\magnification=\magstep1

$$
W_g \quad = \quad \int_0^H \vec{F_g} \cdot d\vec{y} 
   \quad = \quad \int_0^H | m g | \ | dy | \ \cos(180^\circ) 
$$

$$
W_g \quad = \quad -mg \ \int_0^H \ dy 
   \quad = \quad -mg (H - 0) \quad = \quad -mgH
$$

$$
{\rm GPE } \quad = \quad - W_g \quad = \quad 
          - \int_0^H \vec{F_g} \cdot d\vec{y} 
$$

$$
{\rm GPE } \quad = \quad m g y
$$

$$
F_y \quad = \quad - {{\partial {\rm (GPE)}}\over{\partial y}} 
         \quad = \quad - {{\partial (mgy)}\over{\partial y}} \quad = \quad - mg
$$

$$
F_x \quad = \quad - {{\partial {\rm (GPE)}}\over{\partial x}} 
         \quad = \quad - {{\partial (mgy)}\over{\partial x}} \quad = \quad 0
$$

$$
{\rm Force\ in\ x\ dir\ } \quad = \quad - {{\partial {\rm (PE)}}\over{\partial x}} 
$$
$$
{\rm Force\ in\ y\ dir\ } \quad = \quad - {{\partial {\rm (PE)}}\over{\partial y}} 
$$
$$
{\rm Force\ in\ z\ dir\ } \quad = \quad - {{\partial {\rm (PE)}}\over{\partial z}} 
$$

$$
{\rm SPE}(x) \quad = \quad \left( 126 \ {{\rm J}\over{\rm m^2}} \right) \ x^2
$$

$$
F_x \quad \quad - {{\partial {\rm (SPE)}}\over{\partial x}} 
$$
$$
F_x \quad = \quad - \left( 126 \ {{\rm J}\over{\rm m^2}} \right) \ 2 x
           \quad = \quad - \left( 252 \ {{\rm J}\over{\rm m}} \right) x
$$

$$
{\rm PE}(x) \quad =\quad 10 \thinspace  x^2 \thinspace \cos(x) \thinspace e^{-x}
$$


\bye
