```markdown
# 6.2 The damped harmonic oscillator

A mass $m$ on a spring of stiffness $k$, with a damping force proportional to its velocity, obeys

$$m \frac{d^2x}{dt^2} + c \frac{dx}{dt} + kx = 0 \tag{6.1}$$

Substituting $x(t) = e^{rt}$ gives the characteristic equation $mr^2 + cr + k = 0$, whose roots are

$$r = \frac{-c \pm \sqrt{c^2 - 4mk}}{2m} \tag{6.2}$$

When $c^2 < 4mk$ the system is underdamped and oscillates with a decaying amplitude:

$$x(t) = Ae^{-\gamma t} \cos(\omega_d t + \phi), \quad \gamma = \frac{c}{2m}, \quad \omega_d = \sqrt{\omega_0^2 - \gamma^2} \tag{6.3}$$

where $\omega_0 = \sqrt{k/m}$ is the natural frequency. The energy of the oscillation decays as

$$E(t) = \frac{1}{2} k A^2 e^{-2\gamma t} \tag{6.4}$$

and the quality factor of the oscillator is

$$Q = \frac{\omega_0}{2\gamma} = \frac{\sqrt{mk}}{c} \tag{6.5}$$

**Example.** For $m = 0.50$ kg, $k = 200$ N/m and $c = 0.80$ kg/s:

| Quantity          | Symbol | Value      |
|-------------------|--------|------------|
| Natural frequency | $\omega_0$ | 20.0 rad/s |
| Decay rate        | $\gamma$ | 0.80 s$^{-1}$ |
| Damped frequency  | $\omega_d$ | 19.98 rad/s |
| Quality factor    | $Q$ | 12.5 |
```