Practical application of theoretical mechanics. Theoretical mechanics for engineers and researchers

13.05.2020

Within any training course The study of physics begins with mechanics. Not from theoretical, not from applied and not computational, but from good old classical mechanics. This mechanics is also called Newtonian mechanics. According to legend, a scientist was walking in the garden, saw an apple fall, and it was this phenomenon that prompted him to discover the law gravity. Of course, the law has always existed, and Newton only gave it a form understandable to people, but his merit is priceless. In this article, we will not describe the laws of Newtonian mechanics in as much detail as possible, but we will outline the basics, basic knowledge, definitions and formulas that can always play into your hands.

Mechanics is a branch of physics, a science that studies the movement of material bodies and the interactions between them.

The word itself is of Greek origin and translates as "the art of building machines". But before building machines, we still have a long way to go, so let's follow in the footsteps of our ancestors, and we will study the movement of stones thrown at an angle to the horizon, and apples falling on heads from a height h.

Why does the study of physics begin with mechanics? Because it is completely natural, not to start it from thermodynamic equilibrium?!

Mechanics is one of the oldest sciences, and historically the study of physics began precisely with the foundations of mechanics. Placed within the framework of time and space, people, in fact, could not start from something else, no matter how much they wanted to. Moving bodies are the first thing we pay attention to.

What is movement?

Mechanical motion is a change in the position of bodies in space relative to each other over time.

It is after this definition that we quite naturally come to the concept of a frame of reference. Changing the position of bodies in space relative to each other. Key words here: relative to each other . After all, a passenger in a car moves relative to a person standing on the side of the road at a certain speed, and rests relative to his neighbor in a seat nearby, and moves at some other speed relative to a passenger in a car that overtakes them.

That is why, in order to normally measure the parameters of moving objects and not get confused, we need reference system - rigidly interconnected reference body, coordinate system and clock. For example, the earth moves around the sun in a heliocentric frame of reference. In everyday life, we carry out almost all our measurements in a geocentric reference system associated with the Earth. The earth is a reference body relative to which cars, planes, people, animals move.

Mechanics, as a science, has its own task. The task of mechanics is to know the position of the body in space at any time. In other words, mechanics builds a mathematical description of motion and finds connections between physical quantities characterizing it.

In order to move further, we need the notion of “ material point ". They say that physics is an exact science, but physicists know how many approximations and assumptions have to be made in order to agree on this very accuracy. No one has ever seen a material point or sniffed an ideal gas, but they do exist! They are just much easier to live with.

A material point is a body whose size and shape can be neglected in the context of this problem.

Sections of classical mechanics

Mechanics consists of several sections

Kinematics
Dynamics
Statics

Kinematics from a physical point of view, studies exactly how the body moves. In other words, this section deals with quantitative characteristics movement. Find speed, path - typical tasks of kinematics

Dynamics solves the question of why it moves the way it does. That is, it considers the forces acting on the body.

Statics studies the equilibrium of bodies under the action of forces, that is, it answers the question: why does it not fall at all?

Limits of applicability of classical mechanics

Classical mechanics no longer claims to be a science that explains everything (at the beginning of the last century, everything was completely different), and has a clear scope of applicability. In general, the laws of classical mechanics are valid for the world familiar to us in terms of size (macroworld). They cease to work in the case of the world of particles, when classical mechanics is replaced by quantum mechanics. Also, classical mechanics is inapplicable to cases where the movement of bodies occurs at a speed close to the speed of light. In such cases, relativistic effects become pronounced. Roughly speaking, within the framework of quantum and relativistic mechanics - classical mechanics, this special case when the dimensions of the body are large and the speed is small.

Generally speaking, quantum and relativistic effects never go away; they also take place during the usual motion of macroscopic bodies at a speed much lower than the speed of light. Another thing is that the action of these effects is so small that it does not go beyond the most accurate measurements. Classical mechanics will thus never lose its fundamental importance.

We will continue to study physical foundations mechanics in the following articles. For a better understanding of the mechanics, you can always refer to our authors, which individually shed light on the dark spot of the most difficult task.

Theoretical mechanics- This is a branch of mechanics, which sets out the basic laws of mechanical motion and mechanical interaction of material bodies.

