Basics of Mechanics for Dummies. Introduction. Basic laws and formulas for theoretical mechanics. Solution of examples Termekh lectures 1 course

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• Statics
  • Basic concepts of statics
  • Types of forces
  • Axioms of statics
  • Connections and their reactions
  • System of converging forces
    • Methods for determining the resultant system of converging forces
    • Equilibrium conditions for a system of converging forces
  • Moment of force relative to the center as a vector
    • Algebraic value of the moment of force
    • Properties of the moment of force about the center (point)
  • The theory of pairs of forces
    • Addition of two parallel forces directed in one direction
    • Addition of two parallel forces directed in opposite directions
    • Couples of forces
    • The pair of forces theorems
    • Equilibrium conditions for a system of pairs of forces
  • Lever arm
  • Arbitrary flat system of forces
    • Cases of Reducing a Plane System of Forces to a Simpler Form
    • Analytical Equilibrium Conditions
  • Center of Parallel Forces. The center of gravity
    • Center of Parallel Forces
    • The center of gravity of a rigid body and its coordinates
    • Center of gravity of volume, plane and line
    • Methods for determining the position of the center of gravity
• Basics of strength calculations
  • Tasks and methods of strength of materials
  • Classification of loads
  • Classification of structural elements
  • Bar deformations
  • Basic hypotheses and principles
  • Internal forces. Section method
  • Voltage
  • Stretching and squeezing
  • Mechanical characteristics of the material
  • Allowable voltage
  • Hardness of materials
  • Plots of longitudinal forces and stresses
  • Shift
  • Geometric characteristics of sections
  • Torsion
  • Bending
    • Differential Bending Constraints
    • Flexural strength
    • Normal voltages. Strength calculation
    • Shear bending stresses
    • Bending stiffness
  • Elements of the general theory of the stress state
  • Strength theories
  • Torsion bend
• Kinematics
  • Point kinematics
    • Point trajectory
    • Ways to set point movement
    • Point speed
    • Point acceleration
  • Rigid body kinematics
    • The translational motion of a rigid body
    • Rotational motion of a rigid body
    • Gear kinematics
    • Plane-parallel movement of a rigid body
  • Complex point movement
• Dynamics
  • Basic laws of dynamics
  • Point dynamics
    • Differential equations of a free material point
    • Two problems of point dynamics
  • Rigid body dynamics
    • Classification of forces acting on a mechanical system
    • Differential equations of motion of a mechanical system
  • General theorems of dynamics
    • The theorem on the motion of the center of mass of a mechanical system
    • Momentum Change Theorem
    • The theorem on the change in the angular momentum
    • The theorem on the change in kinetic energy
• Forces acting in machines
  • Forces in engagement of a spur gear
  • Friction in mechanisms and machines
    • Sliding friction
    • Rolling friction
  • Efficiency
• Machine parts
  • Mechanical transmission
    • Types of mechanical transmissions
    • Basic and derived parameters of mechanical transmissions
    • Gear transmission
    • Flexible link transmissions
  • Shafts
    • Purpose and classification
    • Design calculation
    • Check calculation of shafts
  • Bearings
    • Plain bearings
    • Rolling bearings
  • Connecting machine parts
    • Types of detachable and one-piece connections
    • Keyed connections
• Standardization of norms, interchangeability
  • Tolerances and landings
  • Unified system of tolerances and landings (ESDP)
  • Shape tolerance and position

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Russian language

An example of calculating a spur gear
An example of calculating a spur gear. The choice of material, calculation of permissible stresses, calculation of contact and bending strength were performed.

An example of solving the problem of bending a beam
In the example, diagrams of shear forces and bending moments are constructed, a dangerous section is found and an I-beam is selected. The problem analyzes the construction of diagrams using differential dependencies, a comparative analysis of various cross-sections of the beam is carried out.

An example of solving the problem of shaft torsion
The task is to check the strength of a steel shaft for a given diameter, material and allowable stresses. During the solution, diagrams of torques, shear stresses and torsion angles are plotted. The dead weight of the shaft is not taken into account.

