The main vector of a pair of forces. Couple of forces

A pair of forces is a system of two forces equal in magnitude, parallel and directed in opposite directions, acting on absolutely solid(Fig. 32, a). The system of forces F, F forming a pair is obviously not in equilibrium (these forces are not directed along the same straight line). At the same time, a pair of forces does not have a resultant, since, as will be proven, the resultant of any system of forces is the main vector, i.e., the sum of these forces, and for a pair, therefore, the properties of a pair of forces, as a special measure of the mechanical interaction of bodies, should be considered separately.

The plane passing through the lines of action of a pair of forces is called the plane of action of the pair. The distance d between the lines of action of the forces of a pair is called the shoulder of the pair. The action of a pair of forces on a rigid body is reduced to a certain rotational effect, which is characterized by a quantity called the moment of the pair. This moment is determined by: 1) its module, equal to the product of the position in space of the plane of action of the pair; 3) the direction of rotation of the pair in this plane. Thus, like the moment of force relative to the center, this is a vector quantity.

Let us introduce the following definition: the moment of a pair of forces is a vector (or M), the modulus of which is equal to the product of the modulus of one of the forces of the pair and its shoulder and which is directed perpendicular to the plane of action of the pair in the direction from which the pair is seen trying to turn the body counterclockwise (Fig. 32, b).

Let us also note that since the arm of force F relative to point A is equal to d, and the plane passing through point A and force F coincides with the plane of action of the pair, then at the same time

But unlike the moment of force, the vector, as will be shown below, can be applied at any point (such a vector is called free). The moment of a couple, like the moment of force, is measured in newton meters.

Let us show that the moment of a couple can be given another expression: the moment of a couple equal to the sum moments relative to any center O of the forces forming a pair, i.e.

To prove this, let’s draw radius vectors from an arbitrary point O (Fig. 33)

Then, according to formula (14), what we get and, therefore,

Since the validity of equality (15) has been proven. Hence, in particular, the result already noted above follows:

i.e., that the moment of a couple is equal to the moment of one of its forces relative to the point of application of the other force. Let us also note that the modulus of the moment of the pair

If we accept that the action of a pair of forces on a solid body (its rotational effect) is completely determined by the value of the sum of the moments of the forces of the pair relative to any center O, then from formula (15) it follows that two pairs of forces having the same moments are equivalent, i.e. have the same mechanical effect on the body. Otherwise, this means that two pairs of forces, regardless of where each of them is located in a given plane (or in parallel planes) and what the individual modules of their forces and their shoulders are equal to, if their moments have the same value, will are equivalent. Since the choice of center O is arbitrary, the vector can be considered applied at any point, i.e. it is a free vector.

SHOULDER OF A PAIR OF FORCES the shortest distance between the lines of action of the forces that make up the pair

(Bulgarian language; Български) - ramo for two sili

(Czech language; Čeština) - rameno dvojice sil

(German; Deutsch) - Hebelarm eines Kräftepaares

(Hungarian; Magyar) - erőpár karja

(Mongolian) - xoc khүchniy mөr

(Polish language; Polska) - ramię pary sił

(Romanian language; Român) - braţ al cuplului de forţe

(Serbo-Croatian language; Srpski jezik; Hrvatski jezik) - krak sprega strength

(Spanish; Español) -brazo del par

(English language; English) -arm of couple of forces

(French; Français) - bras de couple des forces

Construction dictionary.

See what “SHOULDER OF A PAIR OF FORCES” is in other dictionaries:

The distance between the straight lines along which the forces forming a pair of forces are directed. Samoilov K.I. Marine dictionary. M.L.: State Naval Publishing House NKVMF USSR, 1941 ... Marine Dictionary

leverage of a couple of forces- The shortest distance between the lines of action of the forces that make up a pair [Terminological dictionary of construction in 12 languages (VNIIIS Gosstroy USSR)] EN arm of couple of forces DE Hebelarm eines Kräftepaares FR bras de couple des forces ...

