A pair of forces is a system of two equal in magnitude, parallel and directed in opposite sides forces acting on an absolutely rigid body (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 pair can be given another expression: the moment of a pair is equal to the sum of the moments relative to any center O of the forces forming the 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 be 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.

With a couple of forces is a system of two forces equal in magnitude, parallel and directed in different directions.

Let's consider the system of forces (R; B"), forming a pair.

A pair of forces causes rotation of the body and its effect on the body is measured by the moment. The forces entering the pair are not balanced, since they are applied to two points (Fig. 4.1).

Their action on the body cannot be replaced by one force (resultant).

The moment of a pair of forces is numerically equal to the product of the force modulus and the distance between the lines of action of the forces (pair's shoulder).

The moment is considered positive if the couple rotates the body clockwise (Fig. 4.1(b)):

M(F;F") = Fa ; M > 0.

The plane passing through the lines of action of the forces of the pair is called plane of action of the pair.

Properties of pairs(without evidence):

1. A pair of forces can be moved in the plane of its action.

2. Equivalence of pairs.

Two pairs whose moments are equal (Fig. 4.2) are equivalent (their effect on the body is similar).

3. Addition of pairs of forces. The system of force pairs can be replaced by a resultant pair.

The moment of the resultant pair is equal to the algebraic sum of the moments of the pairs that make up the system (Fig. 4.3):

4. Equilibrium of pairs.

For equilibrium of pairs, it is necessary and sufficient that the algebraic sum of the moments of the pairs of the system equals zero:

End of work -

This topic belongs to the section:

Theoretical mechanics

Theoretical mechanics.. lecture.. topic: basic concepts and axioms of statics..

If you need additional material on this topic, or you did not find what you were looking for, we recommend using the search in our database of works:

What will we do with the received material:

If this material was useful to you, you can save it to your page on social networks:

Tweet

All topics in this section:

Problems of theoretical mechanics
Theoretical mechanics is the science of the mechanical motion of material solid bodies and their interaction. Mechanical motion is understood as the movement of a body in space and time from

Third axiom
Without disturbing the mechanical state of the body, you can add or remove a balanced system of forces (the principle of discarding a system of forces equivalent to zero) (Fig. 1.3). P,=P2 P,=P.

Corollary to the second and third axioms
The force acting on a solid body can be moved along the line of its action (Fig. 1.6).

Connections and reactions of connections
All laws and theorems of statics are valid for free solid. All bodies are divided into free and bound. Free bodies are bodies whose movement is not limited.

Hard rod
In the diagrams, the rods are depicted as a thick solid line (Fig. 1.9). The rod can

Fixed hinge
The attachment point cannot be moved. The rod can rotate freely around the hinge axis. The reaction of such a support passes through the hinge axis, but

Plane system of converging forces
A system of forces whose lines of action intersect at one point is called convergent (Fig. 2.1).

Resultant of converging forces
The resultant of two intersecting forces can be determined using a parallelogram or triangle of forces (4th axiom) (vis. 2.2).

Equilibrium condition for a plane system of converging forces
When the system of forces is in equilibrium, the resultant must be equal to zero, therefore, when geometric construction the end of the last vector must coincide with the beginning of the first. If

Solving equilibrium problems using a geometric method
It is convenient to use the geometric method if there are three forces in the system. When solving equilibrium problems, consider the body to be absolutely solid (solidified). Procedure for solving problems:

Solution
1. The forces arising in the fastening rods are equal in magnitude to the forces with which the rods support the load (5th axiom of statics) (Fig. 2.5a). We define possible directions reactions

Projection of force on the axis
The projection of the force onto the axis is determined by the segment of the axis, cut off by perpendiculars lowered onto the axis from the beginning and end of the vector (Fig. 3.1).

Strength in an analytical way
The magnitude of the resultant is equal to the vector (geometric) sum of the vectors of the system of forces. We determine the resultant geometrically. Let's choose a coordinate system, determine the projections of all tasks

Converging forces in analytical form
Based on the fact that the resultant is zero, we obtain: Condition

Moment of force about a point
A force that does not pass through the point of attachment of the body causes rotation of the body relative to the point, therefore the effect of such a force on the body is estimated as a moment. Moment of force rel.

