question_answer

Show that moment of a couple does not depend on the point about which you take the moments.

Answer:

                Fig. shows a couple acting on a rigid body. The forces\[-\vec{F}\] and \[\vec{F}\] act at points A and B respectively. Let \[{{\vec{r}}_{1}}\] and \[{{\vec{r}}_{2}}\] be the position vectors of these points with respect to origin O. We consider the moments of the forces about the origin O. Moment of the couple \[=\]Sum of the moments of the two forces \[={{\vec{r}}_{1}}\times (-\vec{F})+{{\vec{r}}_{2}}\times \vec{F}={{\vec{r}}_{2}}\times \vec{F}-{{\vec{r}}_{1}}\times \vec{F}\] \[=({{\vec{r}}_{2}}-{{\vec{r}}_{1}})\times \vec{F}=\overrightarrow{AB}\times \vec{F}\] \[[\because {{\vec{r}}_{1}}+\overrightarrow{AB}={{\vec{r}}_{2}}]\] Clearly, this moment is independent of the point O about which we have considered the moments of the forces.