leverage of a couple of forces- jėgų dvejeto petys statusas T sritis fizika atitikmenys: engl. arm of couple; moment arm vok. Arm des Kräftepaares, f rus. leverage of a couple of forces, n pranc. bras de levier du couple, m; bras du couple, m; bras du couple de forces, m … Fizikos terminų žodynas

shoulder of the internal force pair- z - [English-Russian dictionary for the design of building structures. MNTKS, Moscow, 2011] Topics building structures Synonyms z EN lever arm of internal forces ... Technical Translator's Guide

shoulder of an internal pair of forces in the cross section of a reinforced masonry element under the action of a bending moment or eccentric compression- z - [English-Russian dictionary for the design of building structures. MNTKS, Moscow, 2011] Topics building structures Synonyms z EN lever arm ... Technical Translator's Guide

couple's shoulder- The distance between the lines of action of the forces of the pair. [Collection of recommended terms. Issue 102. Theoretical mechanics. Academy of Sciences of the USSR. Committee of Scientific and Technical Terminology. 1984] Topics theoretical mechanics General terms kinetics EN... ... Technical Translator's Guide

couple's shoulder- The distance between the lines of action of the pair forces... Polytechnic terminological explanatory dictionary

P. moment of force (see the corresponding article) or momentum around a given point is the shortest distance of force or direction of speed from this point. The length of a pair of forces is the length of the shortest distance between the forces of the pair. P. inertia of some body... ... Encyclopedic Dictionary F.A. Brockhaus and I.A. Ephron

Two equal in magnitude and opposite in direction parallel forces applied to one body. A pair of forces has no resultant. The shortest distance between the lines of action of the forces forming a pair of forces is called the shoulder of the pair. The action of the couple... ... encyclopedic Dictionary

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Any kinematic state of bodies that have a point or axis of rotation can be described by a moment of force that characterizes the rotational effect of the force.

Moment of force about the center- this is the vector product of the radius - the vector of the point of application of the force by the force vector.

Shoulder of power- the shortest distance from the center to the line of action of the force (perpendicular from the center to the line of action of the force).

The vector is directed according to the vector product rule: the moment of the force relative to the center (point) as a vector is directed perpendicular to the plane in which the force and the center are located so that from its end it can be seen that the force is trying to rotate the body around the center counterclockwise.

Unit of measurement of moment of force there is 1

Moment of force relative to the center in the plane- an algebraic quantity that is equal to the product of the modulus of force by the shoulder relative to the same center, taking into account the sign.

The sign of the moment of the force depends on the direction in which the force tries to rotate about the center:

counterclockwise -„−” (negative)
clockwise -„+” (positive);

Properties of the moment of force relative to the center (point).

The modulus of the moment of force relative to a point is equal to twice the area of the triangle constructed on vectors.
The moment of a force relative to a point does not change when a force is transferred along its line of action, since the arm of the force remains unchanged.
The moment of force relative to the center (point) is equal to zero if:

force is zero F = 0;
arm of force h = 0, i.e. the line of action of the force passes through the center.

Varignon's theorem (about the moment of the resultant).

The moment of the resultant plane system of converging forces relative to any center is equal to the algebraic sum of the moments of the component forces of the system relative to the same center.

Force couple theory

The addition of two parallel forces directed in the same direction.

The resultant of a system of two parallel forces directed in one direction is equal in modulus to the sum of the moduli of the component forces, is parallel to them and directed in the same direction.

The line of action of the resultant passes between the points of application of the components at distances from these points inversely proportional to the forces

Addition of two parallel forces directed in different directions (the case of forces of different magnitudes)

The resultant of two parallel, unequal in magnitude, oppositely directed forces is parallel to them and directed in the direction of the greater force and is equal in magnitude to the difference in the component forces.

The line of action of the resultant passes outside the segment (on the side of the larger force) connecting the points of their application, and is spaced from them at distances inversely proportional to the forces.

Couple of forces- a system of two parallel forces, equal in magnitude and opposite in direction, applied to an absolutely rigid body.

Leverage of force couple- the distance between the lines of action of the forces of the pair, i.e. the length of a perpendicular drawn from an arbitrary point on the line of action of one of the forces of a pair to the line of action of the second force.

Plane of action of a couple of forces- this is the plane in which the lines of action of the pair’s forces are located.
The action of a pair of forces comes down to rotational movement, which is determined by the moment of the couple.

