Example

Question:

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

Solution:

Consider a couple as shown in the figure, acting on a rigid body. The forces \( \mathbf{F} \) and \( -\mathbf{F} \) act at points \( B \) and \( A \) respectively, with position vectors \( \mathbf{r}_1 \) and \( \mathbf{r}_2 \) from the origin \( O \).
The moment of the couple is the sum of the moments of the two forces about the origin: \[ \text{Moment} = \mathbf{r}_1 \times (-\mathbf{F}) + \mathbf{r}_2 \times \mathbf{F} \] \[ = \mathbf{r}_2 \times \mathbf{F} - \mathbf{r}_1 \times \mathbf{F} \] \[ = (\mathbf{r}_2 - \mathbf{r}_1) \times \mathbf{F} \] But \( \mathbf{r}_2 - \mathbf{r}_1 = \overrightarrow{AB} \), so: \[ \text{Moment of the couple} = \overrightarrow{AB} \times \mathbf{F} \] This is independent of the origin—the moment does not depend on the point about which we take the moments of the forces.