PX155 - D7 - damped oscillations

model motion of our oscillator through a viscous fluid with a drag force proportional to $\dot{x}$ , ie: $- b \dot{x}$
- PX155 - A2 - newton's second law: $m \ddot{x} = - k x - b \dot{x}$
- dividing by m on both sides:$$\ddot x + \omega^{2}x + \gamma \dot x =0$$
  where, $γ = \frac{b}{m}$
- switch to complex notation:

\ddot{z} + ω^{2} z + γ \dot{z} = 0 . . . [1]

- physically motivated:

z = a e^{p t}

where, $p$ might be complex
- $p$ is imaginary - oscillation
- $p$ is real and negative - decay
$z = a e^{p t}$
$\dot{z} = p a e^{p t} = p z$
$\ddot{z} = p^{2} a e^{p t} = p^{2} z$
- in $[1]$ : $p^{2} z + ω^{2} z + γ p z = 0$
$z (p^{2} + γ p + ω^{2}) = 0$
$e i t h e r z = 0$ boring
$o r p^{2} + γ p + ω^{2} = 0$

p = \frac{- γ \pm \sqrt{γ^{2} - 4 ω^{2}}}{2}

p = - \frac{γ}{2} \pm \sqrt{{(\frac{γ}{2})}^{2} - ω^{2}}

- $p$ can be complex, or real and negative but never real and positive