Couple moment is called a vector with the following characteristics:

it is perpendicular to the plane of the pair;
directed in the direction from which the rotation performed by the pair is visible counterclockwise;
its modulus is equal to the product of the modulus of one of the forces of the pair and the arm of the pair, taking into account the sign

Sign of the moment of a couple of forces:

“+” - counterclockwise rotation
„-„ - clockwise rotation

The moment of a pair of forces is equal to the product of the modulus of one of the forces of the pair and the arm of the pair.

The moment of a couple is a free vector - for it neither the point of application nor the line of action are designated, they can be arbitrary.

Property of the moment of a pair of forces: the moment of the pair is equal to the moment of one of the forces relative to the point of application of the second force.

Couple force theorems

Theorem 1. A pair of forces does not have a resultant, i.e. A pair of forces cannot be replaced by one force.

Theorem 2. A pair of forces is not a system of balanced forces.

Consequence: a pair of forces acting on an absolutely rigid body tries to rotate it.

Theorem 3. The sum of the moments of forces of a pair relative to arbitrary center(points) in space is a constant quantity and represents the vector-moment of this pair.

Theorem 4. The sum of the moments of forces that make up a pair relative to an arbitrary center in the plane of action of the pair does not depend on the center and is equal to the product of the force by the arm of the pair, taking into account the sign, i.e. the very moment of the couple.

Theorem 5 - about the equivalence of pairs. Pairs of forces whose moments are equal in number and sign are equivalent. Those. a pair of forces can only be replaced or balanced by another equivalent pair of forces.

Theorem 6 is about the balance of a pair of forces. A pair of forces constitutes a balanced system of forces if and only if the moment of the pair is zero.

Theorem 7 - about the possibilities of moving a pair of forces in the plane of its action. The force pair obtained by moving the pair to any place in the plane of its action is equivalent to the provided pair.

Theorem 8 is about adding pairs of forces in the plane. The moment of a pair equivalent to the provided system of pairs in the plane is equal to the algebraic sum of the moments of the constituent pairs. Those. To add pairs of forces, you need to add their moments.

Conditions for the equilibrium of a system of pairs of forces.

Pairs of forces in a plane are balanced if the algebraic sum of their moments is equal to zero.

Language: Russian, Ukrainian

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The shortest distance between the lines of action of the forces forming a pair of forces is called shoulder couples.

Properties

Illustration. The solid body is shown in blue.

The action of a pair of forces on a body is characterized by the moment of the pair of forces - the product of the module of one of the forces by the shoulder, . Like anyone mechanical moment, the moment of a pair of forces is a pseudo-vector quantity and is directed perpendicular to the plane defined by parallel straight lines on which the force vectors lie: (in this case, the direction of the shoulder vector should conditionally be set towards To point of application of the force selected from the pair).

The moment of a couple of forces has no point of application(Varignon’s second theorem): no matter what parts of a rigid body forces are applied, for a given magnitude and direction of the moment of force it will rotate the same.

The action of a force applied to a solid body at a certain distance d from the center of mass (at the point to which a vector can be drawn from the center of mass) is equivalent to the action of the same force applied directly to the center of mass, combined with a certain pair of forces such that , that is, with a moment equal to the moment of force relative to the center of mass (in particular, if , we can set , in this case one of the forces will be applied at the same point as the original one and will amount to ).

Sources

// Encyclopedic Dictionary of Brockhaus and Efron: In 86 volumes (82 volumes and 4 additional ones). - St. Petersburg. , 1890-1907.
- article from Physical encyclopedic dictionary (1983)

Wikimedia Foundation.

2010.

