Oscillations

Hooke's Law & Simple Harmonic Motion: Spring restoring force is $F = -kx$. For an ideal spring, natural angular frequency is $\omega_0 = \sqrt{k/m}$ with period $T = 2\pi\sqrt{m/k}$.

Damped Harmonic Oscillator: Differential equation $m\ddot{x} + c\dot{x} + kx = 0$ has damping ratio $\zeta = \frac{c}{2\sqrt{km}}$. Underdamped ($\zeta < 1$) oscillates with decaying amplitude; critically damped ($\zeta = 1$) returns to equilibrium fastest without overshoot.

Phase Space: Trajectories in $(x, \dot{x})$ phase space form stable spirals (damped) or closed ellipses (undamped).

Operating Instructions: Use the sidebar tab switcher to simulate Simple Pendulums and Spring Mass Oscillations with customizable spring constant $k$ and damping $c$.