Springs & Damping

Second-order equations

An interactive lesson with 6 steps, about 5 minutes: you predict, drag and solve, with instant feedback on every answer. It runs in your browser and needs JavaScript.

What you’ll learn

• $x'' = -\omega^2 x$ gives oscillation $x = \cos(\omega t)$, with $\omega = \sqrt{k/m}$.
• Damping: $x'' + 2\gamma x' + \omega^2 x = 0$.
• Underdamped oscillates and decays. Critical returns fastest. Overdamped creeps back.

Steps

1. Newton’s second law is a differential equation
2. Period of a spring
3. Adding friction
4. Dying away
5. Natural frequency
6. Shock absorbers