3.8: Torque
( \newcommand{\kernel}{\mathrm{null}\,}\)
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.
\boldsymbol\tau = \boldsymbol\tau_{C} + \overline{\textbf{r}} \times \textbf{F} \nonumber
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} \nonumber
therefore
\qquad \boldsymbol\tau = \boldsymbol\tau_{C} +\overline{\textbf{r}} \times \textbf{F} \nonumber