Theoretical mechanics is a science in which the movements of bodies over time (mechanical movements) are studied. It serves as the basis for other sections of mechanics (the theory of elasticity, resistance of materials, the theory of plasticity, the theory of mechanisms and machines, hydroaerodynamics) and many technical disciplines.

mechanical movement- this is a change over time in the relative position in space of material bodies.

Mechanical interaction- this is such an interaction, as a result of which the mechanical movement changes or the relative position of body parts changes.

Rigid body statics

Statics- This is a branch of theoretical mechanics, which deals with the problems of equilibrium of solid bodies and the transformation of one system of forces into another, equivalent to it.

Basic concepts and laws of statics

Absolutely rigid body(solid body, body) is a material body, the distance between any points in which does not change.
Material point is a body whose dimensions, according to the conditions of the problem, can be neglected.
loose body is a body, on the movement of which no restrictions are imposed.
Non-free (bound) body is a body whose movement is restricted.
Connections- these are bodies that prevent the movement of the object under consideration (a body or a system of bodies).
Communication reaction is a force that characterizes the action of a bond on a rigid body. If we consider the force with which a rigid body acts on a bond as an action, then the reaction of the bond is a counteraction. In this case, the force - action is applied to the connection, and the reaction of the connection is applied to the solid body.
mechanical system is a set of interconnected bodies or material points.
Solid can be considered as a mechanical system, the positions and distance between the points of which do not change.
Power is a vector quantity characterizing the mechanical action of one material body on another.
Force as a vector is characterized by the point of application, the direction of action and the absolute value. The unit of measure for the modulus of force is Newton.
line of force is the straight line along which the force vector is directed.
Concentrated Power is the force applied at one point.
Distributed forces (distributed load)- these are forces acting on all points of the volume, surface or length of the body.
The distributed load is given by the force acting per unit volume (surface, length).
The dimension of the distributed load is N / m 3 (N / m 2, N / m).
External force is a force acting from a body that does not belong to the considered mechanical system.
inner strength is the force acting on a material point mechanical system from the side of another material point belonging to the system under consideration.
Force system is the totality of forces acting on a mechanical system.
Flat system of forces is a system of forces whose lines of action lie in the same plane.
Spatial system of forces is a system of forces whose lines of action do not lie in the same plane.
Converging force system is a system of forces whose lines of action intersect at one point.
Arbitrary system of forces is a system of forces whose lines of action do not intersect at one point.
Equivalent systems of forces- these are systems of forces, the replacement of which one for another does not change the mechanical state of the body.
Accepted designation: .
Equilibrium A state in which a body remains stationary or moves uniformly in a straight line under the action of forces.
Balanced system of forces- this is a system of forces that, when applied to a free solid body, does not change its mechanical state (does not unbalance it).
.
resultant force is a force whose action on a body is equivalent to the action of a system of forces.
.
Moment of power is a value that characterizes the rotational ability of the force.
Power couple is a system of two parallel equal in absolute value oppositely directed forces.
Accepted designation: .
Under the action of a couple of forces, the body will perform a rotational motion.
Projection of Force on the Axis- this is a segment enclosed between perpendiculars drawn from the beginning and end of the force vector to this axis.
The projection is positive if the direction of the segment coincides with the positive direction of the axis.
Projection of Force on a Plane is a vector on a plane enclosed between the perpendiculars drawn from the beginning and end of the force vector to this plane.
Law 1 (law of inertia). An isolated material point is at rest or moves uniformly and rectilinearly.
The uniform and rectilinear motion of a material point is a motion by inertia. The state of equilibrium of a material point and a rigid body is understood not only as a state of rest, but also as a movement by inertia. For a rigid body, there are various types of inertia motion, for example, uniform rotation of a rigid body around fixed axle.
Law 2. A rigid body is in equilibrium under the action of two forces only if these forces are equal in magnitude and directed in opposite directions along a common line of action.
These two forces are called balanced.
In general, forces are said to be balanced if the rigid body to which these forces are applied is at rest.
Law 3. Without violating the state (the word "state" here means the state of motion or rest) of a rigid body, one can add and discard balancing forces.
Consequence. Without disturbing the state of a rigid body, the force can be transferred along its line of action to any point of the body.
Two systems of forces are called equivalent if one of them can be replaced by another without disturbing the state of the rigid body.
Law 4. The resultant of two forces applied at one point is applied at the same point, is equal in absolute value to the diagonal of the parallelogram built on these forces, and is directed along this
diagonals.
The modulus of the resultant is:
Law 5 (law of equality of action and reaction). The forces with which two bodies act on each other are equal in magnitude and directed in opposite directions along one straight line.
It should be borne in mind that action- force applied to the body B, and opposition- force applied to the body A, are not balanced, since they are attached to different bodies.
Law 6 (the law of hardening). The equilibrium of a non-solid body is not disturbed when it solidifies.
It should not be forgotten that the equilibrium conditions, which are necessary and sufficient for a rigid body, are necessary but not sufficient for the corresponding non-rigid body.
Law 7 (the law of release from bonds). A non-free solid body can be considered as free if it is mentally freed from bonds, replacing the action of bonds with the corresponding reactions of bonds.

