through energy density (energy stored per unit mass or volume).

For a coil spring made of spring steel, the maximum energy storage per unit volume is approximately:

Where σ is the maximum allowable stress (before yield) and E is the Young’s modulus. For high-quality spring steel, σ ≈ 1,200 MPa and E ≈ 200 GPa, giving U/V ≈ 3,600 J/m³, or about 460 J/kg (using steel density 7,800 kg/m³).

This is far less than chemical energy storage:

Storage type	Energy density (J/kg, approx.)
High-strength steel spring	400-700
Rubber band	2,000-5,000
Lithium-ion battery	500,000-900,000
Gasoline	44,000,000

Springs are not competitive for bulk energy storage. Their advantage lies in power density (energy release rate), reliability, temperature tolerance, and cycle life. A spring can deliver its energy almost instantaneously, can operate at -200°C or +500°C, and can complete millions of cycles with minimal degradation, which batteries and fuel cells cannot match.