Couple (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] Another term for a couple is a pure moment. Its effect is to create rotation without translation, or more generally without any acceleration of the centre of mass.

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.


Simple couple

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 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:

\tau = F \times d \,


τ is the torque
F is the magnitude of one of the forces
d is the perpendicular distance between the forces, sometimes called the arm of the couple

The magnitude of the torque is always equal to Fd, with the direction of the torque given by the unit vector \hat{e}, 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.

Independence of reference point

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 F1, F2, etc. that form a couple, with position vectors (about some origin P) r1, r2, etc., respectively. The moment about P is

M = \mathbf{r}_1\times \mathbf{F}_1 + \mathbf{r}_2\times \mathbf{F}_2 + \cdots

Now we pick a new reference point P' that differs from P by the vector r. The new moment is

M' = (\mathbf{r}_1+\mathbf{r})\times \mathbf{F}_1 + (\mathbf{r}_2+\mathbf{r})\times \mathbf{F}_2 + \cdots

Now the distributive property of the cross product implies

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

\mathbf{F}_1 + \mathbf{F}_2 + \cdots = 0.


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

Force and couple.PNG

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]

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.[4]


Couples are very important in mechanical engineering and the physical sciences. A few examples are:

  • The forces exerted by your hand on a screw-driver
  • The forces exerted by the tip of a screw-driver on the head of a screw
  • Drag forces acting on a spinning propeller
  • Forces on an electric dipole in a uniform electric field.
  • The reaction control system on a spacecraft.

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

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.[5]

See also


  1. ^ a b c Dynamics, Theory and Applications by T.R. Kane and D.A. Levinson, 1985, pp. 90-99: Free download
  2. ^ Physics for Engineering by Hendricks, Subramony, and Van Blerk, page 148, Web link
  3. ^ Engineering Mechanics: Equilibrium, by C. Hartsuijker, J. W. Welleman, page 64 Web link
  4. ^ a b c Augustus Jay Du Bois (1902). The mechanics of engineering, Volume 1. Wiley. p. 186. 
  5. ^ J.L. Ericksen (1979) Timoshenko Acceptance Speech at site for mechanicians
  • H.F. Girvin (1938) Applied Mechanics, §28 Couples, pp 33,4, Scranton Pennsylvania: International Textbook Company.

Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Couple — (  pronunciation (help·info)) may refer to: Two items of a type. Two members of an intimate relationship Couple (mechanics), a system of forces with a resultant moment but no resultant force Thermocouple, a type of temp …   Wikipedia

  • couple — ► NOUN 1) two individuals of the same sort considered together. 2) (treated as sing. or pl. ) two people who are married or otherwise closely associated romantically or sexually. 3) informal an indefinite small number. 4) Mechanics a pair of… …   English terms dictionary

  • Mechanics — This article is about an area of scientific study. For other uses, see Mechanic (disambiguation). Mechanics (Greek Μηχανική) is the branch of physics concerned with the behavior of physical bodies when subjected to forces or displacements, and… …   Wikipedia

  • Mechanics of planar particle motion — Classical mechanics Newton s Second Law History of classical mechanics  …   Wikipedia

  • couple — coupleable, adj. /kup euhl/, n., v., coupled, coupling. n. 1. two of the same sort considered together; pair. 2. two persons considered as joined together, as a married or engaged pair, lovers, or dance partners: They make a handsome couple. 3.… …   Universalium

  • couple — /ˈkʌpəl / (say kupuhl) noun 1. a combination of two; a pair. 2. two of the same sort connected or considered together. 3. two people in a romantic relationship: a young married couple. 4. two people associated as partners in a dance, etc. 5. a… …   Australian English dictionary

  • couple — noun 1》 two individuals of the same sort considered together. 2》 [treated as sing. or plural] two people who are married or otherwise closely associated romantically or sexually. 3》 informal an indefinite small number: a couple of days ago. 4》… …   English new terms dictionary

  • couple — cou·ple || kÊŒpl n. pair, duo, group of two persons or things; (Mechanics) two equal forces acting in opposite directions (causing rotation) v. connect, join, link; marry; mate, engage in sexual intercourse …   English contemporary dictionary

  • Stress (mechanics) — Continuum mechanics …   Wikipedia

  • Continuum mechanics — Continuum mechanics …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”

We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.