 Solution

In a frame centered at the fixed point or centre of mass, a rotating material point has velocity $\dot{r}=\omega \times r$. The identity $|\omega \times r{|}^2=|\omega {|}^2|r{|}^2-(\omega \cdot r{)}^2$ converts the rotational kinetic energy into $T=\frac{1}{2}{\omega }_i{I}_{ij}{\omega }_j$, where the inertia tensor isLikewise integrating $r\times (\omega \times r)$ gives the angular momentum $\boxed{L=I\omega }$. These expressions refer to rotational motion about the chosen origin.