Big Encyclopedic Dictionary

A system of two forces P and P acting on a TV. body equal in abs. magnitude and directed parallel, but in opposite directions, i.e. P = R.P.s. does not have a resultant, i.e. it cannot be replaced (and therefore cannot be balanced) by one... ... Physical encyclopedia

Two equal and parallel forces directed in opposite directions. P.S. acting on some body causes rotation of this body around an axis perpendicular to the plane in which the pair of forces is located. Samoilov K. I. Marine dictionary.... ... Marine dictionary

couple of forces- a couple of forces; pair A system of two parallel forces, equal in magnitude and directed in opposite directions... Polytechnic terminological explanatory dictionary

PAIR OF FORCE- two equal in absolute value and oppositely directed parallel forces applied to one solid body. P.S. tends to cause rotation of the body to which it is applied, and has no (see) force. The distance between the lines of action of P. with ... Big Polytechnic Encyclopedia

PAIR OF FORCES, two equal and oppositely directed parallel forces. Their action leads to the generation of torque... Scientific and technical encyclopedic dictionary

couple of forces- Two coplanar parallel forces, equal in magnitude and opposite in direction, applied to a solid body at some distance from each other [Terminological dictionary of construction in 12 languages (VNIIIS Gosstroy USSR)] EN couple... ... Technical Translator's Guide

A system of two forces P and P acting on a rigid body, equal friend each other in absolute value, parallel and directed in opposite directions (i.e. P = P; see figure). P.S. has no resultant, i.e. its action on the body does not... ... Great Soviet Encyclopedia

Two equal in a6c. value (modulus) and parallel forces F and F opposite in direction (see figure). appl. to the same solid body. The shortest distance l between the lines of action of the forces of a pair is called. her shoulder. P.S. seeks to evoke... Big Encyclopedic Polytechnic Dictionary

The action of a pair of forces on a body is characterized by: 1) the magnitude of the moment modulus of the pair, 2) the plane of action, 3) the direction of rotation in this plane. When considering pairs that do not lie in the same plane, all three of these elements will need to be specified to characterize each pair. This can be done if we agree, by analogy with the moment of a force, to represent the moment of a couple in an appropriate way, constructed by a vector, namely: we will represent the moment of a couple with a vector m or M, the modulus of which is equal (on the chosen scale) to the modulus of the moment of the couple, i.e. the product of one of its forces on the shoulder, and which is directed perpendicular to the plane of action of the pair in the direction from which the rotation of the pair is seen occurring counterclockwise (Fig. 38).

Rice. 38

As is known, the modulus of the moment of a pair is equal to the moment of one of its forces relative to the point where another force is applied, i.e.; in the direction the vectors of these moments coincide. Hence .

Moment of force about the axis.

To move on to solving statics problems for the case of an arbitrary spatial system of forces, it is necessary to introduce the concept of the moment of force relative to the axis.

The moment of force about an axis characterizes the rotational effect created by a force tending to rotate a body around a given axis. Consider a rigid body that can rotate around some axis z(Fig. 39).

Fig.39

Let this body be acted upon by a force applied at a point A. Let's draw through the point A plane xy, perpendicular to the z axis, and decompose the force into components: , parallel to the z axis, and , lying in the xy plane (is simultaneously a projection of the force on the plane xy). Force directed parallel to the axis z, obviously cannot rotate the body around this axis (it only tends to move the body along the axis z). The entire rotational effect created by the force will coincide with the rotational effect of its component. From here we conclude that , where the symbol denotes the moment of force relative to the axis z.

For a force lying in a plane perpendicular to the axis z, the rotational effect is measured by the product of the magnitude of this force and its distance h from the axis. But the same quantity measures the moment of force relative to a point ABOUT, in which the axis z intersects with the plane xy. Therefore, or, according to the previous equality,

As a result, we arrive at the following definition: the moment of a force relative to an axis is a scalar quantity equal to the moment of projection of this force onto a plane perpendicular to the axis, taken relative to the point of intersection of the axis with the plane.

From the drawing (Fig. 40) it is clear that when calculating the moment, the plane xy can be drawn through any point on the axis z. Thus, to find the moment of force about the axis z(Fig. 40) you need to:

1) draw a plane xy, perpendicular to the axis z(anywhere);

2) project the force onto this plane and calculate the value;

3) lower from the point ABOUT intersection of the axis with the plane perpendicular to the direction and find its length h;

4) calculate the product ;

5) determine the sign of the moment.

When calculating moments, the following special cases must be kept in mind:

1) If the force is parallel to the axis, then its moment relative to the axis is zero (since F xy = 0).

2) If the line of action of the force intersects the axis, then its moment relative to the axis is also zero (since h = 0).

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