An example of solving the problem of tension-compression of a bar
The task is to check the strength of a steel bar at a given allowable stress. During the solution, diagrams of longitudinal forces, normal stresses and displacements are plotted. The self-weight of the bar is not taken into account

Application of the kinetic energy conservation theorem
An example of solving the problem on the application of the theorem on the conservation of kinetic energy of a mechanical system

Determination of the speed and acceleration of a point according to the given equations of motion
An example of solving a problem to determine the speed and acceleration of a point according to the given equations of motion

Determination of the speeds and accelerations of points of a rigid body during plane-parallel motion
An example of solving the problem of determining the velocities and accelerations of points of a rigid body with plane-parallel motion

Determination of forces in the bars of a flat truss
An example of solving the problem of determining the forces in the bars of a flat truss by the Ritter method and the method of cutting nodes

Theoretical mechanics - this is a section 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 movement of bodies over time (mechanical movements) is studied. It serves as the basis for other branches of mechanics (theory of elasticity, resistance of materials, theory of plasticity, 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 mechanical movement changes or the relative position of body parts changes.

Rigid body statics

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

Basic concepts and laws of statics
• Absolutely solid (solid, 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.
• Free body Is a body, the movement of which is not subject to any restrictions.
• Unfree (bound) body Is a body whose movement is restricted.
• Connections - these are bodies that prevent the movement of the object under consideration (body or system of bodies).
• Communication reaction Is a force that characterizes the effect 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 bond reaction is a reaction. In this case, the force - the action is applied to the bond, and the bond reaction is applied to the solid.
• 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.
• Force Is a vector quantity that characterizes the mechanical action of one material body on another.
  Force as a vector is characterized by the point of application, direction of action and absolute value. The unit of measure for the modulus of force is Newton.
• Force action line Is a straight line along which the force vector is directed.
• Concentrated power - force applied at one point.
• Distributed forces (distributed load) - these are the forces acting on all points of the volume, surface or length of the body.
  The distributed load is set by the force acting on a 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 a force acting on a material point of a mechanical system from another material point belonging to the system under consideration.
• Force system Is a set of forces acting on a mechanical system.
• Flat system of forces Is a system of forces whose lines of action lie in one plane.
• Spatial system of forces Is a system of forces whose lines of action do not lie in the same plane.
• System of converging forces Is a system of forces whose lines of action intersect at one point.
• Arbitrary force system Is a system of forces, the lines of action of which do not intersect at one point.
• Equivalent systems of forces - these are systems of forces, replacing them with one another does not change the mechanical state of the body.
  Accepted designation:.
• Equilibrium - this is a state in which the body under the action of forces remains stationary or moves uniformly in a straight line.
• Balanced system of forces Is a system of forces that, when applied to a free solid, does not change its mechanical state (does not unbalance).
  .
• Resultant force Is a force, the action of which on the body is equivalent to the action of the system of forces.
  .
• Moment of power Is a value that characterizes the rotational ability of a force.
• A couple of forces Is a system of two parallel, equal in magnitude, oppositely directed forces.
  Accepted designation:.
  Under the action of a pair of forces, the body will rotate.
• Axis force projection 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 line segment coincides with the positive direction of the axis.
• Force projection onto plane Is a vector on a plane, enclosed between 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 evenly and rectilinearly.
  The uniform and rectilinear motion of a material point is motion by inertia. The state of equilibrium between a material point and a rigid body is understood not only as a state of rest, but also as motion by inertia. For a rigid body, there are various types of inertial motion, for example, uniform rotation of a rigid body around a fixed axis.
• Law 2. A solid body is in equilibrium under the action of two forces only if these forces are equal in magnitude and directed in opposite directions along the common line of action.
  These two forces are called balancing forces.
  In general, forces are called balancing if the rigid body to which these forces are applied is at rest.
• Law 3. Without disturbing the state (the word "state" here means the state of motion or rest) of a rigid body, one can add and drop counterbalancing forces.
  Consequence. Without violating the state of a rigid body, force can be transferred along its line of action to any point in the body.
  Two systems of forces are called equivalent if one of them can be replaced by another without violating the state of a rigid body.
• Law 4. The resultant of two forces applied at one point, applied at the same point, is equal in magnitude to the diagonal of the parallelogram built on these forces, and is directed along this
  diagonals.
  The modulus of the resultant is equal to:
  