Connections and their reactions

Smooth surface restricts movement along the normal to the support surface. The reaction is directed perpendicular to the surface.
Articulated movable support limits the movement of the body along the normal to the reference plane. The reaction is directed along the normal to the support surface.
Articulated fixed support counteracts any movement in a plane perpendicular to the axis of rotation.
Articulated weightless rod counteracts the movement of the body along the line of the rod. The reaction will be directed along the line of the rod.
Blind termination counteracts any movement and rotation in the plane. Its action can be replaced by a force presented in the form of two components and a pair of forces with a moment.

Kinematics

Kinematics- a branch of theoretical mechanics that deals with general geometric properties mechanical movement as a process occurring in space and time. Moving objects are considered as geometric points or geometric bodies.

Basic concepts of kinematics

The law of motion of a point (body) is the dependence of the position of a point (body) in space on time.
Point trajectory is the locus of the positions of a point in space during its movement.
Point (body) speed- this is a characteristic of the change in time of the position of a point (body) in space.
Point (body) acceleration- this is a characteristic of the change in time of the speed of a point (body).

Determination of the kinematic characteristics of a point

Point trajectory
In the vector reference system, the trajectory is described by the expression: .
In the coordinate reference system, the trajectory is determined according to the law of point motion and is described by the expressions z = f(x,y) in space, or y = f(x)- in the plane.
V natural system reference trajectory is predetermined.
Determining the speed of a point in a vector coordinate system
When specifying the movement of a point in a vector coordinate system, the ratio of movement to the time interval is called the average value of the speed in this time interval: .
Taking the time interval as an infinitesimal value, we obtain the speed value at a given time (instantaneous speed value): .
Vector average speed is directed along the vector in the direction of the point movement, the instantaneous velocity vector is directed tangentially to the trajectory in the direction of the point movement.
Conclusion: the speed of a point is a vector quantity equal to the derivative of the law of motion with respect to time.
Derivative property: the time derivative of any value determines the rate of change of this value.
Determining the speed of a point in a coordinate reference system
Rate of change of point coordinates:
.
The module of the full speed of a point with a rectangular coordinate system will be equal to:
.
The direction of the velocity vector is determined by the cosines of the steering angles:
,
where are the angles between the velocity vector and the coordinate axes.
Determining the speed of a point in a natural reference system
The speed of a point in a natural reference system is defined as a derivative of the law of motion of a point: .
According to the previous conclusions, the velocity vector is directed tangentially to the trajectory in the direction of the point movement and in the axes is determined by only one projection .

