Last updated: 2026-07-01

Mass-Spring Oscillator (SHM) Calculator

A mass attached to a spring is the textbook example of simple harmonic motion. Pull the mass aside and release it, and the spring pulls it back with a force proportional to how far it has moved — Hooke's law, $F = -kx$. That linear restoring force makes the mass oscillate back and forth at a steady rate set by just two quantities: the mass $m$ and the spring constant $k$.

This calculator returns the period $T$ (the time for one complete cycle), the frequency $f$ (cycles per second) and the angular frequency $\omega$ (radians per second) from the mass and the spring constant.

What sets the rhythm

The oscillation speeds up when the spring is stiff and slows down when the mass is large:

Because both quantities sit under a square root, their effect is gentle: quadrupling the mass only doubles the period, and quadrupling the stiffness only halves it. Notice that gravity does not appear anywhere — a horizontal spring and a vertical spring oscillate at exactly the same rate, and the system keeps perfect time even in orbit. That is why a balance-spring, not a pendulum, regulates a wristwatch.

Formula

Worked example

A 0.5 kg block sits on a spring with a stiffness of 20 N/m. The angular frequency is:

The period is one full cycle:

and the frequency is its reciprocal, $f = 1/T = 1.007$ Hz — just over one full oscillation each second. Entering a mass of 0.5 kg and a spring constant of 20 N/m reproduces these results.

Mass-spring versus pendulum

A mass-spring system and a simple pendulum are both simple harmonic oscillators, and both have a period of the form $2\pi\sqrt{\text{inertia}/\text{restoring strength}}$. The difference is what supplies the restoring force. A pendulum is restored by gravity, so its period $2\pi\sqrt{L/g}$ depends on length and the local gravity. A spring supplies its own restoring force, so its period $2\pi\sqrt{m/k}$ depends on mass and stiffness and is unaffected by gravity. The two share the deeper trait of every simple harmonic oscillator: the period does not depend on the amplitude.

Energy in the oscillation

As the mass swings, energy moves between two forms. At the extremes the mass is momentarily still and all the energy is stored elastically in the spring, $\tfrac{1}{2}kA^2$, where $A$ is the amplitude. At the midpoint the spring is relaxed and all the energy is kinetic, $\tfrac{1}{2}mv_\text{max}^2$. In an ideal frictionless system the total stays constant and the two forms trade back and forth twice per cycle.

Limitations

The formulas assume an ideal spring that obeys Hooke's law over the whole range of motion, a massless spring, and no friction or air resistance. A real spring has its own mass, which slightly lowers the frequency, and damping gradually shrinks the amplitude without changing the period much for light damping. Stretch the spring far enough that it no longer responds linearly and the motion is no longer simple harmonic, so these results no longer apply.