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Couple (mechanics)

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Pair of equal magnitude but opposite direction forces
Part of a series on
Classical mechanics
F=dpdt{\displaystyle {\textbf {F}}={\frac {d\mathbf {p} }{dt}}}

Inphysics, acouple is a pair of forces that are equal in magnitude but opposite in their direction of action. A couple produce a purerotational motion without anytranslational form.

Two forces acting on opposite direction with equal magnitude.

Simple couple

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The simplest kind of couple consists of two equal and oppositeforces whoselines of action do not coincide. This is called a "simple couple".[1] The forces have a turning effect or moment called atorque about an axis which isnormal (perpendicular) to the plane of the forces. TheSI unit for the torque of the couple isnewton metre.

If the two forces areF and−F, then themagnitude of the torque is given by the following formula:τ=Fd{\displaystyle \tau =Fd}where

  • τ{\displaystyle \tau } 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 toF •d, with the direction of the torque given by theunit vectore^{\displaystyle {\hat {e}}}, which is perpendicular to the plane containing the two forces and positive being a counter-clockwise couple. Whend is taken as a vector between the points of action of the forces, then the torque is thecross product ofd andF, i.e.τ=|d×F|.{\displaystyle \mathbf {\tau } =|\mathbf {d} \times \mathbf {F} |.}

Independence of reference point

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The moment of a force is only defined with respect to a certain pointP (it is said to be the "moment aboutP") and, in general, whenP is changed, the moment changes. However, the moment (torque) of acouple isindependent of the reference pointP: Any point will give the same moment.[1] In other words, a couple, unlike any more general moments, is a "free vector". (This fact is calledVarignon's Second Moment Theorem.)[2]

Theproof of this claim is as follows: Suppose there are a set of force vectorsF1,F2, etc. that form a couple, with position vectors (about some originP),r1,r2, etc., respectively. The moment aboutP is

M=r1×F1+r2×F2+⋯{\displaystyle M=\mathbf {r} _{1}\times \mathbf {F} _{1}+\mathbf {r} _{2}\times \mathbf {F} _{2}+\cdots }

Now we pick a new reference pointP' that differs fromP by the vectorr. The new moment is

M′=(r1+r)×F1+(r2+r)×F2+⋯{\displaystyle M'=(\mathbf {r} _{1}+\mathbf {r} )\times \mathbf {F} _{1}+(\mathbf {r} _{2}+\mathbf {r} )\times \mathbf {F} _{2}+\cdots }

Now thedistributive property of thecross product implies

M′=(r1×F1+r2×F2+⋯)+r×(F1+F2+⋯).{\displaystyle 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).}

However, the definition of a force couple means that

F1+F2+⋯=0.{\displaystyle \mathbf {F} _{1}+\mathbf {F} _{2}+\cdots =0.}

Therefore,

M′=r1×F1+r2×F2+⋯=M{\displaystyle M'=\mathbf {r} _{1}\times \mathbf {F} _{1}+\mathbf {r} _{2}\times \mathbf {F} _{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

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A forceF applied to a rigid body at a distanced from the center of mass has the same effect as the same force applied directly to the center of mass and a coupleCℓ = Fd. The couple produces anangular acceleration of the rigid body at right angles to the plane of the couple.[3] The force at the center of mass accelerates the body in the direction of the force without change in orientation. The general theorems are:[3]

A single force acting at any pointO′ of a rigid body can be replaced by an equal and parallel forceF acting at any given pointO and a couple with forces parallel toF whose moment isM = Fd,d being the separation ofO andO′. 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.[3]

Applications

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Couples are very important inengineering and the physical sciences. A few examples are:

See also

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References

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  1. ^abDynamics, Theory and Applications by T.R. Kane and D.A. Levinson, 1985, pp. 90–99:Free download
  2. ^Engineering Mechanics: Equilibrium, by C. Hartsuijker, J. W. Welleman, page 64Web link
  3. ^abcAugustus Jay Du Bois (1902).The mechanics of engineering, Volume 1. Wiley. p. 186.
  • H.F. Girvin (1938)Applied Mechanics, §28 Couples, pp 33,4, Scranton Pennsylvania: International Textbook Company.
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