Rigid Body Kinematics

In the kinematics of rigid bodies, two main problems are solved:
1) task of movement and determination of the kinematic characteristics of the body as a whole;
2) determination of the kinematic characteristics of the points of the body.
Translational motion of a rigid body
Translational motion is a motion in which a straight line drawn through two points of the body remains parallel to its original position.
Theorem: in translational motion, all points of the body move along the same trajectories and at each moment of time have the same speed and acceleration in absolute value and direction.
Conclusion: forward movement of a rigid body is determined by the movement of any of its points, and therefore, the task and study of its movement is reduced to the kinematics of a point.
Rotational motion of a rigid body around a fixed axis
The rotational motion of a rigid body around a fixed axis is the motion of a rigid body in which two points belonging to the body remain motionless during the entire time of movement.
The position of the body is determined by the angle of rotation. The unit of measurement for an angle is radians. (A radian is the central angle of a circle whose arc length is equal to the radius, the full angle of the circle contains 2π radian.)
The law of rotational motion of a body around a fixed axis.
The angular velocity and angular acceleration of the body will be determined by the differentiation method:
— angular velocity, rad/s;
— angular acceleration, rad/s².
If we cut the body by a plane perpendicular to the axis, choose a point on the axis of rotation WITH and arbitrary point M, then the point M will describe around the point WITH radius circle R. During dt there is an elementary rotation through the angle , while the point M will move along the trajectory for a distance .
Linear speed module:
.
point acceleration M with a known trajectory is determined by its components:
,
where .
As a result, we get formulas
tangential acceleration: ;
normal acceleration: .

Dynamics

Dynamics is a branch of theoretical mechanics that deals with mechanical movement material bodies, depending on the causes that cause them.

Basic concepts of dynamics

inertia is the property of material bodies to maintain a state of rest or uniform rectilinear motion until external forces change this state.
Weight is a quantitative measure of the inertia of a body. The unit of mass is kilogram (kg).
Material point is a body with a mass, the dimensions of which are neglected in solving this problem.
Center of mass of a mechanical system — geometric point, whose coordinates are determined by the formulas:

where m k , x k , y k , z k- mass and coordinates k- that point of the mechanical system, m is the mass of the system.
In a uniform field of gravity, the position of the center of mass coincides with the position of the center of gravity.
Moment of inertia of a material body about the axis is a quantitative measure of inertia during rotational motion.
The moment of inertia of a material point about the axis is equal to the product of the mass of the point and the square of the distance of the point from the axis:
.
The moment of inertia of the system (body) about the axis is equal to arithmetic sum moments of inertia of all points:
The force of inertia of a material point is a vector quantity equal in absolute value to the product of the mass of a point and the module of acceleration and directed opposite to the acceleration vector:
Force of inertia of a material body is a vector quantity equal in absolute value to the product of the body mass and the module of acceleration of the center of mass of the body and directed opposite to the acceleration vector of the center of mass: ,
where is the acceleration of the center of mass of the body.
Elemental Force Impulse is a vector quantity, equal to the product force vector for an infinitesimal time interval dt:
.
The total impulse of force for Δt is equal to the integral of elementary impulses:
.
Elementary work of force is a scalar dA, equal to the scalar

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Collection of scientific and methodical articles on theoretical mechanics. Issue 2. M.: Vyssh. school, 1971 (djvu)
Collection of scientific and methodical articles on theoretical mechanics. Issue 3. M.: Vyssh. school, 1972 (djvu)
Collection of scientific and methodical articles on theoretical mechanics. Issue 4. M.: Vyssh. school, 1974 (djvu)
Collection of scientific and methodical articles on theoretical mechanics. Issue 5. M.: Vyssh. school, 1975 (djvu)
Collection of scientific and methodical articles on theoretical mechanics. Issue 6. M.: Vyssh. school, 1976 (djvu)
Collection of scientific and methodical articles on theoretical mechanics. Issue 7. M.: Vyssh. school, 1976 (djvu)
Collection of scientific and methodical articles on theoretical mechanics. Issue 8. M.: Vyssh. school, 1977 (djvu)
Collection of scientific and methodical articles on theoretical mechanics. Issue 9. M.: Vyssh. school, 1979 (djvu)
Collection of scientific and methodical articles on theoretical mechanics. Issue 10. M.: Vyssh. school, 1980 (djvu)
Collection of scientific and methodical articles on theoretical mechanics. Issue 11. M.: Vyssh. school, 1981 (djvu)
Collection of scientific and methodical articles on theoretical mechanics. Issue 12. M.: Vyssh. school, 1982 (djvu)
Collection of scientific and methodical articles on theoretical mechanics. Issue 13. M.: Vyssh. school, 1983 (djvu)
Collection of scientific and methodical articles on theoretical mechanics. Issue 14. M.: Vyssh. school, 1983 (djvu)
Collection of scientific and methodical articles on theoretical mechanics. Issue 15. M.: Vyssh. school, 1984 (djvu)
Collection of scientific and methodical articles on theoretical mechanics. Issue 16. M.: Vyssh. school, 1986

