Question:
The vector sum of a system of non-collinear forces acting on a rigid body is given to be non-zero. If the vector sum of all the torques due to the system of forces about a certain point is found to be zero, does this mean that it is necessarily zero about any arbitrary point?
Solution:
The vector sum of the torques is zero. But the resultant force is not zero. The mathematical explanation is given as:
$G_{i} \sum_{i=1}^{n} F_{t} \neq 0$
$\tau=\tau_{1}+\tau_{2}+\ldots+\tau_{n}=\sum_{i=1}^{n} \underset{r_{i}}{\rightarrow} \times \overrightarrow{F_{i}}=0$
$\sum_{i_{1}}^{n}\left(\rightarrow_{r_{i}}-a\right) \times F_{t}=\sum_{i=1}^{n} \underset{r}{\rightarrow} \times F_{i}-a \sum_{i=1}^{n} F_{i}$