🔬 Spring Potential Energy Calculator

Elastic potential energy is the energy stored in a stretched or compressed spring. The formula U = ½ × k × x² gives the stored energy in joules, where k is the spring constant (N/m) and x is the displacement from the equilibrium position (m). This quadratic relationship means doubling the stretch stores four times as much energy—a principle critical to everything from archery bows to mechanical watches and car suspensions.

When you compress a spring in a toy dart gun or wind a mechanical watch mainspring, you are storing elastic potential energy. When released, this energy converts to kinetic energy. This calculator helps physics students, engineers, and designers quantify how much energy a spring system can store or release.

The calculator uses U = ½ × k × x² to solve for any one unknown given the other two. Since energy depends on x², a spring stretched twice as far stores four times the energy, and a spring stretched three times as far stores nine times the energy.

1. Enter any two of: Spring Constant (k), Displacement (x), or Potential Energy (U).
2. Leave the third field blank—that is what will be calculated.
3. Click the Calculate button.
4. Review the result for the unknown value.

Why is there a ½ in the formula?

The factor ½ arises because the force increases linearly from zero to kx as the spring is stretched. The average force during stretching is ½kx, and work = average force × distance = ½kx × x = ½kx².

What is the difference between spring force and spring energy?

Force (F = kx) tells you how hard the spring pushes or pulls at a given displacement. Energy (U = ½kx²) tells you how much work was done to reach that displacement, or how much work the spring can do when released.

Can potential energy be negative?

No. Since x is squared in U = ½kx², the energy is always non-negative. Both stretching and compression store positive energy. The sign of x itself matters for force direction, but not for energy.

How is spring energy used in real devices?

Mechanical watches store energy in a wound mainspring. Archery bows store energy when drawn. Pogo sticks and trampolines use spring energy to launch people upward. Car suspensions absorb road shocks by compressing springs.

What if my spring is non-linear?

The formula U = ½kx² only applies to linear (Hookean) springs. Non-linear springs require integrating force over displacement numerically: U = ∫F(x)dx. Many real materials behave non-linearly at large deformations.

How does spring energy relate to simple harmonic motion?

In a spring-mass oscillator, energy alternates between potential (U = ½kx²) and kinetic (K = ½mv²). At maximum displacement, all energy is potential; at equilibrium, all energy is kinetic. The total E = ½kA² is conserved (without friction).

Use SI units throughout: k in N/m, x in metres, U in joules. The stored energy is always positive regardless of whether the spring is stretched or compressed, because x is squared. At maximum compression or extension, all mechanical energy in a spring-mass system is potential; at the equilibrium position, all energy is kinetic. The total mechanical energy E = ½kA², where A is the amplitude of oscillation.