\magnification=\magstep2
\baselineskip=25pt


$$
\eqalign{
   {\rm KE \ } &= 
          \thinspace\thinspace  {{1}\over{2}} M_1 v_1^2 \cr
          &= 
          \thinspace\thinspace  {{1}\over{2}} M_1 {( {R_1 \omega} )}^2 \cr
          &= 
          \thinspace\thinspace  {{1}\over{2}} M_1 {R_1}^2 {\omega}^2 \cr
          &= 
          \thinspace\thinspace  {{1}\over{2}} ( M_1 {R_1}^2) {\omega}^2  \cr
}
$$

$$
      I \thinspace = \thinspace M R^2
$$

$$
\eqalign{
   {\rm KE \ } &= 
          \thinspace\thinspace  {{1}\over{2}} I_1 {\omega}^2 \cr
}
$$

$$
\eqalign{
   {\rm KE \ } &= 
          \thinspace\thinspace  {{1}\over{2}} M_1 v_1^2 
           + \thinspace\thinspace  {{1}\over{2}} M_2 v_2^2 
           + \thinspace\thinspace  {{1}\over{2}} M_3 v_3^2 \cr
          &= 
          \thinspace\thinspace  {{1}\over{2}} ( M_1 {R_1}^2) {\omega}^2  
           + \thinspace\thinspace  {{1}\over{2}} ( M_2 {R_2}^2) {\omega}^2  
           + \thinspace\thinspace  {{1}\over{2}} ( M_3 {R_3}^2) {\omega}^2  \cr
          &= 
          \thinspace\thinspace  {{1}\over{2}} I_1 {\omega}^2  
           + \thinspace\thinspace  {{1}\over{2}} I_2 {\omega}^2  
           + \thinspace\thinspace  {{1}\over{2}} I_3 {\omega}^2  \cr
          &= 
          \thinspace\thinspace  {{1}\over{2}} ( I_1 + I_2 + I_3) {\omega}^2   \cr
          &= 
          \thinspace\thinspace  {{1}\over{2}} I_{tot} {\omega}^2   \cr
}
$$

$$
  \lambda(x) \thinspace = \thinspace 3 {\rm {{kg}\over{m}} }
                            + (3 {\rm  {{kg}\over{m}} }) 
                                ( { {x}\over{10 \rm m} } )
$$

$$
({\rm KE - E_{US}}) \quad = \quad   {{1}\over{2}} \thinspace I \thinspace \omega_{\rm new}^2
$$

$$
\lambda \quad = \quad {{M}\over{L}}
$$

$$
dm \quad = \quad \lambda \thinspace dx
$$

$$
dI \quad = \quad {\rm (mass) (distance\ from\ rotation\ axis)^2} \quad 
         = \quad dm \ x^2
$$

$$
I \quad = \quad \int dI \quad = \quad \int_{x=0}^{x=L} dm \ x^2 
             \quad = \quad \int_{x=0}^{x=L} \lambda \thinspace dx \ x^2
$$

$$
I \quad = \quad \lambda \int_0^L x^2 dx  \quad = \quad 
         \lambda \ \left( {{1}\over{3}} L^3 \right)
         \quad = \quad {{M}\over{L}} \ {{1}\over{3}} \thinspace L^3
         \quad = \quad {{1}\over{3}} \thinspace M L^2
$$


$$
I \quad = \quad \int dI \quad = \quad \int_{x=-L/2}^{x=L/2} dm \ x^2 
             \quad = \quad \int_{x=-L/2}^{x=L/2} \lambda \thinspace dx \ x^2
$$

$$
I \quad = \quad \lambda \ \left( {{1}\over{3}} {{L^3}\over{8}} \ -
                          \left[ -{{1}\over{3}} {{L^3}\over{8}} \right] \right)
         \quad = \quad {{M}\over{L}} \ \left( {{2}\over{24}} \thinspace L^3 \right) 
         \quad = \quad {{1}\over{12}} \thinspace M L^2
$$

$$
\lambda(x) \quad = \quad A \ + \ Bx
$$


$$
dm \quad = \quad \lambda \thinspace dx \quad = \quad A \thinspace dx
                 \ + \ B \thinspace x dx
$$

$$
dI \quad = \quad dm \ x^2 \quad = \quad (A \thinspace dx) \thinspace x^2
                 \ + \ (B \thinspace x dx) \thinspace x^2
$$

$$
I \quad = \quad \int dI \quad = \quad \int_{x=0}^{x=L} 
                 \ (A \thinspace x^2 dx) 
             \ + \  \int_{x=0}^{x=L} 
                 \ (B \thinspace x^3 dx) 
$$

$$
I \quad = \quad {{1}\over{3}} \thinspace A L^3 \ + \ 
           {{1}\over{4}} \thinspace B L^4
$$





\bye
