3.8: Torque

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

$$\boldsymbol\tau = \boldsymbol\tau_{C} + \overline{\textbf{r}} \times \textbf{F} \nonumber$$

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

$$\qquad \boldsymbol\tau = \boldsymbol\tau_{C} +\overline{\textbf{r}} \times \textbf{F} \nonumber$$