• Law 5 (the 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 act - force applied to the body Band opposition - force applied to the body Aare not balanced, since they are attached to different bodies.
• Law 6 (law of solidification)... 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 solid, are necessary, but not sufficient for the corresponding non-solid.
• Law 7 (the law of release from ties). An unfree solid 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 constrains movement normal to the support surface. The reaction is directed perpendicular to the surface.
• Articulated movable support constrains 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 bar. The reaction will be directed along the line of the bar.
• Blind termination counteracts any movement and rotation in the plane. Its action can be replaced by a force represented in the form of two components and a pair of forces with a moment.

Kinematics

Kinematics - a section of theoretical mechanics, which considers the general geometric properties of mechanical motion, as a process that occurs 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 geometric location 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 kinematic characteristics of a point
• Point trajectory
  In the vector frame of reference, the trajectory is described by the expression:.
  In the coordinate system, the trajectory is determined by the law of motion of a point and is described by the expressions z \u003d f (x, y) - in space, or y \u003d f (x) - in the plane.
  In the natural frame of reference, the trajectory is set in advance.
• 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 infinitely small value, the speed value is obtained at a given time (instantaneous speed value): .
  The average velocity vector 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 quantity determines the rate of change of this quantity.
• Determining the speed of a point in a coordinate system
  Point coordinates change rates:
  .
  The modulus of the full speed of a point with a rectangular coordinate system will be:
  .
  The direction of the velocity vector is determined by the cosines of the direction angles:
  ,
  where are the angles between the velocity vector and the coordinate axes.
• Determining the speed of a point in a natural frame of reference
  The speed of a point in the natural frame of reference 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 motion of the point and in the axes is determined by only one projection.

Rigid body kinematics
• In the kinematics of solids, two main tasks are solved:
  1) the task of movement and the determination of the kinematic characteristics of the body as a whole;
  2) determination of the kinematic characteristics of body points.
• The translational motion of a rigid body
  A translational motion is a motion in which a straight line drawn through two points of the body remains parallel to its original position.
  Theorem: during translational motion, all points of the body move along the same trajectories and at each moment of time have the same velocity and acceleration in magnitude and direction.
  Conclusion: the translational motion of a rigid body is determined by the motion of any of its points, in connection with which, the task and study of its motion is reduced to the kinematics of the point.
• Rotational movement of a rigid body around a fixed axis
  The rotational movement of a rigid body around a fixed axis is the movement 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 angle unit is radians. (Radian is the central angle of a circle whose arc length is equal to the radius, the total angle of the circle contains 2π radians.)
  The law of rotational motion of a body around a fixed axis.
  The angular velocity and angular acceleration of the body is determined by the differentiation method:
  - angular velocity, rad / s;
  - angular acceleration, rad / s².
  If you cut the body with a plane perpendicular to the axis, select the point on the axis of rotation FROM and an arbitrary point Mthen point M will describe around the point FROM circle radius R... During dt an elementary turn through an angle occurs, while the point M will move along the trajectory a distance .
  Linear speed module:
  .
  Point acceleration M with a known trajectory, it is determined by its components:
  ,
  Where .
  As a result, we get the formulas
  tangential acceleration: ;
  normal acceleration: .

Dynamics

Dynamics - this is a section of theoretical mechanics, which studies the mechanical movements of material bodies, depending on the reasons that cause them.

