To Prove that the Moment of a Couple is Independent of the Axis of Rotation

This sub-section proves algebraically that the torque of a given couple does not depend on where the (fixed) axis of rotation happens to be placed.

Consider a rectangular sheet, free to rotate only about a fixed axis perpendicular to its plane, acted on by a couple of forces $\vec{F}$ and $-\vec{F}$ at different locations (Fig 4.9). In case (a), the axis of rotation lies BETWEEN the two lines of action, at perpendicular distances x (from the +F line) and y (from the -F line); here both individual torques act in the SAME rotational sense (both anticlockwise, say, viewed from the top), so the net torque is their SUM: $\tau=xF+yF=(x+y)F=rF$, using $x+y=r$ (the fixed separation between the lines).

In case (b), the axis of rotation lies OUTSIDE both lines of action (to one side), at perpendicular distances q (from the nearer force) and p (from the farther force); here the individual torques act in OPPOSITE rotational senses (one clockwise, one anticlockwise), so the net torque is their DIFFERENCE: $\tau=qF-pF=(q-p)F=rF$, using $q-p=r$ once again.

Both placements of the axis -- one between the forces, one outside them -- give the identical net torque $rF$. Since this holds for any choice of axis location, it establishes generally that the torque (moment) of a couple is completely independent of the axis of rotation, unlike the torque of a single force, which does depend on where the axis is chosen. …