A Couple Is Two Equal and Opposite Forces That Leave Pure Rotation
When you turn a steering wheel, one hand pushes up while the other pulls down. The two hands' forces are equal in size and opposite in direction. So they do not shove the wheel anywhere; they only turn it. Two forces like this, equal in size, opposite in direction, and acting along parallel lines a distance apart, are called a couple. Adding the two forces gives zero, so they do not move the body, yet a moment that tends to rotate it clearly remains. What is more, this moment is the same no matter where you put the axis. So the moment of a couple is a free quantity, not tied to a location. Start by growing the two forces and confirm that the resultant is zero while the rotation stays alive.
Two forces of equal size and opposite direction sit a distance apart. The upper force points right, the lower one points left. Grow the two forces. They always grow together and point opposite, so adding them as vectors gives exactly zero every time. With a zero resultant they drag the body in no direction at all. And yet, like the curved arrow in the middle, the effect that tends to rotate the body is clearly alive. This is the heart of a couple: the sum of forces is zero, but the moment is not zero. It is a combination of forces that produces a pure turning effect, rotation with no translation.
Why pure rotation? It becomes clear by comparing with a single force. Switch between the two with the buttons. A single force acting off the body's center both shoves the body along and spins it; translation and rotation are mixed. A couple, by contrast, has a zero resultant, so it cannot move the center at all. What remains is rotation only. So you get a pure spin, turning in place. The rotation a figure skater makes with both arms, the force your fingers give when turning a screw with a screwdriver, are all close to couples. A couple is what you use when you want to turn something without moving it.
A couple has a special property that an ordinary moment lacks: the size of its moment is the same no matter where you place the axis. The moment of a single force changes depending on the point you measure it about, but a couple does not. Drag the reference point P. Whether P sits between the two forces, outside them, or far away, the moment of the couple is always F times the distance d between the two forces. As one force moves farther from the reference its moment grows, while the other moves closer and its moment shrinks, and the two changes cancel exactly. So the moment of a couple is a free vector, not tied to a point of application. It is simply a turning effect with a size and a sense.
The size of a couple is simple. Take the size F of one of the forces and multiply by the perpendicular distance d between the two forces. M = F d. The key is that d is the spacing between the paired forces, not the distance of a single force from some point. Use the two sliders to change F and d. Grow the force, or spread the two forces farther apart, and the couple grows the same way. So even a small force can make a large turning effect if it is spread far enough. This is why gripping both handles of a large wrench and turning them in opposite directions is easier than pushing with one hand. The unit is N·m, the same as a moment, and the sign follows the same convention: counterclockwise positive, clockwise negative.
Finally, a very useful trick. Sometimes you want to move a force, parallel to itself, to a different spot off its line of action. Just moving it would change the turning effect. So you attach a couple that compensates for the move, and then the effect stays exactly the same as the original force. This is called a force-couple system. Drag the distance e you move the force. The force at the new spot is unchanged, and you can see a couple of size M = F e come along with it. Because it lets you turn a complicated load into the clean form of one force at a point plus one couple, it is used very often in rigid-body analysis.