Poinsot's theorem on parallel transfer of forces
A force can be transferred parallel to the line of its action; in this case, it is necessary to add a pair of forces with a moment equal to the product of the modulus of the force and the distance over which the force is transferred.

Distributed forces
The lines of action of an arbitrary system of forces do not intersect at one point, therefore, to assess the state of the body, such a system should be simplified. To do this, all the forces of the system are transferred into one arbitrarily

Influence of reference point
The reference point is chosen arbitrarily. When the position of the reference point changes, the value of the main vector will not change. The magnitude of the main moment when moving the reduction point will change,

Flat force system
1. At equilibrium, the main vector of the system is zero. Analytical determination of the main vector leads to the conclusion:

Types of loads
According to the method of application, loads are divided into concentrated and distributed. If the actual load transfer occurs on a negligibly small area (at a point), the load is called concentrated

Moment of force about the axis
The moment of force relative to the axis is equal to the moment of projection of the force onto a plane perpendicular to the axis, relative to the point of intersection of the axis with the plane (Fig. 7.1 a). MOO

Vector in space
In space, the force vector is projected onto three mutually perpendicular coordinate axes. Vector projections form edges rectangular parallelepiped, the force vector coincides with the diagonal (Fig. 7.2

Spatial convergent system of forces
A spatial convergent system of forces is a system of forces that do not lie in the same plane, the lines of action of which intersect at one point. The resultant of the spatial system

Bringing an arbitrary spatial system of forces to the center O
Dana spatial system forces (Fig. 7.5a). Let's bring it to the center O. The forces must be moved in parallel, and a system of pairs of forces is formed. The moment of each of these pairs is equal

Center of gravity of homogeneous flat bodies
(flat figures) Very often it is necessary to determine the center of gravity of various flat bodies and geometric flat figures of complex shape. For flat bodies we can write: V =

Determining the coordinates of the center of gravity of plane figures
Note. The center of gravity of a symmetrical figure is on the axis of symmetry. The center of gravity of the rod is at the middle of the height. Positions of the centers of gravity of simple geometric shapes can

Kinematics of a point
Have an idea of ​​space, time, trajectory, path, speed and acceleration. Know how to specify the movement of a point (natural and coordinate). Know the designations

Distance traveled
The path is measured along the trajectory in the direction of travel. Designation - S, units of measurement - meters. Equation of motion of a point: Equation defining

Travel speed
The vector quantity that currently characterizes the speed and direction of movement along the trajectory is called speed. Velocity is a vector directed at any moment towards

Point acceleration
A vector quantity that characterizes the rate of change in speed in magnitude and direction is called the acceleration of a point. Speed ​​of the point when moving from point M1

Uniform movement
Uniform motion is motion at a constant speed: v = const. For rectilinear uniform motion (Fig. 10.1 a)

Equally alternating motion
Equally variable motion is motion with constant tangential acceleration: at = const. For rectilinear uniform motion

Forward movement
Translational is the movement of a rigid body in which any straight line on the body during movement remains parallel to its initial position (Fig. 11.1, 11.2). At

Rotational movement
During rotational motion, all points of the body describe circles around a common fixed axis. Fixed axis, around which all points of the body rotate, is called the axis of rotation.

Special cases of rotational motion
Uniform rotation (angular velocity is constant): ω =const The equation (law) of uniform rotation in this case has the form:

Velocities and accelerations of points of a rotating body
The body rotates around point O. Let us determine the parameters of motion of point A, located at a distance RA from the axis of rotation (Fig. 11.6, 11.7). Path

Solution
1. Section 1 - uneven accelerated movement, ω = φ’; ε = ω’ 2. Section 2 - speed is constant - uniform movement, . ω = const 3.

