\magnification=\magstep2


$$
 a_y \quad = \quad { {d^2 y} \over {d t^2}} \quad = 
$$


$$
  y(t) \quad = \quad  A \cos \left( \omega t + \phi \right) 
$$


$$
  {{ d \thinspace f(x) } \over {dx } } \quad = \quad  A
$$


$$
  {{ d \thinspace f(x) } \over {dx } } \quad = \quad  0
$$

$$
  {{ d \thinspace f(x) } \over {dx } } \quad = \quad  f(x)
$$


$$
  {{ d \thinspace f(x) } \over {dx } } \quad = \quad - f(x)
$$

$$
  {{ d \thinspace f(x) } \over {dx } } \quad = \quad  A f(x)
$$


$$
  {{ d^2 \thinspace f(x) } \over {d x^2 } } \quad = \quad  0
$$

$$
  {{ d^2 \thinspace f(x) } \over {d x^2 } } \quad = \quad  A
$$

$$
  {{ d^2 \thinspace f(x) } \over {d x^2 } } \quad = \quad  A f(x)
$$


$$
  {{ d^2 \thinspace f(x) } \over {d x^2 } } \quad = \quad  -A f(x)
$$


$$
\eqalign{
  I \quad &= \quad I_{\rm com} \quad \ \ \ \ + \quad M x^2   \cr
   \cr
          &= \quad {{1}\over{12}} ML^2 \ \ + \quad M x^2 \cr
}
$$

$$
\tau \quad = \quad Mgx \sin(\theta)
$$

$$
\eqalign{
 \alpha \quad &= \quad {{ \tau }\over{I_{\rm tot}}} \cr
  \cr
         &= \quad \left( {{ M g x } \over { {{1}\over{12}} M L^2 + M x^2 }} 
                 \right)  \sin(\theta) 
}
$$


$$
 \vec{\alpha}  \quad   \simeq \quad - \left( {{ M g x } \over { {{1}\over{12}} M L^2 + M x^2 }} 
                 \right)  \vec{\theta}
$$


$$
  \sin(\theta) \quad \simeq \quad \theta  \quad\quad\quad\quad 
       {\rm in\ radians}
$$

$$
\alpha \quad = \quad {{d^2 \theta}\over{d t^2}}
$$


$$
 {{d^2 \theta}\over{dt^2}}  \quad   \simeq \quad - \left( {{ M g x } \over { {{1}\over{12}} M L^2 + M x^2 }} 
                 \right)  \theta
$$


\bye
