\magnification=\magstep2


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

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



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

$$
\vec{r} \quad = \quad r_x \thinspace \hat{\imath} \ + \ 
                r_y \thinspace \hat{\jmath} \ + \ 
                r_z \thinspace \hat{k} 
$$

$$
\vec{F} \quad = \quad F_x \thinspace \hat{\imath} \ + \ 
                F_y \thinspace \hat{\jmath} \ + \ 
                F_z \thinspace \hat{k} 
$$

$$
\vec{r} \quad = \quad 3 \thinspace {\rm m} \thinspace \hat{\imath} \ + \ 
                0 \thinspace {\rm m} \hat{\jmath} \ - \ 
                5 \thinspace {\rm m} \thinspace \hat{k} 
$$

$$
\vec{F} \quad = \quad 6  \thinspace{\rm N} \thinspace \hat{\imath} \ + \ 
                2 \thinspace {\rm N} \thinspace \hat{\jmath} \ + \ 
                0 \thinspace {\rm N} \hat{k} 
$$

$$
5 \ \times \ 3 \quad = \quad 15
$$

$$
\eqalign{
\vec{A} \quad &= \quad 5 \hat{\imath} \ + \ 8 \hat{\jmath}  \cr
\vec{B} \quad &= \quad 1 \hat{\imath} \ - \ 3 \hat{\jmath}  \cr
}
$$


$$
| A \times B | \quad = \quad | A | \thinspace | B | \thinspace \sin(\theta)
$$

$$
\eqalign{
\vec{A} \quad &= \quad 5 \hat{\imath} \ + \ 8 \hat{\jmath} \ + \ 0 \hat{k} \cr
\vec{B} \quad &= \quad 1 \hat{\imath} \ - \ 3 \hat{\jmath} \ + \ 0 \hat{k} \cr
}
$$


$$
\eqalign{
A \times B \quad &= \quad \left[ (8)(0) - (0)(-3) \right] \thinspace \hat{\imath}  \ - \ 
                       \left[ (0)(5) - (1)(0) \right] \thinspace \hat{\jmath} \ + \ 
                       \left[ (5)(-3) - (8)(1) \right] \thinspace \hat{k}  \cr
    &= \quad 0 \hat{\imath} \ + \ 0 \hat{\jmath} \ - \ 23 \hat{k} \cr
}
$$

$$
\eqalign{
\vec{A} \quad &= \quad 1 \hat{\imath} \ + \ 2 \hat{\jmath} \ + \ 3 \hat{k} \cr
\vec{B} \quad &= \quad 3 \hat{\imath} \ + \ 2 \hat{\jmath} \ + \ 1 \hat{k} \cr
}
$$



\bye
