Simple Harmonic Motion — Sine Waves Everywhere
Simple Harmonic Motion
A child on a swing, a guitar string vibrating, your heartbeat, the AC current in your wall outlet — they all follow the same mathematical curve: the sine wave.
$$x(t) = A \sin(2\pi f t + \phi)$$
where $A$ = amplitude (how far it moves), $f$ = frequency (oscillations per second, Hz), $t$ = time, and $\phi$ = phase (starting position).
The period is $T = 1/f$, and the angular frequency is $\omega = 2\pi f$.
This is the sine function from trigonometry, transformed with the amplitude, frequency, and phase parameters you learn about in class.
The Pendulum
For small swings, a pendulum's period depends only on its length:
$$T = 2\pi\sqrt{\frac{L}{g}}$$
This uses $\pi$, a square root, and division. A 1-meter pendulum swings with period $T = 2\pi\sqrt{1/9.8} \approx 2.01$ seconds — this is how grandfather clocks keep time.
The note "middle C" on a piano has frequency $f = 261.6$ Hz. What is its period and wavelength in air (speed of sound = 343 m/s)?
Period: $T = \frac{1}{f} = \frac{1}{261.6} \approx 0.00382$ s $= 3.82$ ms
Wavelength: $\lambda = \frac{v}{f} = \frac{343}{261.6} \approx 1.31$ m
The sound wave for middle C is about 1.3 meters long. Division — that's all you need.
Resonance — When Math Gets Dangerous
Every object has a natural frequency. If you push it at that frequency, oscillations grow without bound — this is resonance. It's why:
- Opera singers can shatter glass
- The Tacoma Narrows Bridge collapsed in 1940
- Soldiers break step when crossing bridges
- Earthquake damage depends on building height
The sine function from trigonometry describes every vibration in nature. Frequency, period, and wavelength are connected by simple division. Understanding trig functions means understanding music, earthquakes, electronics, and light.