```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:

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

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

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

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             |

```