\magnification=\magstep2

$$
\eqalign{
x \quad &= \quad R \thinspace \theta \cr
v \quad &= \quad R \thinspace \omega \cr
a \quad &= \quad R \thinspace \alpha \cr
}
$$


$$
 \alpha = { {\tau } \over { I } }
$$


$$
\eqalign{
 \alpha \quad &= \quad { {R (Mg) \sin(\theta) } \over { {{2}\over{5}} M R^2  } } \cr
}
$$


$$
{\rm Rot\  KE\ } = \thinspace \thinspace { {1}\over{2} } I \omega^2
$$

$$
\eqalign{
{\rm Rot\ KE}_{\rm f} \quad &= \quad {\rm Rot\ KE}_{\rm i} \quad 
                         + \quad {\rm Work}_{\rm net} \cr
{\rm Rot\ KE}_{\rm f} \quad &= \quad {\rm Rot\ KE}_{\rm i} \quad 
                         + \quad \int{ \vec{\tau} \cdot \vec{d \theta} } \cr
}
$$


$$
 \alpha \quad = \quad { {R (Mg) \sin(\theta) } \over { {{5}\over{3}} M R^2  } } \quad = \quad {{3}\over{5}} \ {{ g \thinspace \sin(\theta) }\over{R}} 
$$

$$
a \quad = \quad \alpha \ R \quad = \quad {{3}\over{5}} \ g \thinspace \sin(\theta) 
$$

$$
v_f^2 \quad = \quad v_i^2 \ + \ 2 a L \quad = \quad 0 \ + \ 
              2 \thinspace {{3}\over{5}} \thinspace g \sin(\theta) \thinspace L
  \quad = \quad 
              {{6}\over{5}} \thinspace g \thinspace h
$$

$$
v_f \quad = \quad \sqrt{ {{6}\over{5}} g h }
$$

$$
L \quad = \quad {{1}\over{2}} \thinspace a t^2 \quad = \quad
             {{1}\over{2}} \ {{3}\over{5}} \thinspace g \sin(\theta) \thinspace t^2 
        \quad = \quad 
             {{3}\over{10}} \thinspace  g \sin(\theta) \thinspace t^2
$$

$$
L \quad = \quad {{3}\over{10}} \thinspace g \thinspace 
              \left( {{h}\over{L}} \right) \thinspace t^2
$$

$$
t \quad = \quad \sqrt{ { {{10}\over{3}} L^2} \over{\thinspace g h}  }
$$


\bye
