Couple

A couple is a pair of parallel forces equal in magnitude but in opposite directions. The forces are separated by a perpendicular distance, known as the couple's arm. The couple causes a rotation force or moment that rotates the body about an axis perpendicular to the plane of the forces. The resulting moment is referred to as the couple moment. The SI unit of a couple moment is the Newton-meter (N-m).

A typical example to understand this concept is tightening a bolt with a lug wrench. A force couple is formed when two equal and opposite forces are applied to the wrench at points A and B separated by a perpendicular distance. The respective position vectors are directed from the origin of a coordinate system to the point of application. The couple moment about the origin is the sum of the cross-product of the position vectors and the forces. The difference between the position vectors of both the opposite forces gives the position vector r, which is the distance between the opposing forces.

Interestingly, the couple moment only depends on the position vector r between the forces, not the individual position vectors for the two forces. The position vector r is independent of the origin. It implies computing the moments of the same forces around a different point would yield the same result. The couple moment is a free vector. Unlike a couple moment, the moment of a force depends on the position at which it is measured. If two couples produce a moment of equal magnitude and direction, then these two couples are equivalent. Equivalent couples have equal torque or turning effects regardless of their positions, directions, or magnitudes. 

Calculating the couple moment and understanding its effects is crucial in designing stable and robust systems that can withstand external forces.