couple (mechanics)
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In
mechanics Mechanics (from Ancient Greek: μηχανική, ''mēkhanikḗ'', "of machines") is the area of mathematics and physics concerned with the relationships between force, matter, and motion among physical objects. Forces applied to object ...
, a couple is a system of forces with a
resultant In mathematics, the resultant of two polynomials is a polynomial expression of their coefficients, which is equal to zero if and only if the polynomials have a common root (possibly in a field extension), or, equivalently, a common factor (ove ...
(a.k.a.
net Net or net may refer to: Mathematics and physics * Net (mathematics), a filter-like topological generalization of a sequence * Net, a linear system of divisors of dimension 2 * Net (polyhedron), an arrangement of polygons that can be folded up ...
or sum)
moment of force In physics and mechanics, torque is the rotational equivalent of linear force. It is also referred to as the moment of force (also abbreviated to moment). It represents the capability of a force to produce change in the rotational motion of the ...
but no resultant force.''Dynamics, Theory and Applications'' by T.R. Kane and D.A. Levinson, 1985, pp. 90-99
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/ref> A better term is force couple or pure moment. Its effect is to impart
angular momentum In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
but no linear momentum. In
rigid body dynamics In the physical science of dynamics, rigid-body dynamics studies the movement of systems of interconnected bodies under the action of external forces. The assumption that the bodies are ''rigid'' (i.e. they do not deform under the action of ...
, 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.


Simple couple

;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 Lines of Action (or LOA) is an abstract strategy board game for two players invented by Claude Soucie. The objective is to connect all of one's pieces into a single group. The game was recommended by the Spiel des Jahres in 1988. Rules Goal The ...
do not coincide. This is called a "simple couple".''Dynamics, Theory and Applications'' by T.R. Kane and D.A. Levinson, 1985, pp. 90-99
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/ref> The forces have a turning effect or moment called a
torque In physics and mechanics, torque is the rotational equivalent of linear force. It is also referred to as the moment of force (also abbreviated to moment). It represents the capability of a force to produce change in the rotational motion of th ...
about an axis which is
normal Normal(s) or The Normal(s) may refer to: Film and television * ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson * ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie * ''Norma ...
(perpendicular) to the plane of the forces. The
SI unit The International System of Units, known by the international abbreviation SI in all languages and sometimes pleonastically as the SI system, is the modern form of the metric system and the world's most widely used system of measurement. ...
for the torque of the couple is
newton metre The newton-metre (also newton metre or newton meter; symbol N⋅m or N m) is the unit of torque (also called ) in the International System of Units (SI). One newton-metre is equal to the torque resulting from a force of one newton applie ...
. If the two forces are and , then the
magnitude Magnitude may refer to: Mathematics *Euclidean vector, a quantity defined by both its magnitude and its direction *Magnitude (mathematics), the relative size of an object *Norm (mathematics), a term for the size or length of a vector *Order of ...
of the torque is given by the following formula: \tau = F d where *\tau is the moment of couple * is the magnitude of the force * is the perpendicular distance (moment) between the two parallel forces The magnitude of the torque is equal to , with the direction of the torque given by the
unit vector In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat"). The term ''direction v ...
\hat, which is perpendicular to the plane containing the two forces and positive being a counter-clockwise couple. When is taken as a vector between the points of action of the forces, then the torque is the cross product of and , i.e. \mathbf = , \mathbf \times \mathbf , .


Independence of reference point

The moment of a force is only defined with respect to a certain point (it is said to be the "moment about ") and, in general, when is changed, the moment changes. However, the moment (torque) of a ''couple'' is ''independent'' of the reference point : 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''.)''Engineering Mechanics: Equilibrium'', by C. Hartsuijker, J. W. Welleman, page 6
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/ref> The proof of this claim is as follows: Suppose there are a set of force vectors , , etc. that form a couple, with position vectors (about some origin ), , , etc., respectively. The moment about is :M = \mathbf_1\times \mathbf_1 + \mathbf_2\times \mathbf_2 + \cdots Now we pick a new reference point that differs from by the vector . The new moment is :M' = (\mathbf_1+\mathbf)\times \mathbf_1 + (\mathbf_2+\mathbf)\times \mathbf_2 + \cdots Now the
distributive property In mathematics, the distributive property of binary operations generalizes the distributive law, which asserts that the equality x \cdot (y + z) = x \cdot y + x \cdot z is always true in elementary algebra. For example, in elementary arithmet ...
of the cross product implies :M' = \left(\mathbf_1\times \mathbf_1 + \mathbf_2\times \mathbf_2 + \cdots\right) + \mathbf\times \left(\mathbf_1 + \mathbf_2 + \cdots \right). However, the definition of a force couple means that :\mathbf_1 + \mathbf_2 + \cdots = 0. Therefore, :M' = \mathbf_1\times \mathbf_1 + \mathbf_2\times \mathbf_2 + \cdots = M This proves that the moment is independent of reference point, which is proof that a couple is a free vector.


Forces and couples

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. The force at the center of mass accelerates the body in the direction of the force without change in orientation. The general theorems are: :A single force acting at any point ''O′'' of a rigid body can be replaced by an equal and parallel force ''F'' acting at any given point ''O'' and a couple with forces parallel to ''F'' whose moment is ''M = Fd'', ''d'' being the separation of ''O'' and ''O′''. Conversely, a couple and a force in the plane of the couple can be replaced by a single force, appropriately located. :Any couple can be replaced by another in the same plane of the same direction and moment, having any desired force or any desired arm.


Applications

Couples are very important in
mechanical engineering Mechanical engineering is the study of physical machines that may involve force and movement. It is an engineering branch that combines engineering physics and mathematics principles with materials science, to design, analyze, manufacture, an ...
and the physical sciences. A few examples are: * The forces exerted by one's hand on a screw-driver * The forces exerted by the tip of a screwdriver on the head of a screw * Drag forces acting on a spinning propeller * Forces on an
electric dipole The electric dipole moment is a measure of the separation of positive and negative electrical charges within a system, that is, a measure of the system's overall polarity. The SI unit for electric dipole moment is the coulomb- meter (C⋅m). T ...
in a uniform electric field. * The
reaction control system A reaction control system (RCS) is a spacecraft system that uses thrusters to provide attitude control and translation. Alternatively, reaction wheels are used for attitude control. Use of diverted engine thrust to provide stable attitude cont ...
on a spacecraft. * Force exerted by hands on steering wheel.


See also

*
Traction (engineering) Traction, or tractive force, is the force used to generate motion between a body and a tangential surface, through the use of dry friction, though the use of shear force of the surface is also commonly used. Traction can also refer to the ''maxim ...
*
Torque In physics and mechanics, torque is the rotational equivalent of linear force. It is also referred to as the moment of force (also abbreviated to moment). It represents the capability of a force to produce change in the rotational motion of th ...
*
Moment (physics) In physics, a moment is a mathematical expression involving the product of a distance and physical quantity. Moments are usually defined with respect to a fixed reference point and refer to physical quantities located some distance from the ref ...
* Force


References

{{reflist * H.F. Girvin (1938) ''Applied Mechanics'', §28 Couples, pp 33,4, Scranton Pennsylvania: International Textbook Company. Physical quantities Mechanics