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Simple Harmonic Motion

24 min Vibrations & Waves Math

Simple Harmonic Motion (SHM)

Springs, pendulums, vibrating strings — they all follow the sine wave, the most fundamental waveform in mathematics.

SHM Equation

$$x(t) = A\sin(\omega t + \phi)$$

$A$ = amplitude, $\omega = 2\pi f$ = angular frequency, $\phi$ = phase angle.

Period: $T = 1/f = 2\pi/\omega$.

Mass-Spring System

$$\omega = \sqrt{\frac{k}{m}}, \qquad T = 2\pi\sqrt{\frac{m}{k}}$$

$k$ = spring constant (N/m), $m$ = mass (kg).

Simple Pendulum

$$T = 2\pi\sqrt{\frac{L}{g}}$$

$L$ = length, $g$ = 9.8 m/s². Period depends only on length, not mass.

Example 1

A 0.5 kg mass on a spring ($k = 200$ N/m). Find the frequency and period.

$\omega = \sqrt{200/0.5} = \sqrt{400} = 20$ rad/s.

$f = \omega/(2\pi) = 20/(2\pi) \approx 3.18$ Hz.

$T = 1/f \approx 0.314$ s.

Example 2

A pendulum is 1.0 m long. Find its period.

$T = 2\pi\sqrt{1.0/9.8} = 2\pi(0.3194) \approx 2.01$ s.

Example 3

$x(t) = 3\sin(4\pi t)$. Find $A$, $f$, and $T$.

$A = 3$, $\omega = 4\pi$ → $f = 4\pi/(2\pi) = 2$ Hz, $T = 0.5$ s.

1. Mass-spring: $k = 50$ N/m, $m = 2$ kg. Find $T$.

2. Pendulum: $L = 0.25$ m. Find $T$.

3. $x(t) = 5\cos(10t)$. Find $f$.

4. What spring constant gives $f = 5$ Hz with a 0.1 kg mass?

5. A grandfather clock pendulum has $T = 2$ s. Find $L$.

6. Does pendulum period depend on amplitude (for small swings)?

Show Answer Key

1. $T = 2\pi\sqrt{2/50} = 2\pi(0.2) \approx 1.26$ s

2. $T = 2\pi\sqrt{0.25/9.8} \approx 1.00$ s

3. $\omega = 10$ → $f = 10/(2\pi) \approx 1.59$ Hz

4. $\omega = 2\pi(5) = 10\pi$, $k = m\omega^2 = 0.1(100\pi^2) \approx 98.7$ N/m

5. $L = gT^2/(4\pi^2) = 9.8(4)/(4\pi^2) \approx 0.993$ m ≈ 1 m

6. No — period is independent of amplitude for small oscillations.