Elastic Potential Energy Calculator
Calculate elastic potential energy stored in a spring using E = ½kx². Switch modes to solve for energy, spring constant, stretch/compression, or combine two springs. Use metric or US units — every value has its own unit box. Includes quick picks, step-by-step, and a mini spring + energy visual.
Background
When a spring is stretched or compressed by a distance x, it stores energy. For an ideal (linear) spring, Hooke’s law is F = kx and the stored energy is E = ½kx². (Same formula for compression or stretch — just use the magnitude of x.)
How to use this calculator
- Choose a mode: solve for E, k, x, or keq.
- Type each value and pick its unit in the box next to it: x in m, cm, mm, in or ft; k in N/m, N/cm or lbf/in; E in J, kJ or ft·lbf.
- Choose which unit you want the answer in — it is also shown in all the other units.
- Optional: enter x_max to see limit warnings and an E_max notch on the energy bar.
- Click Calculate (or press Enter in a box) to see outputs, steps, and the visual.
How this calculator works
- Hooke’s law: F = kx
- Elastic potential energy: E = ½kx²
- Solve for k: k = 2E/x²
- Solve for x: x = √(2E/k)
- Combine springs: Parallel k_eq = k₁ + k₂, Series 1/k_eq = 1/k₁ + 1/k₂
- Everything is converted to SI (meters, N/m, joules) internally, then shown back in the units you chose.
Formula & Equation Used
Elastic energy: E = ½kx²
Hooke’s law: F = kx
Rearranged: k = 2E/x², x = √(2E/k)
Energy limit from x_max: E_max = ½k x_max²
Example Problems & Step-by-Step Solutions
Example 1 — Find energy from k and x
A spring has k = 200 N/m and is stretched x = 0.10 m. Find E.
- E = ½kx²
- E = 0.5 × 200 × (0.10)²
- E = 100 × 0.01 = 1.0 J
Example 2 — Solve for k
A spring stores E = 2.0 J when stretched x = 0.20 m. Find k.
- k = 2E/x²
- k = (2×2.0) / (0.20)² = 4 / 0.04
- k = 100 N/m
Example 3 — Solve for x
A spring with k = 400 N/m stores E = 5.0 J. Find x.
- x = √(2E/k)
- x = √(10/400) = √(0.025)
- x ≈ 0.158 m (about 15.8 cm)
Frequently Asked Questions
Q: Does compression use the same formula as stretching?
Yes. Use the magnitude of displacement: E = ½k|x|².
Q: Why is there a ½ in the energy formula?
Because the force increases from 0 to kx as you stretch the spring. The average force is ½kx, so work is (½kx)×x.
Q: What does a larger k mean physically?
A larger k is a stiffer spring. For the same x, it stores more energy.
Q: When does this calculator stop being accurate?
If the spring isn’t linear (it doesn’t follow F=kx), then E=½kx² may not match reality.
Q: Which units can I use?
Stretch in m, cm, mm, in or ft; spring constant in N/m, N/cm or lbf/in; energy in J, kJ or ft·lbf. Pick the unit next to each value, and choose the unit you want the answer in.
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