Differential Equations B — Oscillation & Damping

The harmonic oscillator — y″ = −ω²·y

ELI5: when a thing's acceleration (second derivative) points back toward home in proportion to how far it has strayed, it can't settle — it swings. Springs, pendulums, quartz watches, vibrating beams. The natural angular frequency is ω₀ = √(k/m): stiffer spring → faster; heavier mass → slower.

k_{s} := 200 N/m

= 200 N / m spring stiffness

m_{s} := 0.5 kg

= 0.5 kg hanging mass

\omega_{0} = \sqrt{\frac{k_{s}}{m_{s}}}

= 20 Hz natural angular frequency (rad/s)

f_{osc} = \frac{\omega_{0}}{2 \cdot \pi}

= 3.1831 Hz cycles per second (Hz)

Real-world hook: ω₀ is the note a wine glass rings at, the sway period a skyscraper is tuned away from, and the frequency a quartz watch counts to keep time.

Try it yourself: a spring–mass system (in SI units) has k/m = 400. What is its natural frequency ω₀ = √(k/m)?

w0_{you} = 20

✏️ Your turn: find ω₀ = √(k/m) for k/m = 400. The check confirms ω₀² = 400 — so it never states the root.

✓ pass

abs(w0_{you}^2 - 400) < 1e-6 and w0_{you} > 0

green when ω₀² = 400