Basic concepts of dynamics
• Inertia - this 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 body inertia. The unit of measure for mass is kilogram (kg).
• Material point Is a body with a mass, the dimensions of which are neglected when solving this problem.
• Center of gravity of the mechanical system - geometric point, the coordinates of which are determined by the formulas:
  
  Where m k, x k, y k, z k - mass and coordinates k-th point of the mechanical system, m Is the mass of the system.
  In a homogeneous gravity field, the position of the center of mass coincides with the position of the center of gravity.
• The 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 point's mass by the square of the point's distance from the axis:
  .
  The moment of inertia of the system (body) about the axis is equal to the arithmetic sum of the moments of inertia of all points:
  
• The force of inertia of a material point Is a vector quantity equal in magnitude to the product of the point mass by the acceleration modulus and directed opposite to the acceleration vector:
• The force of inertia of a material body Is a vector quantity equal in magnitude to the product of body mass by the modulus of acceleration of the center of mass of the body and directed opposite to the vector of acceleration of the center of mass:,
  where is the acceleration of the center of mass of the body.
• Elementary Force Impulse Is a vector quantity equal to the product of the force vector by an infinitely small time interval dt:
  .
  The total impulse of force for Δt is equal to the integral of elementary impulses:
  .
• Elementary power work Is a scalar dAequal to scalar proi

state autonomous institution

Kaliningrad region

professional educational organization

College of Service and Tourism

A course of lectures with examples of practical assignments

"Foundations of Theoretical Mechanics"

by disciplineTechnical mechanics

for students3 course

specialty02/20/04 Fire safety

Kaliningrad

APPROVED

Deputy Director for UR GAU KO VET KSTN. Myasnikova

APPROVED

Methodological Council of GAU KO POO KST

CONSIDERED

At a meeting of the PCC

Editorial team:

Kolganova A.A., methodologist

Falaleeva A.B., teacher of Russian language and literature

Tsvetaeva L.V., Chairman of the PCCgeneral mathematical and natural science disciplines

Compiled by:

I.V. Nezvanova lecturer at GAU KO VET KST

Content

1. Theoretical information

1. Theoretical information

1. Examples of solving practical problems

Dynamics: basic concepts and axioms

1. Theoretical information

1. Examples of solving practical problems

Bibliography

Statics: basic concepts and axioms.

1. Theoretical information

Statics - a section of theoretical mechanics, which considers the properties of forces applied to the points of a rigid body and the conditions for their equilibrium. Main goals:

1. Transformations of systems of forces into equivalent systems of forces.

2. Determination of equilibrium conditions for systems of forces acting on a rigid body.

Material point called the simplest model of a material body

any shape, the dimensions of which are small enough and which can be taken as a geometric point having a certain mass. Any collection of material points is called a mechanical system. An absolutely rigid body is a mechanical system, the distances between the points of which do not change with any interactions.

Force Is a measure of the mechanical interaction of material bodies with each other. Force is a vector quantity, since it is determined by three elements:

numerical value;

direction;

point of application (A).

Force unit - Newton (N).

Figure 1.1

A system of forces is a combination of forces acting on a body.

A balanced (equal to zero) system of forces is called a system that, being applied to a body, does not change its state.

The system of forces acting on the body can be replaced by one resultant, acting as a system of forces.

Axioms of statics.

Axiom 1: If a balanced system of forces is applied to the body, then it moves uniformly and rectilinearly or is at rest (the law of inertia).

Axiom 2: An absolutely rigid body is in equilibrium under the action of two forces if and only if these forces are equal in magnitude, act in one straight line and are directed in opposite directions. Figure 1.2

Axiom 3: The mechanical state of a body will not be disturbed if a balanced system of forces is added to or subtracted from the system of forces acting on it.

Axiom 4: The resultant of the two forces applied to the body is equal to their geometric sum, that is, it is expressed in magnitude and direction by the diagonal of the parallelogram built on these forces as on the sides.

Figure 1.3.

Axiom 5: The forces with which two bodies act on each other are always equal in magnitude and directed along one straight line in opposite directions.

Figure 1.4.

Types of bonds and their reactions

Links any restrictions that prevent the movement of a body in space are called. The body, striving under the action of the applied forces to carry out the movement, which is impeded by the connection, will act on it with some force, called force of pressure on communication ... According to the law of equality of action and reaction, the connection will act on the body with the same modulus, but oppositely directed force.
The force with which this connection acts on the body, preventing one or another movement, is called force of reaction (reaction) of connection .
One of the main provisions of mechanics is bond release principle : any non-free body can be considered as free if one discards connections and replaces their action with reactions of connections.