Basic definitions
A complex movement is a movement that can be broken down into several simple ones. Simple movements consider translational and rotational. To consider the complex motion of points

Plane-parallel motion of a rigid body
Plane-parallel, or flat, motion of a rigid body is called such that all points of the body move parallel to some fixed one in the reference system under consideration

Translational and rotational
Plane-parallel motion is decomposed into two motions: translational with a certain pole and rotational relative to this pole. Decomposition is used to determine

Speed ​​Center
The speed of any point on the body can be determined using the instantaneous center of velocities. In this case, complex movement is represented as a chain of rotations around different centers. Task

Axioms of dynamics
The laws of dynamics generalize the results of numerous experiments and observations. The laws of dynamics, which are usually considered as axioms, were formulated by Newton, but the first and fourth laws were also

The concept of friction. Types of friction
Friction is the resistance that occurs when one rough body moves over the surface of another. When bodies slide, sliding friction occurs, and when they roll, rolling friction occurs. Nature support

Rolling friction
Rolling resistance is associated with mutual deformation of the soil and the wheel and is significantly less than sliding friction. Usually the soil is considered softer than the wheel, then the soil is mainly deformed, and

Free and non-free points
A material point whose movement in space is not limited by any connections is called free. Problems are solved using the basic law of dynamics. Material then

Inertia force
Inertia is the ability to maintain one’s state unchanged; this is an internal property of all material bodies. Inertia force is a force that arises during the acceleration or braking of bodies

Solution
Active forces: driving force, friction force, gravity force. Reaction in the support R. We apply the inertial force in the opposite direction from the acceleration. According to d'Alembert's principle, the system of forces acting on the platform

Work done by resultant force
Under the action of a system of forces, a point with mass m moves from position M1 to position M 2 (Fig. 15.7). In the case of movement under the influence of a system of forces, use

Power
To characterize the performance and speed of work, the concept of power was introduced. Power is work performed per unit of time:

Rotating power
Rice. 16.2 The body moves along an arc of radius from point M1 to point M2 M1M2 = φr Work of force

Efficiency
Each machine and mechanism, when doing work, spends part of its energy to overcome harmful resistances. Thus, the machine (mechanism) except useful work also performs an additional

Momentum change theorem
The momentum of a material point is a vector quantity equal to the product of the mass of the point and its speed mv. The vector of momentum coincides with

Theorem on the change of kinetic energy
Energy is the ability of a body to do mechanical work. There are two forms of mechanical energy: potential energy, or positional energy, and kinetic energy.

Fundamentals of the dynamics of a system of material points
A set of material points connected by interaction forces is called a mechanical system. Any material body in mechanics is considered as a mechanical

Basic equation for the dynamics of a rotating body
Let a rigid body, under the action of external forces, rotate around the Oz axis with angular velocity

Voltages
The section method makes it possible to determine the value of the internal force factor in the section, but does not make it possible to establish the distribution law internal forces by section. To assess the strength of n

Internal force factors, tensions. Construction of diagrams
Have an idea about longitudinal forces ah, about normal stresses in cross sections. Know the rules for constructing diagrams of longitudinal forces and normal stresses, the distribution law

Longitudinal forces
Let us consider a beam loaded with external forces along its axis. The beam is fixed in the wall (fastening “fixing”) (Fig. 20.2a). We divide the beam into loading areas. Loading area with

Geometric characteristics of flat sections
Have an idea about physical sense and the procedure for determining axial, centrifugal and polar moments of inertia, about the main central axes and the main central moments of inertia.

Static moment of sectional area
Let's consider an arbitrary section (Fig. 25.1). If we divide the section into infinitesimal areas dA and multiply each area by the distance to the coordinate axis and integrate the resulting

Centrifugal moment of inertia
Centrifugal moment The inertia of a section is the sum of the products of elementary areas taken over both coordinates:

Axial moments of inertia
The axial moment of inertia of a section relative to a certain yard lying in the same plane is called the sum of the products of elementary areas taken over the entire area by the square of their distance

Polar moment of inertia of the section
The polar moment of inertia of a section relative to a certain point (pole) is the sum of the products of elementary areas taken over the entire area by the square of their distance to this point:

Moments of inertia of the simplest sections
Axial moments of inertia of a rectangle (Fig. 25.2) Imagine directly

Polar moment of inertia of a circle
For a circle, first calculate the polar moment of inertia, then the axial ones. Let's imagine a circle as a collection of infinitely thin rings (Fig. 25.3).

Torsional Deformation
Torsion of a round beam occurs when it is loaded with pairs of forces with moments in planes perpendicular to the longitudinal axis. In this case, the generatrices of the beam are bent and rotated through an angle γ,

Hypotheses for torsion
1. The hypothesis is fulfilled flat sections: the cross section of the beam, flat and perpendicular to the longitudinal axis, after deformation remains flat and perpendicular to the longitudinal axis.

