Elastic Potential Energy Calculator — PE = ½kx²
Calculate elastic potential energy stored in a spring using Hooke's Law PE = ½kx². Covers spring constant, compression vs extension, and real applications.
Springs store energy when compressed or stretched, and release it when you let go. That's the simple version. The interesting part is how much energy: it scales with the square of displacement, which means stretching a spring twice as far stores four times as much energy. This property makes springs useful in everything from mousetraps to car suspension systems.
The CalcHub elastic potential energy calculator gives you stored energy from spring constant and displacement.
The Formula
PE = ½ × k × x²- PE = elastic potential energy (Joules, J)
- k = spring constant (N/m) — how stiff the spring is
- x = displacement from equilibrium (meters)
Spring Constant Reference Values
| Spring Type | Typical k (N/m) |
|---|---|
| Soft cushion spring | 100–500 |
| Pen click mechanism | ~200 |
| Car suspension | 15,000–30,000 |
| Engine valve spring | 20,000–80,000 |
| Stiff industrial spring | 100,000+ |
How to Use the Calculator
Enter the spring constant (in N/m or N/cm) and displacement (in meters or centimeters). The calculator returns elastic PE in joules. You can also enter force and displacement to back-calculate the spring constant.
Worked Example
A toy catapult stretches a rubber band that has an effective k = 800 N/m by 15 cm (0.15 m). How much energy is stored?
PE = 0.5 × 800 × 0.15² = 0.5 × 800 × 0.0225 = 9 J
When released, that 9 J converts to kinetic energy: ½mv² = 9 J. For a 10 g projectile:
v = √(2 × 9 / 0.01) = √1800 ≈ 42.4 m/s — about 153 km/h. Catapults are surprisingly energetic.
What's the elastic limit?
Hooke's Law only applies within the elastic limit. Beyond that, the spring deforms permanently (plastic deformation). The spring won't return to its original length. The formula PE = ½kx² only applies while the spring behaves linearly.
Does PE = ½kx² apply to compression and extension?
Yes — the formula uses x², so it doesn't matter whether x is positive (extension) or negative (compression). The stored energy is the same for equal magnitudes of stretch and squeeze (assuming the spring behaves symmetrically).
How is spring PE related to oscillation frequency?
For a mass m on a spring with constant k, the oscillation frequency is f = (1/2π) × √(k/m). Stiffer spring and lighter mass → higher frequency. A baby bouncer has low k and high m → low, slow bouncing frequency. The pendulum calculator covers the analogous case for gravity.