The bond reaction is directed in the direction opposite to the one where the bond does not allow the body to move. The main types of bonds and their reactions are shown in Table 1.1.

Table 1.1

Types of bonds and their reactions

Communication name

Symbol

1

Smooth surface (support) - surface (support), friction on which the given body can be neglected.
With free support, the reaction is directed perpendicular to the tangent drawn through the pointA body contact1 with support surface2 .

2

Thread (flexible, non-extensible). The connection, implemented in the form of an inextensible thread, does not allow the body to move away from the suspension point. Therefore, the reaction of the thread is directed along the thread to the point of its suspension.

3

Weightless rod - a rod whose weight can be neglected in comparison with the perceived load.
The reaction of a weightless, hingedly attached rectilinear rod is directed along the axis of the rod.

4

Movable hinge, hinge-movable support. The reaction is directed along the normal to the support surface.

7

Rigid termination. In the plane of the rigid termination there will be two components of the reaction, and the moment of a pair of forceswhich prevents the beam from turning1 relative to pointA .
Rigid fixation in space takes away from body 1 all six degrees of freedom - three displacements along the coordinate axes and three rotations about these axes.
In spatial rigid termination there will be three components, , and three moments of pairs of forces.

System of converging forces

A system of converging forces is called a system of forces whose lines of action intersect at one point. Two forces converging at one point, according to the third axiom of statics, can be replaced by one force -resultant .
The main vector of the system of forces - a value equal to the geometric sum of the forces of the system.

The resultant plane system of converging forces can be determinedgraphically and analytically.

Addition of the system of forces . The addition of a flat system of converging forces is carried out either by successive addition of forces with the construction of an intermediate resultant (Fig. 1.5), or by constructing a force polygon (Fig. 1.6).

Figure 1.5 Figure 1.6

Axis force projection - an algebraic quantity equal to the product of the modulus of the force by the cosine of the angle between the force and the positive direction of the axis.
ProjectionF x(Figure 1.7) axle forces xpositive if angle α is acute, negative if angle α is obtuse. If strengthis perpendicular to the axis, then its projection onto the axis is zero.

Figure 1.7

Force projection onto plane Ooh- vector enclosed between the projections of the beginning and end of the forceonto this plane. Those. the projection of the force onto the plane is a vector quantity, characterized not only by a numerical value, but also by the direction in the planeOoh (Figure 1.8).

Figure 1.8

Then the projection module on the plane Ooh will be equal to:

F xy \u003d Fcosα,

where α is the angle between the direction of the forceand its projection.
An analytical way of setting forces . For an analytical way of setting strengthit is necessary to select a coordinate systemOhyz, in relation to which the direction of the force in space will be determined.
Vector depicting strength, can be plotted if the modulus of this force and the angles α, β, γ, which the force forms with the coordinate axes, are known. DotAforce application set separately by its coordinatesx, at, z... You can set the strength of its projectionsFx, Fy, Fzon the coordinate axes. The modulus of force in this case is determined by the formula:

and the direction cosines are:

, .

Analytical way of adding forces : the projection of the vector of the sum onto some axis is equal to the algebraic sum of the projections of the terms of the vectors onto the same axis, i.e., if:

then,,.
Knowing Rx, Ry, Rz, we can define the module

and direction cosines:

, , .

Figure 1.9

For the equilibrium of the system of converging forces, it is necessary and sufficient that the resultant of these forces be equal to zero.
1) Geometric equilibrium condition for a converging system of forces : for the equilibrium of the system of converging forces, it is necessary and sufficient that the power polygon built from these forces,

was closed (the end of the vector of the last term

force must be combined with the beginning of the vector of the first term of the force). Then the main vector of the system of forces will be equal to zero ()
2) Analytical Equilibrium Conditions . The modulus of the main vector of the system of forces is determined by the formula. \u003d 0. Because the , then the radical expression can be equal to zero only if each term simultaneously vanishes, i.e.

