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Concept
Couple (mechanics)
Summary
In mechanics, a couple is a system of forces with a resultant (a.k.a. net or sum) moment of force but no resultant force.
A better term is force couple or pure moment. Its effect is to impart angular momentum but no linear momentum. In rigid body dynamics, 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 a special case of moment. A couple has the property that it is independent of reference point.
Definition
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". 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
is the moment of couple
F is the magnitude of the force
d is the perpendicular distance (moment) between the two parallel forces
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 counter-clockwise 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. In other words, a couple, unlike any more general moments, is a "free vector". (This fact is called Varignon's Second Moment Theorem.)
The proof of this claim is as follows: Suppose there are a set of force vectors F_1, F_2, etc.
Official source
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