Internal force factors during torsion
Torsion is a loading in which only one internal force factor appears in the cross section of the beam - torque. External loads are also two

Torque diagrams
Torque moments can vary along the axis of the beam. After determining the values ​​of the moments along the sections, we construct a graph of the torques along the axis of the beam.

Torsional stress
We draw a grid of longitudinal and transverse lines on the surface of the beam and consider the pattern formed on the surface after Fig. 27.1a deformation (Fig. 27.1a). Pop

Maximum torsional stresses
From the formula for determining stresses and the diagram of the distribution of tangential stresses during torsion, it is clear that the maximum stresses occur on the surface. Let's determine the maximum voltage

Types of strength calculations
There are two types of strength calculations: 1. Design calculation - the diameter of the beam (shaft) in the dangerous section is determined:

Stiffness calculation
When calculating rigidity, the deformation is determined and compared with the permissible one. Let us consider the deformation of a round beam under the action of an external pair of forces with a moment t (Fig. 27.4).

Basic definitions
Bending is a type of loading in which an internal force factor—a bending moment—appears in the cross section of the beam. Beam working on

Internal force factors during bending
Example 1. Consider a beam that is acted upon by a pair of forces with a moment m and external force F (Fig. 29.3a). To determine internal power factors we use the method with

Bending moments
A transverse force in a section is considered positive if it tends to rotate it

Differential dependencies for direct transverse bending
The construction of diagrams of shear forces and bending moments is greatly simplified by using differential relationships between the bending moment, shear force and uniform intensity

Using the section method The resulting expression can be generalized
The transverse force in the section under consideration is equal to the algebraic sum of all forces acting on the beam up to the section under consideration: Q = ΣFi Since we are talking

Voltages
Let us consider the bending of a beam clamped to the right and loaded with a concentrated force F (Fig. 33.1).

Stress state at a point
The stress state at a point is characterized by normal and tangential stresses that arise on all areas (sections) passing through this point. Usually it is enough to determine for example

The concept of a complex deformed state
The set of deformations occurring in different directions and in different planes passing through a point determines the deformed state at this point. Complex deformation

Calculation of a round beam for bending with torsion
In the case of calculating a round beam under the action of bending and torsion (Fig. 34.3), it is necessary to take into account normal and tangential stresses, since the maximum stress values ​​in both cases arise

The concept of stable and unstable equilibrium
Relatively short and massive rods are designed for compression, because they fail as a result of destruction or residual deformations. Long rods with a small cross-section for action

Stability calculation
The stability calculation consists of determining the permissible compressive force and, in comparison with it, the acting force:

Calculation using Euler's formula
The problem of determining the critical force was mathematically solved by L. Euler in 1744. For a rod hinged on both sides (Fig. 36.2), Euler’s formula has the form

Critical stresses
Critical stress is the compressive stress corresponding to the critical force. The stress from the compressive force is determined by the formula

Limits of applicability of Euler's formula
Euler's formula is valid only within the limits of elastic deformations. Thus, the critical stress must be less than the elastic limit of the material. Prev

1. Plane system of converging forces

The system of converging forces is in equilibrium, when the algebraic sums of the projections of its terms onto each of the two coordinate axes are equal to zero.

Projection of force onto the axis.

Axis called a straight line to which a certain direction is assigned. The projection of the vector onto the axis is a scalar quantity.

The projection of a vector is considered positive (+) if the direction from the beginning to its end coincides with positive direction axes. A vector projection is considered negative (-) if the direction from the beginning of the projection to its end is opposite to the positive direction of the axis.

If the force coincides with the positive direction of the axis, but the angle is obtuse, then the projection of the force onto the axis will be negative.

So, the projection of the force onto the coordinate axis is equal to the product of the force modulus and the cosine or sine of the angle between the force vector and the positive direction of the axis.

The force located on the xOy plane can be projected onto two coordinate axes Ox and Oy:

; ; .

Projection of a vector sum onto an axis.

The geometric sum, or resultant, of these forces

determined by the closing side of the force polygon: ,

Where p – number terms of vectors.

So, the projection of a vector sum or resultant onto any axis is equal to the algebraic sum of the projections of the summands of the vectors onto the same axis.

