24.10: Relating Angular Momentum to Angular Velocity
- Page ID
- 30521
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It’s easy to check that the angular momentum vector is
\(L_{i}=I_{i j} \Omega_{j}\)
since
\begin{equation}
\mathbf{L}=\sum \vec{r}_{n} \times m_{n} \vec{v}_{n}=\sum m_{n} \vec{r}_{n} \times\left(\vec{\Omega} \times \vec{r}_{n}\right)=\vec{\Omega} \sum m_{n} r_{n}^{2}-\sum m_{n} \vec{r}_{n}\left(\vec{\Omega} \cdot \vec{r}_{n}\right)=\mathbf{I} \vec{\Omega}
\end{equation}
Exercise: verify this by putting in all the suffixes.