As part of any curriculum, the study of physics begins with mechanics. Not from theoretical, not from applied and not computational, but from good old classical mechanics. This mechanics is also called Newtonian mechanics. According to legend, the scientist was walking in the garden, saw an apple fall, and it was this phenomenon that prompted him to discover the law of universal gravitation. Of course, the law has always existed, and Newton only gave it a form understandable to people, but his merit is priceless. In this article, we will not describe the laws of Newtonian mechanics in as much detail as possible, but we will outline the basics, basic knowledge, definitions and formulas that can always play into your hands.

Mechanics is a branch of physics, a science that studies the movement of material bodies and the interactions between them.

Why does the study of physics begin with mechanics? Because it is completely natural, not to start it from thermodynamic equilibrium?!

What is movement?

Mechanical motion is a change in the position of bodies in space relative to each other over time.

Mechanics, as a science, has its own task. The task of mechanics is to know the position of the body in space at any time. In other words, mechanics constructs a mathematical description of motion and finds connections between the physical quantities that characterize it.

A material point is a body whose size and shape can be neglected in the context of this problem.

Sections of classical mechanics

Mechanics consists of several sections

Kinematics
Dynamics
Statics

Kinematics from a physical point of view, studies exactly how the body moves. In other words, this section deals with the quantitative characteristics of movement. Find speed, path - typical tasks of kinematics

Dynamics solves the question of why it moves the way it does. That is, it considers the forces acting on the body.

Statics studies the equilibrium of bodies under the action of forces, that is, it answers the question: why does it not fall at all?

Limits of applicability of classical mechanics

Classical mechanics no longer claims to be a science that explains everything (at the beginning of the last century, everything was completely different), and has a clear scope of applicability. In general, the laws of classical mechanics are valid for the world familiar to us in terms of size (macroworld). They cease to work in the case of the world of particles, when classical mechanics is replaced by quantum mechanics. Also, classical mechanics is inapplicable to cases where the movement of bodies occurs at a speed close to the speed of light. In such cases, relativistic effects become pronounced. Roughly speaking, within the framework of quantum and relativistic mechanics - classical mechanics, this is a special case when the dimensions of the body are large and the speed is small.

We will continue to study the physical foundations of mechanics in future articles. For a better understanding of the mechanics, you can always refer to our authors, which individually shed light on the dark spot of the most difficult task.

Statics is a branch of theoretical mechanics that studies the equilibrium conditions for material bodies under the action of forces, as well as methods for converting forces into equivalent systems.

Under the state of equilibrium, in statics, is understood the state in which all parts of the mechanical system are at rest relative to some inertial coordinate system. One of the basic objects of statics are forces and points of their application.

The force acting on a material point with a radius vector from other points is a measure of the influence of other points on the considered point, as a result of which it receives acceleration relative to the inertial reference frame. Value strength is determined by the formula:
,
where m is the mass of the point - a value that depends on the properties of the point itself. This formula is called Newton's second law.

Application of statics in dynamics

An important feature of the equations of motion of an absolutely rigid body is that forces can be converted into equivalent systems. With such a transformation, the equations of motion retain their form, but the system of forces acting on the body can be transformed into a simpler system. Thus, the point of application of force can be moved along the line of its action; forces can be expanded according to the parallelogram rule; forces applied at one point can be replaced by their geometric sum.

An example of such transformations is gravity. It acts on all points of a rigid body. But the law of motion of the body will not change if the force of gravity distributed over all points is replaced by a single vector applied at the center of mass of the body.

It turns out that if we add an equivalent system to the main system of forces acting on the body, in which the directions of the forces are reversed, then the body, under the action of these systems, will be in equilibrium. Thus, the task of determining equivalent systems of forces is reduced to the problem of equilibrium, that is, to the problem of statics.