Rx= 0, Ry= 0, Rz \u003d 0.

Therefore, for the equilibrium of the spatial system of converging forces, it is necessary and sufficient that the sums of the projections of these forces on each of the three coordinates of the axes be equal to zero:

For the equilibrium of a flat system of converging forces, it is necessary and sufficient that the sums of the projections of the forces on each of the two coordinate axes be equal to zero:

The addition of two parallel forces directed in one direction.

Figure 1.9

Two parallel forces directed in one direction are reduced to one resultant force, parallel to them and directed in the same direction. The magnitude of the resultant is equal to the sum of the magnitudes of these forces, and the point of its application C divides the distance between the lines of action of the forces in an internal way into parts inversely proportional to the magnitudes of these forces, that is

B A C

R \u003d F 1 + F 2

The addition of two unequal parallel forces directed in opposite directions.

Two not equal in magnitude antiparallel forces are reduced to one resultant force parallel to them and directed towards the greater force. The magnitude of the resultant is equal to the difference in the magnitudes of these forces, and the point of its application, C, divides the distance between the lines of action of the forces externally into parts inversely proportional to the magnitudes of these forces, that is

A pair of forces and a moment of force relative to a point.

A moment of power relative to point O is called, taken with the appropriate sign, the product of the magnitude of the force by the distance h from point O to the line of action of the force ... This product is taken with a plus sign if the strength tends to rotate the body counterclockwise, and with the sign - if the force tends to rotate the body clockwise, that is ... The length of the perpendicular h is calledshoulder of strength point O. Effect of force action i.e. the angular acceleration of the body is greater, the greater the value of the moment of force.

Figure 1.11

With a couple of forces is called a system consisting of two equal in magnitude parallel forces directed in opposite directions. The distance h between the lines of action of forces is calledshoulder pair . A moment of steam m (F, F ") is the product of the magnitude of one of the forces that make up the pair on the shoulder of the pair, taken with the appropriate sign.

It is written like this: m (F, F ") \u003d ± F × h, where the product is taken with a plus sign, if a pair of forces tends to rotate the body counterclockwise and with a minus sign, if a pair of forces tends to rotate the body clockwise.

The theorem on the sum of the moments of the forces of a pair.

The sum of the moments of forces of the pair (F, F ") relative to any point 0 taken in the plane of the action of the pair does not depend on the choice of this point and is equal to the moment of the pair.

Equivalent Pairs Theorem. Consequences.

Theorem. Two pairs, the moments of which are equal to each other, are equivalent, i.e. (F, F ") ~ (P, P")

Corollary 1 ... A pair of forces can be transferred to any place in the plane of its action, as well as rotated at any angle and change the shoulder and the magnitude of the forces of the pair, while maintaining the moment of the pair.

Corollary 2. The pair of forces has no resultant and cannot be balanced by one force lying in the plane of the pair.

Figure 1.12

Addition and equilibrium condition for a system of pairs on a plane.

1. The theorem on the addition of pairs lying in the same plane. A system of pairs, arbitrarily located in the same plane, can be replaced by one pair, the moment of which is equal to the sum of the moments of these pairs.

2. A theorem on the equilibrium of a system of pairs on a plane.

In order for an absolutely rigid body to be at rest under the action of a system of pairs, arbitrarily located in one plane, it is necessary and sufficient that the sum of the moments of all pairs be equal to zero, that is

The center of gravity

The force of gravity - the resultant of the forces of attraction to the Earth, distributed throughout the body.

Body center of gravity - this is such a point invariably connected with this body through which the line of action of the force of gravity of this body passes at any position of the body in space.

Methods for finding the center of gravity

1. Symmetry method:

1.1. If a homogeneous body has a plane of symmetry, then the center of gravity lies in this plane

1.2. If a homogeneous body has an axis of symmetry, then the center of gravity lies on this axis. The center of gravity of a uniform body of revolution lies on the axis of rotation.

1.3 If a homogeneous body has two axes of symmetry, then the center of gravity is at the point of their intersection.

2. Method of splitting: The body is split into the smallest number of parts, the forces of gravity and the position of the centers of gravity of which are known.

