Elastic Potential Energy Formula (U = ½kx²): Calculator & Examples

Calculate spring energy with U = ½kx². Link to F = kx, unit traps (cm vs m), worked examples, and free Hooke's law and PE calculators.

What Elastic Potential Energy Asks

Elastic potential energy is the energy stored in a stretched or compressed spring. Intro physics writes it as U = ½ k x² (also written PEₛ = ½kx² or Us = ½kx²).

k is the spring constant in N/m. x is the displacement from the unstretched length in meters — not the full coil length. Energy comes out in joules.

Hooke's law F = kx gives the force; elastic PE stores the work done while stretching. Check force problems on the free Hooke's law calculator, then use U = ½kx² for energy.

The Formula

U = ½ k x² — elastic potential energy in joules.

If you know U and k: |x| = √(2U / k).

If you know U and x: k = 2U / x².

Because x is squared, stretch and compression store the same energy for the same |x|. Doubling the stretch multiplies stored energy by four.

How U Connects to F = kx

The spring force grows linearly with x. The work to stretch from 0 to x is the area under the F-vs-x line: a triangle with area ½ × base × height = ½ × x × (kx) = ½kx². That work becomes elastic potential energy in an ideal spring.

For gravitational PE = mgh instead of spring energy, see how to calculate potential energy (mgh).

Units

k → N/m.

x → m (convert cm first: divide by 100).

U → J (1 J = 1 N·m).

Worked Examples

Example 1: find U

k = 200 N/m, x = 0.15 m. U = ½(200)(0.15)² = 100 × 0.0225 = 2.25 J.

Example 2: leave x in cm by mistake

Same spring, x written as 15 instead of 0.15. Fake U = ½(200)(15)² = 22,500 J — wrong by a factor of 10,000. Always convert centimeters to meters.

Example 3: find stretch from energy

U = 0.80 J, k = 160 N/m. x² = 2U/k = 1.6/160 = 0.010. |x| = 0.10 m = 10 cm.

Example 4: compare to Hooke force

k = 200 N/m, x = 0.15 m. Force magnitude F = kx = 30 N, but energy U = 2.25 J. Do not treat F and U as interchangeable.

When the Formula Fails

Past the elastic limit (permanent deformation) — F ≠ kx and U ≠ ½kx².

Non-ideal springs with friction or plasticity inside the coil.

Using gravitational mgh when the problem is about a spring, or the reverse.

Common Mistakes

Writing U = kx or U = ½kx instead of ½kx².

Leaving x in centimeters while k is in N/m.

Using total spring length instead of stretch from equilibrium.

Confusing elastic PE with kinetic energy ½mv² or gravitational mgh.

Check Related Tools

Hooke's law calculator — F = kx for force, k, or x.

Potential energy calculator — gravitational PE = mgh.

Kinetic energy calculator — KE = ½mv².

SHM calculator — mass-spring period T = 2π√(m/k).

Learn U = ½kx² and keep force problems on F = kx.

Articles

All articles · Browse all calculators