The main task of statics is the establishment of laws for the transformation of a system of forces into equivalent systems. Thus, the methods of statics are used not only in the study of bodies in equilibrium, but also in the dynamics of a rigid body, in the transformation of forces into simpler equivalent systems.

Material point statics

Consider a material point that is in equilibrium. And let n forces act on it, k = 1, 2, ..., n.

If the material point is in equilibrium, then the vector sum of the forces acting on it is equal to zero:
(1) .

In equilibrium, the geometric sum of the forces acting on a point is zero.

Geometric interpretation. If the beginning of the second vector is placed at the end of the first vector, and the beginning of the third is placed at the end of the second vector, and then this process is continued, then the end of the last, nth vector will be combined with the beginning of the first vector. That is, we get a closed geometric figure, the lengths of the sides of which are equal to the modules of the vectors. If all vectors lie in the same plane, then we get a closed polygon.

It is often convenient to choose rectangular coordinate system Oxyz. Then the sums of the projections of all force vectors on the coordinate axes are equal to zero:

If you choose any direction defined by some vector , then the sum of the projections of the force vectors on this direction is equal to zero:
.
We multiply equation (1) scalarly by the vector:
.
Here is the scalar product of the vectors and .
Note that the projection of a vector onto the direction of the vector is determined by the formula:
.

Rigid body statics

Moment of force about a point

Determining the moment of force

Moment of force, applied to the body at point A, relative to the fixed center O, is called a vector equal to the vector product of the vectors and:
(2) .

Geometric interpretation

The moment of force is equal to the product of the force F and the arm OH.

Let the vectors and be located in the plane of the figure. According to property vector product, the vector is perpendicular to the vectors and , that is, it is perpendicular to the plane of the figure. Its direction is determined by the right screw rule. In the figure, the moment vector is directed towards us. The absolute value of the moment:
.
Because , then
(3) .

Using geometry, one can give another interpretation of the moment of force. To do this, draw a straight line AH through the force vector . From the center O we drop the perpendicular OH to this line. The length of this perpendicular is called shoulder of strength. Then
(4) .
Since , formulas (3) and (4) are equivalent.

In this way, absolute value of the moment of force relative to the center O is product of force on the shoulder this force relative to the chosen center O .

When calculating moment, it is often convenient to decompose the force into two components:
,
where . The force passes through the point O. Therefore, its momentum is zero. Then
.
The absolute value of the moment:
.

Moment components in rectangular coordinates

If we choose a rectangular coordinate system Oxyz centered at the point O, then the moment of force will have the following components:
(5.1) ;
(5.2) ;
(5.3) .
Here are the coordinates of point A in the selected coordinate system:
.
The components are the values of the moment of force about the axes, respectively.

Properties of the moment of force about the center

The moment about the center O, from the force passing through this center, is equal to zero.

If the point of application of the force is moved along a line passing through the force vector, then the moment, during such a movement, will not change.

The moment from the vector sum of forces applied to one point of the body is equal to the vector sum of the moments from each of the forces applied to the same point:
.

The same applies to forces whose extension lines intersect at one point.

If the vector sum of the forces is zero:
,
then the sum of the moments from these forces does not depend on the position of the center, relative to which the moments are calculated:
.

Power couple

Power couple are two forces, equal in absolute value and having opposite directions, applied to different points body.

A pair of forces is characterized by the moment they create. Since the vector sum of the forces included in the pair is zero, the moment created by the couple does not depend on the point relative to which the moment is calculated. From the point of view of static equilibrium, the nature of the forces in the pair is irrelevant. A pair of forces is used to indicate that a moment of forces acts on the body, having a certain value.

Moment of force about a given axis

Often there are cases when we do not need to know all the components of the moment of force about a selected point, but only need to know the moment of force about a selected axis.

The moment of force about the axis passing through the point O is the projection of the vector of the moment of force, about the point O, on the direction of the axis.

Properties of the moment of force about the axis

The moment about the axis from the force passing through this axis is equal to zero.

The moment about an axis from a force parallel to this axis is zero.

Calculation of the moment of force about an axis

Let a force act on the body at point A. Let us find the moment of this force relative to the O′O′′ axis.

