Examples of solving problems in statics. Pair of forces and moments of forces Resultant of which pair of forces does not exist
This publication will help you systematize previously acquired knowledge, as well as prepare for an exam or test and pass it successfully.
* * *
by liters company.
5. Couple of forces. Moment of power
With a couple of forces is a system of two forces equal in magnitude, parallel and directed in different directions.
A pair of forces causes the body to rotate, 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.
The action of these forces on a body cannot be replaced by one resultant force.
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 couple's shoulder.
The moment is considered positive if the couple rotates the body clockwise.
M(f,f") = Fa; M > 0.
The plane passing through the lines of action of the forces of the pair is called the plane of action of the pair.
Properties of force pairs.
1. A pair of forces can be moved in the plane of its action.
2. Equivalence of pairs. Two pairs whose moments are equal 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:
M Σ = F 1 a 1 + F 2 a 2 + F 3 a 3 + … + F n a 1 ;
Equilibrium of couples. For equilibrium of pairs, it is necessary and sufficient that the algebraic sum of the moments of the pairs of the system equals zero:
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.
The moment of a force relative to a point is numerically equal to the product of the modulus of the force and the distance from the point to the line of action of the force. A perpendicular dropped from a point onto the line of action of a force is called shoulder of strength.
The moment is indicated:
M O = (F) or m O (F).
The moment is considered positive if the force turns clockwise.
* * *
The given introductory fragment of the book Technical mechanics. Crib (Aurika Lukovkina, 2009) provided by our book partner -
PAIR OF FORCE
PAIR OF FORCE
A system of two forces P and P", acting on a solid body, equal in absolute value and directed parallel, but in opposite directions, i.e. P" = -P. P.S. does not have a resultant, i.e. it cannot be replaced (and therefore cannot be balanced) by one force.
The distance l between the lines of action of a pair of forces is called. shoulder P. s. The action provided by P. s. on TV. body, characterized by its moment, which is represented by the vector M, equal in abs. the value of Pl and directed perpendicular to the plane of action of the P. s. to the side from where the rotation made by the P. s. is seen to occur counterclockwise (in the right coordinate system). The main property of P. s.: the effect it has on a given TV. body does not change if P. s. transfer anywhere in the plane of the pair or in a plane parallel to it, and also if the abs. the magnitude of the forces of the pair and the length of its arm, keeping the moment of the P. s. unchanged. Thus, the moment of P. s. can be considered applied to any point of the body. Two P. s. with the same moments M, applied to the same TV. body, are mechanically equivalent to one another. Any PS system attached to this TV. body, mechanically equivalent to one P. s. with a moment equal to geom. the sum of vectors - the moments of these P. s. If geom. the sum of vectors - moments of some system of P. s. is equal to zero, then this system of P. s. yavl. balanced.
Physical encyclopedic dictionary. - M.: Soviet Encyclopedia. . 1983 .
PAIR OF FORCE
A system of two forces of equal magnitude, parallel and directed in opposite directions. In Fig. P. s. is depicted. ( R, R"), Where R"= - R. P.S. has no resultant, i.e. it on the telon can be mechanically equivalent to the action of the c.p. one force; respectivelyP. With. cannot be balanced by force alone.
Distance l between the lines of action of the forces of the pairs called. shoulder P. s. The action provided by P. s. on a solid body M, equal in modulus Rl and directed perpendicular to the plane of action of the P. s. in the direction from which the turn the P is trying to make. pp., is seen happening counterclockwise (in the right coordinate system). Basic property of P. s. is that the action exerted by P. With. on a given solid body does not change if P. s. transfer anywhere in the plane of the pair or in a plane parallel to it, and also if you arbitrarily change the modules of the forces of the pair and the length of its arm, keeping the moment of the P. s. Thus, the moment of P. s. - free vector: it can be considered applied at any point of the body. Two P. s. with the same moments M
, applied to the same solid body, are mechanically equivalent to one another. Any system of parametric systems applied to a given solid body CM. Targ.
Physical encyclopedia. In 5 volumes. - M.: Soviet Encyclopedia. Editor-in-Chief A. M. Prokhorov. 1988 .
