\magnification=\magstep2

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


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


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

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

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

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

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


$$
 I  \quad = \quad \sum { m_i \thinspace R_i^2 } 
$$

$$
{\rm mass\ of\ each\ little\ piece\ } \quad = \quad dm
$$

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


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

$$
\vec{F} \quad = \quad m \thinspace \vec{a}
$$

$$
\vec{\tau} \quad = \quad I \thinspace \vec{\alpha}
$$

$$
{\rm units\ of\ } \tau \quad = \quad 
                                   ({\rm kg * m^2}) 
                                \thinspace ({\rm {{rad}\over{s^2}} })
           \quad = \quad 
                        ({\rm {{kg * m}\over{s^2}} }) \thinspace ({\rm m})
           \quad = \quad 
                        {\rm N * m}
$$

$$
| \tau | \quad = \quad | r | \thinspace | F | \thinspace \sin(\theta) 
$$

$$
I_{\rm off-center} \quad = \quad I_{\rm central} \ + \ M d^2
$$

$$
I_{\rm central} \quad = \quad {{1}\over{12}} \thinspace M \thinspace \left( L^2 + L^2 \right)
  \quad = \quad 
                        {{1}\over{6}}  \thinspace M \thinspace L^2 
$$

$$
d \quad = \quad {{\sqrt{2}}\over{2}} \thinspace L
$$


$$
I_{\rm off-center} \quad = \quad {{1}\over{6}} M L^2 \ + \ {{1}\over{2}} M L^2
     \quad = \quad
                     {{2}\over{3}} M L^2
$$



\bye