Let's build a rectangular coordinate system. Let the Oz axis coincide with O′O′′ . From the point A we drop the perpendicular OH to O′O′′ . Through the points O and A we draw the axis Ox. We draw the axis Oy perpendicular to Ox and Oz. We decompose the force into components along the axes of the coordinate system:
.
The force crosses the O′O′′ axis. Therefore, its momentum is zero. The force is parallel to the O′O′′ axis. Therefore, its moment is also zero. By formula (5.3) we find:
.

Note that the component is directed tangentially to the circle whose center is the point O . The direction of the vector is determined by the right screw rule.

Equilibrium conditions for a rigid body

In equilibrium, the vector sum of all forces acting on the body is equal to zero and the vector sum of the moments of these forces relative to an arbitrary fixed center is equal to zero:
(6.1) ;
(6.2) .

We emphasize that the center O , relative to which the moments of forces are calculated, can be chosen arbitrarily. Point O can either belong to the body or be outside it. Usually the center O is chosen to make the calculations easier.

The equilibrium conditions can be formulated in another way.

In equilibrium, the sum of the projections of forces on any direction given by an arbitrary vector is equal to zero:
.
The sum of moments of forces about an arbitrary axis O′O′′ is also equal to zero:
.

Sometimes these conditions are more convenient. There are times when, by choosing axes, calculations can be made simpler.

Center of gravity of the body

Consider one of the most important forces- gravity. Here, the forces are not applied at certain points of the body, but are continuously distributed over its volume. For each part of the body with an infinitesimal volume ∆V, the gravitational force acts. Here ρ is the density of the substance of the body, is the acceleration of free fall.

Let be the mass of an infinitely small part of the body. And let the point A k defines the position of this section. Let us find the quantities related to the force of gravity, which are included in the equilibrium equations (6).

Let's find the sum of gravity forces formed by all parts of the body:
,
where is the mass of the body. Thus, the sum of the gravity forces of individual infinitesimal parts of the body can be replaced by one gravity vector of the entire body:
.

Let's find the sum of the moments of the forces of gravity, relative to the chosen center O in an arbitrary way:

.
Here we have introduced point C which is called center of gravity body. The position of the center of gravity, in a coordinate system centered at the point O, is determined by the formula:
(7) .

So, when determining static equilibrium, the sum of the gravity forces of individual sections of the body can be replaced by the resultant
,
applied to the center of mass of the body C , whose position is determined by formula (7).

The position of the center of gravity for various geometric shapes can be found in the relevant guides. If the body has an axis or plane of symmetry, then the center of gravity is located on this axis or plane. So, the centers of gravity of a sphere, circle or circle are located in the centers of the circles of these figures. Centers of gravity cuboid, rectangle or square are also located in their centers - at the intersection points of the diagonals.

Uniformly (A) and linearly (B) distributed load.

There are also cases similar to the force of gravity, when the forces are not applied at certain points of the body, but are continuously distributed over its surface or volume. Such forces are called distributed forces or .

(Figure A). Also, as in the case of gravity, it can be replaced by the resultant force of magnitude , applied at the center of gravity of the diagram. Since the diagram in figure A is a rectangle, the center of gravity of the diagram is in its center - point C: | AC | = | CB |.

(picture B). It can also be replaced by the resultant. The value of the resultant is equal to the area of the diagram:
.
The point of application is in the center of gravity of the plot. The center of gravity of a triangle, height h, is at a distance from the base. So .

Friction forces

Sliding friction. Let the body be on a flat surface. And let be a force perpendicular to the surface with which the surface acts on the body (pressure force). Then the sliding friction force is parallel to the surface and directed to the side, preventing the body from moving. Its largest value is:
,
where f is the coefficient of friction. The coefficient of friction is a dimensionless quantity.

rolling friction. Let the rounded body roll or may roll on the surface. And let be the pressure force perpendicular to the surface with which the surface acts on the body. Then on the body, at the point of contact with the surface, the moment of friction forces acts, which prevents the movement of the body. The largest value of the friction moment is:
,
where δ is the coefficient of rolling friction. It has the dimension of length.

References:
S. M. Targ, Short course Theoretical Mechanics, Higher School, 2010.