
\magnification=\magstep1


$$
\vec{\tau} \quad = \quad \vec{r} \times \vec{F_g}
$$

$$
\vec{\tau} \quad = \quad \vec{L} \times (m \vec{g})
$$

$$
\vec{\tau} \quad = \quad m g L \thinspace \sin(\theta)
$$

$$
\vec{\tau} \quad = \quad m g L \thinspace \sin(\theta)  
                 \qquad ({\rm into\ page})
$$

$$
I \quad = \quad m L^2
$$

$$
\vec{\tau} \quad = \quad I \vec{\alpha}
$$


$$
\vec{\alpha} \quad = \quad { {\vec{\tau}} \over {I} }
$$

$$
\vec{\alpha} \quad = \quad \left( {g}\over{L} \right) 
                      \thinspace \sin(\theta) 
                      \qquad ({\rm into\ page})
$$

$$
\vec{\alpha} \quad = \quad \left( {g}\over{L} \right) 
                      \thinspace \theta 
                      \qquad ({\rm into\ page})
$$

$$
\vec{\alpha} \quad = \quad - \left( {g}\over{L} \right) 
                      \thinspace \vec{\theta}
$$

$$
{ { d^2 {\vec{\theta}} } \over { d t^2 } }
      \quad = \quad 
     - \left( {g}\over{L} \right) \thinspace \vec{\theta}
$$


$$
{ { d^2 {\vec{x}} } \over { d t^2 } }
      \quad = \quad 
     - \left( {k}\over{m} \right) \thinspace \vec{x}
$$

$$
\theta(t) \quad = \quad A \cos( \omega t + \phi ) 
$$

$$
\omega \quad  = \quad \sqrt{ {g}\over{L} }
$$

                    


\bye
