\magnification=\magstep2


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


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

$$
  x(t) \quad = \quad  A \cos \left( \omega t \right) 
$$


$$
  M \quad = \quad {{4}\over{3}} \pi R_E^3 \thinspace \rho
$$

$$
  a \quad = \quad {{ G M } \over {R_E^2}} 
             \qquad  = \quad {{ 4 \pi G \rho R_E^3} \over {3 R_E^2}} 
             \qquad  = \quad { {{4}\over{3}} \pi G \rho } \thinspace \thinspace R_E 
$$


$$
  m \quad = \quad {{4}\over{3}} \pi x^3 \thinspace \rho
$$

$$
  a \quad = \quad {{ G m } \over {x^2}} 
             \qquad  = \quad {{ 4 \pi G \rho x^3} \over {3 x^2}} 
             \qquad  = \quad { {{4}\over{3}} \pi G \rho } \thinspace \thinspace x 
$$

$$
  \vec{a} \quad = \quad  - \left( { {{4}\over{3}} \pi G \rho }  \right) 
                     \thinspace \thinspace \vec{x}
$$

$$
  {  {d \vec{x}^2} \over { d t^2 }  }
        \quad = \quad  - \left( { {{4}\over{3}} \pi G \rho }  \right) 
                     \thinspace \thinspace \vec{x}
$$

$$
 \omega \quad \equiv  \quad  \sqrt{ {{4}\over{3}} \pi G \rho }
$$

$$
  x(t) \quad = \quad R_E \thinspace \thinspace \cos(\omega t)
$$


\bye