2. Couple of forces

The sum of the projections of a pair of forces on the x-axis and on the y-axis is equal to zero, therefore the pair of forces does not have a resultant. Despite this, the body is in equilibrium under the influence of a pair of forces.

The ability of a pair of forces to produce rotation is determined couple moment, equal to the product of the force and the shortest distance between the lines of action of the forces. Let us denote the moment of the couple M, and the shortest distance between the forces A, then the absolute value of the moment:

The shortest distance between the lines of action of forces is called - couple's shoulder, so we can say that The absolute value of the moment of a pair of forces is equal to the product of one of the forces and its shoulder.

The moment of a pair of forces can be shown by an arc-shaped arrow indicating the direction of rotation.

Two pairs of forces are considered equivalent in the event that after replacing one pair with another, the mechanical state of the body does not change, i.e. the movement of the body does not change or its balance is not disturbed.

The effect of a pair of forces on a rigid body does not depend on its position in the plane. Thus, a pair of forces can be transferred in the plane of its action to any position.

Another property of a pair of forces, which is the basis for adding pairs:

− without disturbing the state of the body, you can change the force modules and the arm of the pair as you like, as long as the moment of the pair remains unchanged.

By definition, pairs of forces are equivalent, i.e. produce the same effect if their moments are equal.

If, by changing the values ​​of the forces and the arm of the new pair, we maintain the equality of their moments M 1 = M 2 or F 1 a = F 2 b, then the state of the body will not be disturbed by such a replacement.

Similar to the forces of a couple, they can be added. A pair that replaces the action of these pairs is called resulting. The action of a pair of forces is completely determined by its moment and direction of rotation. Based on this, the addition of pairs is carried out by algebraic summation of their moments, i.e. the moment of the resulting pair is equal to the algebraic sum of the moments of the component pairs.

The moment of the resulting pair is determined by the formula:

M= M 1 + M 2 +. .. + M p.=

M і ,

Where the moments of pairs rotating clockwise are taken to be positive, and those rotating counterclockwise are taken to be negative. Based on the above rule for adding pairs, the equilibrium condition for a system of pairs lying in the same plane is established, namely: for a system of pairs to be in equilibrium, it is necessary and sufficient that the moment of the resulting pair be equal to zero or that the algebraic sum of the moments of the pairs be equal to zero:

Moment of force about a point and an axis.

The moment of a force relative to a point is determined by the product of the modulus of the force and the length of the perpendicular drawn from the point to the line of action of the force.

When fixing a body at point O, the force

tends to rotate it around this point. The point O about which the moment is taken is called moment center, and the length of the perpendicular a – shoulder relative to the center of moment.

moment of force

relative to O is determined by the product of the force by the shoulder: .

The moment is considered positive if the force tends to rotate the body clockwise, and negative - counterclockwise. There is one significant difference between the moment of a couple and the moment of a force. The numerical value and direction of the moment of a pair of forces does not depend on the position of this pair in the plane. The value and direction (sign) of the moment of force depends on the position of the point relative to which the moment is determined. Therefore, to determine the moment of force relative to an axis, you need to project the force onto a plane perpendicular to the axis and find the moment of projection of the force relative to the point of intersection of the axis with this plane.

3. Kinetostatic method

Let us imagine a material point of mass m, moving with acceleration a under the influence of some system of active and reactive forces, the resultant of which is equal to F.

Let us use one of the formulas known to us (the basic equation of dynamics) in order to write the equations of motion in the form of equilibrium equations (the kinetostatic method):

Let's rewrite this equation as follows:

The expression is denoted by Kin and is called the force of inertia:

The inertial force is a vector equal to the product of the mass of a point and its acceleration and directed in the direction opposite to the acceleration.

This equality is mathematical expression the principle, which bears the name of the French scientist d'Alembert (1717-1783), can be considered as the equilibrium equation of a material point. It should be emphasized that the resulting equality, although called the equilibrium equation, is in fact a modified equation of motion of a material point.

D'Alembert's principle is formulated as follows: active and reactive forces acting on a material point, together with inertial forces, form a system of mutually balanced forces that satisfies all equilibrium conditions.

