Mass on a Spring

How does a restoring force produce simple harmonic motion, and how much damping does it take to stop the oscillation?

Interactive · live simulation

Pluck the mass, then tune stiffness, mass, and damping. The trace is an oscilloscope of position; push the damping up to find the critical value where the oscillation just dies.

Mass on a Spring — simulation render — fig 1. Position vs. time across damping regimes: undamped (ζ=0), underdamped (decaying oscillation), and critically damped (no overshoot, fastest return).

Hooke plus a dashpot

x'' = -(k/m)x - (c/m)v. Two numbers organize everything: the natural frequency \omega = \sqrt{k/m} and the damping ratio \zeta = c/(2\sqrt{mk}). \zeta = 0 is undamped, 0 \lt \zeta \lt 1 underdamped, \zeta = 1 critically damped, \zeta \gt 1 overdamped.

fig 2. Energy. Constant for the undamped run (a conservation check); decaying for the damped runs as the dashpot dissipates mechanical energy.

Critical damping

The interesting threshold is \zeta = 1: the system returns to equilibrium as fast as possible without overshooting — the regime every door-closer and galvanometer is tuned to. Sweep c from 0 through 2\sqrt{mk} and watch the phase plot collapse from a closed loop to a single inward swoop.

Reading ω off the data

With no damping there is a closed form, x(t) = x_0\cos\omega t + (v_0/\omega)\sin\omega t. Measuring the time between zero-crossings in the CSV and comparing to \sqrt{k/m} is a satisfying way to confirm the simulation is telling the truth.

This chapter is drawn from the physics-lab study notes and renders. The longer write-ups and project essays live on the blog.