See what “PAIR OF FORCES” is in other dictionaries:
A pair of forces are two equal in magnitude and opposite in direction forces applied to the same body. The resultant force pair is the zero vector. The shortest distance between the lines of action of the forces forming a pair of forces is called the shoulder of the pair.... ... Wikipedia
Big Encyclopedic Dictionary
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
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
A system of two forces P and P acting on a solid body, equal to 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
Books
- Lovely couple, Hardy Kate, Dance TV show producers have paired children's show presenter Polly Anna Adams with dancer Liam Flynn. Mutual attraction arose almost immediately, but both have compelling reasons... Category: Contemporary foreign prose Series: Romance Publisher: Tsentrpoligraf,
- Lovely couple, Kate, Hardy, Dance TV show producers have paired children's show presenter Polly Anna Adams with dancer Liam Flynn. Mutual attraction arose almost immediately, but both have compelling reasons... Category:
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 a free rigid body. 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, in a 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 determine possible directions of reactions due to
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. The projections of the vector form the edges of a 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
A spatial system of forces is given (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. The positions of the centers of gravity of simple geometric figures 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. The 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 - the speed is constant - the movement is uniform, . ω = const 3.
Basic definitions
A complex movement is a movement that can be broken down into several simple ones. Simple movements are considered to be 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 in the form of 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. 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 - 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), in addition to useful work, also performs additional work.
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 a section, but does not make it possible to establish the law of distribution of internal forces over the section. To assess the strength of n
Internal force factors, tensions. Construction of diagrams
Have an idea of longitudinal forces and 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 of the physical meaning and procedure for determining axial, centrifugal and polar moments of inertia, 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
The centrifugal moment of 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 of flat sections is fulfilled: 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. Timber 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 an external force F (Fig. 29.3a). To determine internal force 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 significantly 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 stressed 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
Moment of power. A couple of forces.
1. Basic concepts and definitions of statics.
Material objects in statics:
material point,
system of material points,
absolutely solid body.
A system of material points, or a mechanical system, is a collection of material points in which the position and movement of each point depends on the position and movement of other points of this system.
Absolutely rigid body is a body whose distance between two points does not change.
A solid body can be in a state of rest or in motion of a certain nature. We will call each of these states kinematic state of the body.
|
Force- a measure of the mechanical interaction of bodies, determining the intensity and direction of this interaction. Force can be applied at a point, then this force is concentrated. Force can act on all points of a given volume or surface of the body, then this force is distributed. |
|
|
System of forces - with the totality of forces acting on a given body. |
|
|
Resultant is called a force equivalent to a certain system of forces. |
|
|
A balancing force is called a force equal in magnitude to the resultant and directed along the line of its action in the opposite direction. |
|
|
A system of mutually balancing forces is a system of forces that, when applied to a solid body at rest, does not remove it from this state. |
Inner forces- these are forces that act between points or bodies of a given system.
External forces- these are forces that act from points or bodies that are not part of a given system.
Statics tasks:
- transformation of systems of forces acting on a solid body into systems equivalent to them;
- study of the equilibrium conditions of bodies under the influence of forces applied to them.
1. Axioms of statics.
|
|
3. Axiom of addition and exclusion of balancing forces. The action of a system of forces on a solid body will not change if a system of mutually balancing forces is added to it or excluded from it. Consequence. Without changing the kinematic state of an absolutely rigid body, the force can be transferred along the line of its action, keeping its modulus and direction unchanged. WITH silt - sliding vector. |
|
|
|
4. Axiom of parallelogram of forces. The resultant of two intersecting forces is applied at the point of their intersection and is represented by the diagonal of a parallelogram constructed on these forces. |
|
5. Axiom of equality of action and reaction. Every action has an equal and opposite reaction.
2. Connections and their reactions
A rigid body is called free if it can move in space in any direction.
A body that limits the freedom of movement of a given rigid body is a connection in relation to it.
A rigid body whose freedom of movement is limited by bonds is called non-free.
All forces acting on a non-free rigid body can be divided into:
- set (active)
- bond reactions
Set force expresses the action on a given body of other bodies that can cause a change in its kinematic state.
Communication reaction - this is the force with which a given connection acts on the body, preventing one or another of its movements.
The principle of liberation of solids from bonds - a non-free solid body can be considered as a free body, on which, in addition to the specified forces, reactions of bonds act.
|
|
|
|
|
|
How to determine the direction of a reaction?
If there are two mutually perpendicular directions on the plane, in one of which the connection prevents the movement of the body, and in the other not, then the direction of its reaction is opposite to the first direction.
In general the reaction of the connection is directed in the direction opposite to that in which the connection does not allow the body to move.
 |
|
|
|
Fixed hinge Mobile |
|
|
3. Moment of force about the center
A moment of power F relative to some fixed center O is a vector located perpendicular to the plane passing through the force vector and the center O directed in that direction so that looking from its end one can see the rotation of the force F relative to the center O counterclockwise.