It should be remembered that the inertial force is applied to the considered material point conditionally, but for the connection causing acceleration, it is in a certain sense real. Possessing the property of inertia, any body tends to maintain its speed in magnitude and direction unchanged, as a result of which it will act on the connection causing acceleration with a force equal to the force of inertia. As an example of the action of inertial forces, we can cite cases of destruction of flywheels when they reach a critical angular velocity. In any rotating body, inertial forces act, since each particle of this body has acceleration, and neighboring particles are connections for it. Note that the weight of a body is the force with which the body, due to the gravity of the Earth, acts on the support (or suspension) that holds it from free fall. If the body and the support are motionless, then the weight of the body is equal to its gravity.

4. Moment of force about a point

Consider a nut that is tightened with a wrench of a certain length, applying muscular force to the end of the wrench. If you take a wrench several times longer, then using the same force, the nut can be tightened much stronger. It follows from this that the same force can have different rotational effects. The rotational action of a force is characterized by a moment of force.

The concept of a moment of force relative to a point was introduced into mechanics by the Italian Renaissance scientist and artist Leonardo da Vinci (1452-1519).

The moment of a force relative to a point is the product of the modulus of the force and its shoulder:

M 0 (¥) = RI.

The point about which the moment is taken is called the center of the moment. The arm of a force relative to a point is the shortest distance from the center of the moment to the line of action of the force.

A system of two forces equal in magnitude, parallel and directed in opposite directions, acting on an absolutely rigid body. The action of a pair of forces on a rigid body is reduced to a certain rotational effect, which is characterized by a value - the moment of the pair.

It is defined:

Its module = F*d. d - the distance between the lines of action of the forces of the pair, is called the shoulder of the pair.

Position in space of the plane of action of the pair.

The direction of rotation of the pair in this plane.

Moment of a couple of forces- vector m (or M), the modulus of which is equal to the product of the modulus of one of the forces of the pair by its shoulder, and which is directed perpendicular to the plane of action of the pair in the direction from which the pair is visible trying to turn the body counterclockwise.

Two pairs lying in || planes and having the same moment are equivalent.

All pairs in intersecting planes can be replaced by one pair with a moment equal to the sum of the moments of these pairs. For an absolutely solid body of steam- free vector, determined only by the moment. The moment is perpendicular to the plane formed by the pair.

The pair can be replaced by a parallel force equal to it and a pair with a moment equal to the product of this force and the distance to the new point of application.

Pair theorems .

1) Two pairs lying in the same plane can be replaced by one pair lying in the same plane, with a moment equal to the sum of the moments of these two pairs. .

2) Two pairs having geometrically equal moments, equivalents.

3) Without disturbing the state of a solid body, a couple of forces can be transferred in the plane of its action. Those. the moment of a couple of forces is a free vector.

4) A system of several pairs of forces is equivalent to one pair, the moment of which is equal to the vector sum of the moments of these pairs. Those. the system of pairs is reduced to one pair, the moment of which is equal to the sum of the moments of all pairs. Condition for equilibrium of pairs of forces: - geometric sum their moments are equal to 0. Pairs of forces located in the same plane will be mutually balanced if the algebraic sum of their moments åM i =0.

Moment of force about a point - a vector numerically equal to the product of the modulus of force by the shoulder and directed perpendicular to the plane containing the force and the point, in such a direction that looking towards it, you can see the force tending to turn counterclockwise. Shoulder "h" is the shortest distance from a point to the line of action of the force. - the moment of force is equal to the vector product of a vector and a vector. Module vector product: R×F×sina = F×h. For flat system usually it is not the torque vector that is found, but only its magnitude: ± F×h, >0 - counterclockwise; x, F y, F z are projections of force on the coordinate axes and point 0 is the origin of coordinates, then

= (yF z - zF y) + (zF x - xF z) + (xF y - yF x), whence the projections of the moment of force on the coordinate axis: М 0 x () = yF z - zF y ; М 0 y () = zF x - xF z ; M 0 z () = xF y - yF x .

Main vector - vector sum all forces applied to the body. Main point relative to the center - the vector sum of the moments of all forces applied to the body relative to the same center.

Theorem (lemma) O parallel transfer strength: force applied at any point on a solid. body, equivalent to the same force applied at any other point of this body, and a pair of forces, the moment of which is equal to the moment of the given force relative to the new point of application.

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

leverage 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. Brockhaus and I.A. Efron

Two equal in size 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