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| {{Classical mechanics|cTopic=Fundamental concepts}}
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| {{Expand Sinhalese|බල යුග්ම|date=March 2013}}
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| In [[mechanics]], a '''couple''' is a system of [[force]]s with a resultant (a.k.a. net or sum) [[torque|moment]] but no resultant force.<ref name=Kane>''Dynamics, Theory and Applications'' by T.R. Kane and D.A. Levinson, 1985, pp. 90-99: [http://ecommons.library.cornell.edu/handle/1813/638 Free download]</ref> A better term is '''force couple''' or '''pure moment'''. Its effect is to create [[rotation]] without [[Translation_(physics)|translation]], or more generally without any acceleration of the [[center of mass|centre of mass]]. In [[rigid body dynamics|rigid body mechanics]], force couples are ''free vectors'', meaning their effects on a body are independent of the point of application.
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| 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.<ref>''Physics for Engineering'' by Hendricks, Subramony, and Van Blerk, page 148, [http://books.google.com/books?id=8Kp-UwV4o0gC&pg=PA148 Web link]</ref> 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. | |
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| ==Simple couple==
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| Definition-
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| A couple is a pair of forces, equal in magnitude, oppositely directed, and displaced by perpendicular distance or moment.
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| The simplest kind of couple consists of two equal and opposite [[force]]s whose [[line of action|lines of action]] do not coincide. This is called a "simple couple".<ref name=Kane/> The forces have a turning effect or moment called a [[torque]] about an axis which is normal to the plane of the forces. The [[SI unit]] for the torque of the couple is [[newton metre]].
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| If the two forces are '''F''' and '''−F''', then the [[Euclidean vector|magnitude]] of the torque is given by the following formula:
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| :<math>\tau = F d \,</math>
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| where
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| :<math>\tau</math> is the torque
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| :''F'' is the magnitude of one of the forces
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| :''d'' is the perpendicular distance between the forces, sometimes called the ''arm'' of the couple
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| The magnitude of the torque is always equal to ''F d'', with the direction of the torque given by the [[unit vector]] <math>\hat{e}</math>, which is perpendicular to the plane containing the two forces. When '''d''' is taken as a vector between the points of action of the forces, then the couple is the [[cross product]] of '''d''' and '''F'''. I.e.,
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| :<math> \mathbf{\tau} = \mathbf{d} \times \mathbf{F} .</math>
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| ==Independence of reference point==
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| 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.<ref name=Kane/> In other words, a torque vector, unlike any other moment vector, is a "free vector".
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| (This fact is called ''Varignon's Second Moment Theorem''.)<ref>''Engineering Mechanics: Equilibrium'', by C. Hartsuijker, J. W. Welleman, page 64 [http://books.google.com/books?id=oPhH90IWW60C&pg=PA64 Web link]</ref>
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| The proof of this claim is as follows: Suppose there are a set of force vectors '''F'''<sub>1</sub>, '''F'''<sub>2</sub>, etc. that form a couple, with position vectors (about some origin ''P'') '''r'''<sub>1</sub>, '''r'''<sub>2</sub>, etc., respectively. The moment about ''P'' is
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| :<math>M = \mathbf{r}_1\times \mathbf{F}_1 + \mathbf{r}_2\times \mathbf{F}_2 + \cdots</math>
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| Now we pick a new reference point ''P''' that differs from ''P'' by the vector '''r'''. The new moment is
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| :<math>M' = (\mathbf{r}_1+\mathbf{r})\times \mathbf{F}_1 + (\mathbf{r}_2+\mathbf{r})\times \mathbf{F}_2 + \cdots</math>
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| Now the [[distributive property]] of the [[cross product]] implies
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| :<math>M' = \left(\mathbf{r}_1\times \mathbf{F}_1 + \mathbf{r}_2\times \mathbf{F}_2 + \cdots\right) + \mathbf{r}\times \left(\mathbf{F}_1 + \mathbf{F}_2 + \cdots \right).</math>
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| However, the definition of a force couple means that
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| :<math>\mathbf{F}_1 + \mathbf{F}_2 + \cdots = 0.</math>
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| Therefore,
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| :<math>M' = \mathbf{r}_1\times \mathbf{F}_1 + \mathbf{r}_2\times \mathbf{F}_2 + \cdots = M</math>
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| This proves that the moment is independent of reference point, which is proof that a couple is a free vector.
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| ==Forces and couples==
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| [[File:Force and couple.PNG |thumb]]
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| 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.<ref name="DuBois">{{cite book |title=The mechanics of engineering, Volume 1 |author=Augustus Jay Du Bois |url=http://books.google.com/books?id=euUeAAAAMAAJ&pg=PA186 |page=186 |publisher=Wiley |year=1902}}
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| </ref> The force at the center of mass accelerates the body in the direction of the force without change in orientation. The general theorems are:<ref name=DuBois/>
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| :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.
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| :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.<ref name=DuBois/>
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| ==Applications==
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| Couples are very important in [[mechanical engineering]] and the physical sciences. A few examples are:
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| * The forces exerted by one's hand on a screw-driver
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| * The forces exerted by the tip of a screw-driver on the head of a screw
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| * Drag forces acting on a spinning propeller
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| * Forces on an [[electric dipole]] in a uniform electric field.
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| * The [[reaction control system]] on a spacecraft.
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| 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
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| :At first glance, it may seem that it is optics or electronics which is involved, rather than mechanics. Actually, the changes in optical behavior, etc. are associated with changes in orientation. In turn, these are produced by couples. Very roughly, it is similar to bending a wire, by applying couples.<ref> J.L. Ericksen (1979) [http://imechanica.org/node/10396 Timoshenko Acceptance Speech] at iMechanica.org site for [[mechanician]]s</ref>
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| ==See also==
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| * [[Traction (engineering)]]
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| * [[Torque]]
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| * [[Moment (physics)]]
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| * [[Force]]
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| ==References==
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| {{reflist}}
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| * H.F. Girvin (1938) ''Applied Mechanics'', §28 Couples, pp 33,4, Scranton Pennsylvania: International Textbook Company.
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| [[Category:Physical quantities]]
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| [[Category:Mechanics]]
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