# 3.8: Torque

- Page ID
- 8380

Notation:

- \( \boldsymbol\tau_{C} \) = vector sum of all the torques about C.
- \( \boldsymbol\tau\)
**=**vector sum of all the torques about the origin O. - \(\textbf{F}\)
**=**vector sum of all the external forces.

Theorem

\[ \boldsymbol\tau = \boldsymbol\tau_{C} + \overline{\textbf{r}} \times \textbf{F} \]

Thus:

\[\begin{align} \boldsymbol\tau &= \sum \textbf{r}_{i} \times \textbf{F}_{i} = \sum (\textbf{r}'_{i} + \overline{\textbf{r}}) \times \textbf{F}_{i} \\ &= \sum \textbf{r}'_{i} \times \textbf{F}_{i} + \overline{\textbf{r}} \sum \textbf{F}_{i} \end{align}\]

therefore

\[\qquad \boldsymbol\tau = \boldsymbol\tau_{C} +\overline{\textbf{r}} \times \textbf{F}\]