\magnification=\magstep2

$$
   I \quad = \quad \sum{ M_i {R_i}^2 }
$$


$$
   I_{\rm central}  =  { 1 \over {12}} m L^2
$$


$$
\eqalign{
   I_{\rm shifted}  &=  I_{\rm central}  +  m x^2    \cr
                    &=  { 1 \over{12}} m L^2  +  m x^2    \cr
}
$$

$$
\eqalign{
   I_{\rm 2-rod}  &=  I_{\rm central}  +  I_{\rm shifted}    \cr
                  &=   ( { 1 \over{12}} m L^2)  +  
                          ( { 1 \over{12}} m L^2  +  M x^2  )  \cr
}
$$

$$
{\rm surface \ density\ \  } \sigma = { M \over {A}} =  
        {M \over { L^2 } }  
$$

$$
dM = \sigma * ({\rm \ area \ }) = \sigma * (L * dx) 
$$

$$
\eqalign{
   dI  &=  ( {1 \over{12}} \thinspace {dM} \thinspace L^2 ) + ({dM} \thinspace x^2)   \cr
  \cr
   dI  &=  ( {1 \over{12}} (\sigma L dx) L^2 ) + ((\sigma L dx) x^2)   \cr
  \cr
   dI  &=  ( {1 \over{12}} \sigma L^3 dx ) + (\sigma L x^2 dx)   \cr
}
$$

$$
 I  \quad = \quad \int { dm \thinspace R^2 } 
$$


$$
 \eqalign{
  W \thinspace &= \thinspace \int{\vec{F}\cdot{d}\vec{x}} \cr \cr W \thinspace &= \thinspace {{1}\over{2}} m v_f^2 - {{1}\over{2}} m v_i^2 \cr \cr P \thinspace &= \thinspace \vec{F} \cdot \vec{v} \cr
 }
 \qquad
 \eqalign{
  W \thinspace &= \thinspace \int{\vec{\tau}\cdot{d}\vec{\theta}} \cr \cr W \thinspace &= \thinspace {{1}\over{2}} I {\omega}_f^2 - {{1}\over{2}} I {\omega}_i^2 \cr \cr P \thinspace &= \thinspace \vec{\tau} \cdot \vec{\omega}  \cr 
 }
$$
\bye
