\magnification=\magstep1

$$
\vec{\tau} \quad = \quad \vec{r} \times \vec{F}_g \ + \ 
                      \vec{r} \times \vec{T}
$$

$$
\vec{\tau} \quad = \quad |L| \thinspace |mg| \thinspace \sin(\theta) 
                          \quad {\rm into\ page}
$$

$$
\vec{\tau} \quad \simeq \quad |L| \thinspace |mg| \thinspace \theta 
                          \quad {\rm into\ page} 
$$

$$
\tau \quad = \quad - L m g \ \theta 
$$

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

$$
m L^2 \thinspace \alpha \quad = \quad - L m g \ \theta
$$

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

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

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

$$
{{d^2 \theta}\over{dt^2}} \quad = \quad - \omega^2 \ \theta
$$

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

$$
P \quad = \quad {{ 2 \pi}\over{\omega}} \quad = \quad 2 \pi 
              \thinspace \sqrt{ {{L}\over{g}} }
$$

$$
g \quad = \quad {{ 4 \pi^2 L} \over {P^2}}
$$

$$
P^2 \quad = \quad \left( {{4 \pi^2}\over{g}} \right) \thinspace L
$$


$$
\vec{\tau} \quad = \quad \vec{R} \times \vec{F}_g \ + \ 
                      \vec{0} \times \vec{T}
$$

$$
\vec{\tau} \quad = \quad |R| \thinspace |mg| \thinspace \sin(\theta) 
                          \quad {\rm into\ page}
$$

$$
\vec{\tau} \quad \simeq \quad |R| \thinspace |mg| \thinspace \theta 
                          \quad {\rm into\ page} 
$$


$$
\tau \quad = \quad - R m g \ \theta 
$$

$$
I \thinspace \alpha \quad = \quad - R m g \ \theta 
$$


$$
I \quad = \quad I_{\rm Center\ of\ Mass} \ + \ m R^2
$$

$$
I \quad = \quad {{1}\over{2}} \thinspace m R^2 \ + \ m R^2
               \quad = \quad {{3}\over{2}} \thinspace m R^2
$$


$$
{{3}\over{2}} \thinspace m R^2 \thinspace \alpha \quad = \quad - R m g \ \theta
$$

$$
\alpha \quad = \quad - \left( {{m g R}\over{I}} \right) \ \theta
$$

$$
{{d^2 \theta}\over{dt^2}} \quad = \quad - \left( {{m g R}\over{I}} \right) \ \theta
$$

$$
\omega \quad = \quad \sqrt{ {{m g R}\over{I}} }
$$

$$
{{d^2 \theta}\over{dt^2}} \quad = \quad - \omega^2 \ \theta
$$

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

$$
P \quad = \quad {{ 2 \pi}\over{\omega}} \quad = \quad 2 \pi 
              \thinspace \sqrt{ {{L}\over{g}} }
$$

$$
g \quad = \quad {{ 4 \pi^2 L} \over {P^2}}
$$

$$
P^2 \quad = \quad \left( {{4 \pi^2}\over{g}} \right) \thinspace L
$$



\bye
