\magnification=\magstep1 

$$
\omega_0 \quad = \quad \sqrt{ {{k}\over{m}} }
$$

$$
\vec{F}({\rm air}) \quad = \quad - b \vec{v}
$$

$$
F(t) \quad = \quad F_{\rm max} \thinspace e^{i \omega t}
$$

$$
m {{d^2x}\over{dt^2}} \ + \ b {{dx}\over{dt}} \ + \ k x 
    \quad = \quad  F_{\rm max} \thinspace e^{i \omega t}
$$

$$
x(t) \quad = \quad A e^{i(\omega t + \phi)}
$$

$$
{{dx}\over{dt}} \quad = \quad i \omega A e^{i(\omega t + \phi)}
                \quad = \quad i \omega \thinspace x(t)
$$

$$
{{d^2x}\over{dt^2}} \quad = \quad - \omega^2 A e^{i(\omega t + \phi)}
                \quad = \quad - \omega^2 \thinspace x(t)
$$

$$
-m \omega^2 A e^{i(\omega t + \phi)} \ + \ i b \omega A e^{i(\omega t + \phi)} 
     \ + \ k A e^{i(\omega t + \phi)} \quad = \quad
     F_{\rm max} e^{i \omega t}
$$


$$
-m \omega^2 A e^{i\phi} \ + \ i b \omega A e^{i \phi} 
     \ + \ k A e^{i \phi} \quad = \quad
     F_{\rm max} 
$$


$$
- \omega^2 A e^{i\phi} \ + \ i { {b \omega}\over{m}} A e^{i \phi} 
     \ + \ {{k}\over{m}} A e^{i \phi} \quad = \quad
     {{F_{\rm max}}\over{m}}
$$


$$
- \omega^2 A e^{i\phi} \ + \ i { {b \omega}\over{m}} A e^{i \phi} 
     \ + \ {\omega_0}^2 A e^{i \phi} \quad = \quad
     {{F_{\rm max}}\over{m}}
$$


$$
- \omega^2 A \ + \ i { {b \omega}\over{m}} A
     \ + \ {\omega_0}^2 A \quad = \quad
     {{F_{\rm max}}\over{m}} e^{-i \phi}
$$


$$
- \omega^2 A \ + \ i { {b \omega}\over{m}} A
     \ + \ {\omega_0}^2 A \quad = \quad
     {{F_{\rm max}}\over{m}} \thinspace \left[ \cos(\phi) - i \sin(\phi) \right]
$$

% this is the real parts only
$$
- \omega^2 A \ + \ {\omega_0}^2 A \quad = \quad
({\omega_0}^2 - \omega^2) A \quad = \quad
     {{F_{\rm max}}\over{m}} \thinspace \cos(\phi)
$$

% this is the imaginary parts only
$$
 { {b \omega}\over{m}} A \quad = \quad
     - {{F_{\rm max}}\over{m}} \thinspace \sin(\phi) 
$$

$$
\cos(\phi) \quad = \quad { {A m (\omega_0^2 - \omega^2)}\over{F_{\rm max}} }
$$

$$
\sin(\phi) \quad = \quad - { {A b \thinspace \omega}\over{F_{\rm max}} }
$$

$$
F_{\rm max} \quad = \quad \sqrt{ A^2 m^2 {( \omega_0^2 - \omega^2)}^2
                  \ + \  A^2 b^2 \omega^2 }
$$

$$
F_{\rm max} \quad = \quad A \sqrt{ m^2 {( \omega_0^2 - \omega^2)}^2
                  \ + \  b^2 \omega^2 }
$$

$$
A \quad = \quad {{ F_{\rm max} } \over 
              { \sqrt{  m^2 {( \omega_0^2 - \omega^2)}^2 \ + \  b^2 \omega^2 } } }
$$

$$
A \quad = \quad {{ F_{\rm max} / m } \over 
              { \sqrt{  {( \omega_0^2 - \omega^2)}^2 \ + \  {{b^2}\over{m^2}} \omega^2 } } }
$$

$$
\phi \quad = \quad \tan^{-1}{\left( { ({{b}\over{m}}) \thinspace \omega}
                     \over{{\omega_0^2 - \omega^2}} \right) }
$$


$$
{{Q}\over{\pi}}  \quad = \quad {\rm number\ of\ cycles\ for\ amplitude\ to\ decay\ to\ } \ \left( {{1}\over{e}} \right)
$$

$$
Q \quad = \quad \pi \ \bigl[ {\rm number\ of\ cycles\ for\ amplitude\ to\ decay\ to\ } \ \left( {{1}\over{e}} \right) \ \bigr] 
$$


\bye 
