\magnification=\magstep2

$$
\vec{F} = - k{\vec{x}}
$$

$$
m {{d^2\vec{x}}\over{dt^2}} = - k{\vec{x}}
$$

$$
m {{d^2\vec{x}}\over{dt^2}} + k{\vec{x}} = 0
$$

$$
\vec{x}(t) = \vec{A} \cos ( \omega t + \phi ) 
$$

$$
\omega = \sqrt{ {k}\over{m} }
$$

$$
\vec{F}_{\rm resist} = - b \vec{v} 
$$

$$
\vec{F} = - k{\vec{x}} - b \vec{v}
$$

$$
m {{d^2\vec{x}}\over{dt^2}} = - k{\vec{x}} - b {{d\vec{x}} \over {dt}}
$$

$$
m {{d^2\vec{x}}\over{dt^2}} + b {{d\vec{x}} \over {dt}} + k{\vec{x}} = 0
$$

$$
x(t) = A \cos ( \omega t + \phi ) e^{- t / \tau }  \qquad {\rm ???}
$$


$$
{{d x}\over{ d t }}  =  
        - {{1}\over{\tau}} A \cos ( \omega t + \phi ) e^{- t / \tau }
        - {\omega } A \sin ( \omega t + \phi ) e^{-t / \tau }
$$

$$\eqalign{
{{d^2 x}\over{ d t^2 }}  = &  
       \phantom{+\ } {{1}\over{\tau^2}} A \cos ( \omega t + \phi ) e^{- t / \tau } \cr
    & + {{\omega}\over{\tau} } A \sin ( \omega t + \phi ) e^{-t / \tau } \cr
    & + {{\omega}\over{\tau} } A \sin ( \omega t + \phi ) e^{-t / \tau } \cr
    & - {\omega^2} A \cos ( \omega t + \phi ) e^{- t / \tau } \cr
}$$


\vfill
\eject

$$
{m \over {\tau^2}} - { m \omega^2} - { {b}\over{\tau}} + k  =  0
$$

$$
{ {2 m \omega}\over{\tau}} - {b \omega} = 0
$$

$$
\tau = {{2 m} \over {b}}
$$

$$
\omega = \sqrt{ {{k}\over{m}} - {{b^2}\over{4 m^2}} }
$$

$$
{{k}\over{m}} - {{b^2}\over{4 m^2}} = 0
$$

$$
{{\rm max\ } \theta (t) } \quad = \quad A \thinspace {e^{-t/\tau} } + \thinspace 0
$$

$$
{{\rm max\ } \theta (t) } \quad = \quad A \thinspace {e^{-t/\tau} } 
$$

\bye