Properties of the moment of force relative to the center:
|
|
1) The modulus of the moment of force relative to the center can be expressed by twice the area of the triangle OAV
2) Moment of force relative to the center equal to zero in the event that the line of action of the force passes through this point, that is h = 0 . |
|
|
3) If from a point ABOUT to the point of application of force A draw a radius vector, then the vector of the moment of force can be expressed as a vector product
|
|
|
4) When a force is transferred along the line of its action, the vector of its moment relative to a given point does not change. |
(1.1)
(1.2)
If several forces lying in the same plane are applied to a rigid body, you can calculate the algebraic sum of the moments of these forces relative to any point in this plane
Moment M O , equal to the algebraic sum of the moments of a given system relative to any point in the same plane, is called the main moment of the system of forces relative to this point.
3. Moment of force about the axis
To determine the moment of force relative to the axis it is necessary:
1) draw a plane perpendicular to the Z axis;
2) determine the point ABOUT intersection of an axis with a plane;
3) project force orthogonally F to this plane;
4) find the moment of force projection F relative to the point O of intersection of the axis with the plane.
Sign rule:
The moment of force relative to the axis is considered positive if, looking towards the Z axis , one can see the projection tending to rotate the plane I around the Z axis in the direction opposite to the clockwise rotation.
|
|
Properties of moment of force relative to the axis 1) The moment of force relative to the axis is represented by a segment plotted along the Z axis from point O in the positive direction if > 0 and in the negative direction if< 0. 2) The value of the moment of force about the axis can be expressed by twice the area Δ
3) The moment of force about the axis is zero in two cases:
|
4. Couple of forces. Vector and algebraic moment of a pair of forces
A system of two equal in magnitude, parallel and oppositely directed forces and is called a couple of forces.
The plane in which the lines of action of the forces and are located is called plane of action of a pair of forces.
Shortest distance h between the lines of action of the forces that make up the pair is called shoulder of a couple of forces.
Moment of a couple of forces is determined by the product of the modulus of one of the forces of the pair and the shoulder.
|
|
Rule of signs
The moment vector M of the pair is directed perpendicular to the plane of action of the pair of forces in such a direction that, looking towards this vector, one can see the pair of forces tending to rotate the plane of its action in the direction opposite to the clockwise rotation.
- 4. Properties of force pairs on a plane
Property 1. Moment vector M pairs in magnitude and direction is equal to the vector product of the radius of the vector AB to that of the forces of this pair, towards the beginning of which the radius vector is directed AB, that is
(1.7)
Property 2. The main moment of the forces that make up a pair relative to an arbitrary point on the plane of action of the pair does not depend on the position of this point and is equal to the moment of this pair of forces.
|
|
5. Conditions for the equivalence of force pairs
Theorem on the condition of equivalence of pairs of forces,
lying in the same plane.
A pair of forces is a system of two equal in magnitude, parallel and directed in opposite directions forces acting on an absolutely rigid body (Fig. 15).
The shortest distance (perpendicular) between the lines of action of forces is called shoulderα pairs.
The action of a pair of forces on a body comes down to a rotational effect, which depends on:
1) on the modulus F of the pair’s forces and the length of its arm α;
2) the position of the plane of action of the pair;
The moment of a couple is a quantity equal to the product of the modulus of one of the forces of the couple and its shoulder, taken with the corresponding sign:
M = ±Fα. (1.7)
The algebraic sum of the moments of a pair of forces relative to any center lying in the plane of its action does not depend on the choice of this center and is equal to the moment of the pair:
m 0 (F) + m 0 (F′) = M.
Theorem on the equivalence of pairs. Without changing the action exerted on the body, a pair of forces applied to an absolutely rigid body can be replaced by any other pair lying in the same plane and having the same moment. The following properties of a pair of forces follow from this theorem:
1) this pair, without changing the effect it exerts on the body, can be transferred anywhere in the plane of action of the pair;
2) for a given pair, without changing the action it exerts on the body, you can arbitrarily change the force module or the length of the arm, keeping its moment unchanged.
Theorem. The action of a pair of forces on a solid body will not change if the pair of forces is transferred from a given plane to any other plane parallel to it.
Addition of pairs lying in the same plane
Theorem on addition of pairs. A system of pairs lying in the same plane is equivalent to one pair lying in the same plane and having a moment equal to the algebraic sum of the moments of the terms of the pairs:
M=Σ m i.
For the equilibrium of a flat system of pairs, it is necessary and sufficient that the algebraic sum of these pairs be equal to zero:
Σ m i= 0 .
This equality is a condition for equilibrium of pairs.
QUESTIONS FOR SELF-CONTROL
1. Is the bond reaction applied to the body or to the bond?
2. List the main types of connections
3. How many reaction components does each type of connection have and where are they directed?
4 Formulate the concept of “algebraic moment of force”.
5. What does “shoulder of power” mean?
6. How is the sign of the algebraic moment of force determined?
7. What is a “power couple”?
8 What does “pair leverage” mean?
9. How is the algebraic moment of a pair and its sign determined?
