question_answer 50)

                  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 necessary zero about any arbitrary point?                

Answer:

                  The net torque about a point O is given to be zero i.e. \[\vec{\tau }=\sum\limits_{i}{{{{\vec{r}}}_{i}}\,\times \,{{{\vec{F}}}_{i}}=0}\]                 and \[\sum\limits_{i}{\,{{{\vec{F}}}_{i}}\ne \,0}\]                 Now consider another point O? whose position vector is \[({{\vec{r}}_{i}}-\vec{R})\]                 Therefore, not torque about O?                 \[=\Sigma \,({{\vec{r}}_{i}}-\vec{R})\,\times \,{{\vec{F}}_{i}}\]                 \[=\Sigma \,{{\vec{r}}_{i}}\times \,{{\vec{F}}_{i}}\,-\vec{R}\times \,\Sigma {{\vec{F}}_{i}}=\,-\vec{R}\,\times \,\Sigma {{\vec{F}}_{i}}\]                 Thus, it is not necessarily zero about any arbitrary point.