Q.Two forces of equal magnitude $F$, but opposite in direction, act along two separate parallel lines of action separated by a perpendicular distance $d$, forming a couple. Show that the net force of the couple on the body is zero while its net torque (moment) is not, and obtain the magnitude of this moment in terms of $F$ and $d$.

Let the two forces be $+F\hat j$ acting at a point $A$ and $-F\hat j$ acting at a point $B$, with $A$ and $B$ separated by perpendicular distance $d$ (say, along the $x$-axis, with $A$ at the origin and $B$ at $(d,0)$).

Net force: $\vec F_{net} = F\hat j + (-F\hat j) = 0$ -- the two forces are equal in magnitude and exactly opposite in direction, so they sum to zero regardless of where they act. The body therefore has no tendency to translate. …

Take the vector sum of the two forces (zero, since equal and opposite), then take the vector sum …

• Assuming zero net force automatically means zero net torque as well -- a couple is exactly the counterexample showing this is false. …