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Classical mechanics 


In mechanics, a couple is a system of forces with a resultant (a.k.a.net or sum) moment but no resultant force.^{[1]}
A better term is force couple or pure moment. Its effect is to create rotation without translation, or more generally without any acceleration of the centre of mass. In rigid body mechanics, force couples are free vectors, meaning their effects on a body are independent of the point of application.
The resultant moment of a couple is called a torque. This is not to be confused with the term torque as it is used in physics, where it is merely a synonym of moment.^{[2]} Instead, torque is a special case of moment. Torque has special properties that moment does not have, in particular the property of being independent of reference point, as described below.
A couple is a pair of forces, equal in magnitude, oppositely directed, and displaced by perpendicular distance or moment.
The simplest kind of couple consists of two equal and opposite forces whose lines of action do not coincide. This is called a "simple couple".^{[1]} The forces have a turning effect or moment called a torque about an axis which is normal (perpendicular) to the plane of the forces. The SI unit for the torque of the couple is newton metre.
If the two forces are F and −F, then the magnitude of the torque is given by the following formula:
where
The magnitude of the torque is equal to F • d, with the direction of the torque given by the unit vector , which is perpendicular to the plane containing the two forces and positive being a counterclockwise couple. When d is taken as a vector between the points of action of the forces, then the torque is the cross product of d and F, i.e.
The moment of a force is only defined with respect to a certain point P (it is said to be the "moment about P") and, in general, when P is changed, the moment changes. However, the moment (torque) of a couple is independent of the reference point P: Any point will give the same moment.^{[1]} In other words, a torque vector, unlike any other moment vector, is a "free vector". (This fact is called Varignon's Second Moment Theorem.)^{[3]}
The proof of this claim is as follows: Suppose there are a set of force vectors F_{1}, F_{2}, etc. that form a couple, with position vectors (about some origin P), r_{1}, r_{2}, etc., respectively. The moment about P is
Now we pick a new reference point P' that differs from P by the vector r. The new moment is
Now the distributive property of the cross product implies
However, the definition of a force couple means that
Therefore,
This proves that the moment is independent of reference point, which is proof that a couple is a free vector.
A force F applied to a rigid body at a distance d from the center of mass has the same effect as the same force applied directly to the center of mass and a couple Cℓ = Fd. The couple produces an angular acceleration of the rigid body at right angles to the plane of the couple.^{[4]} The force at the center of mass accelerates the body in the direction of the force without change in orientation. The general theorems are:^{[4]}
Couples are very important in mechanical engineering and the physical sciences. A few examples are:
In a liquid crystal it is the rotation of an optic axis called the director that produces the functionality of these compounds. As Jerald Ericksen explained