3. Method of negative masses: When determining the center of gravity of a body with free cavities, the method of partitioning should be used, but the mass of free cavities should be considered negative.

The coordinates of the center of gravity of a plane figure:

The positions of the centers of gravity of simple geometric figures can be calculated using known formulas. (Figure 1.13)

Note: The center of gravity of the figure's symmetry is on the axis of symmetry.

The center of gravity of the rod is at mid-height.

1.2. Examples of solving practical problems

Example 1: The load is suspended from a rod and is in equilibrium. Determine the efforts in the rod. (figure 1.2.1)

Decision:

The forces occurring in the fastening rods are equal in magnitude to the forces with which the rods support the load. (5th axiom)

We determine the possible directions of the reactions of the bonds "rigid rods".

Forces are directed along the rods.

Figure 1.2.1.

Let's free point A from connections, replacing the action of connections with their reactions. (Figure 1.2.2)

We start the construction with a known force by drawing the vectorF on some scale.

From the end of the vectorF draw lines parallel to reactionsR 1 andR 2 .

Figure 1.2.2

Crossing lines create a triangle. (Figure 1.2.3.). Knowing the scale of the constructions and measuring the length of the sides of the triangle, you can determine the magnitude of the reactions in the rods.

For more accurate calculations, you can use geometric relationships, in particular the theorem of sines: the ratio of the side of a triangle to the sine of the opposite angle is a constant

For this case:

Figure 1.2.3

Comment: If the direction of the vector (bond reaction) on the given scheme and in the triangle of forces does not coincide, then the reaction on the scheme should be directed in the opposite direction.

Example 2: Determine the magnitude and direction of the resultant flat system of converging forces analytically.

Decision:

Figure 1.2.4

1. Determine the projection of all forces of the system on Ox (Figure 1.2.4)

Adding the projections algebraically, we obtain the projection of the resultant onto the Ox axis.

The sign indicates that the resultant is directed to the left.

2. Determine the projection of all forces on the Oy axis:

Adding the projections algebraically, we obtain the projection of the resultant onto the Oy axis.

The sign indicates that the resultant is directed downward.

3. Determine the modulus of the resultant by the values \u200b\u200bof the projections:

4. Determine the value of the angle of the resultant with the Ox axis:

and the value of the angle with the Oy axis:

Example 3: Calculate the sum of the moments of forces relative to point O (Figure 1.2.6).

OA= AB= IND \u003d DE \u003d CB \u003d 2m

Figure 1.2.6

Decision:

1. The moment of force relative to a point is numerically equal to the product of the modulus and the shoulder of the force.

2. The moment of force is equal to zero if the line of action of the force passes through the point.

Example 4: Determine the position of the center of gravity of the figure shown in Figure 1.2.7

Decision:

We split the figure into three:

1-rectangle

A 1 \u003d 10 * 20 \u003d 200cm 2

2-triangle

A 2 \u003d 1/2 * 10 * 15 \u003d 75cm 2

3-circle

A 3 =3,14*3 2 \u003d 28.3cm 2

CG of figure 1: x 1 \u003d 10cm, y 1 \u003d 5cm

CG of figure 2: x 2 \u003d 20 + 1/3 * 15 \u003d 25cm, y 2 \u003d 1/3 * 10 \u003d 3.3cm

CG of figure 3: x 3 \u003d 10cm, y 3 \u003d 5cm

The y from \u003d 4.5cm

Kinematics: basic concepts.

Basic kinematic parameters

Trajectory - a line outlined by a material point when moving in space. The trajectory can be straight and curved, flat and spatial.

Trajectory equation for plane motion: y \u003df ( x)

Distance traveled. The path is measured along the path in the direction of travel. Designation -S, units of measurement - meters.

Point motion equation Is an equation that determines the position of a moving point as a function of time.

Figure 2.1

The position of the point at each moment of time can be determined by the distance traveled along the trajectory from some fixed point, considered as the origin (Figure 2.1). This way of setting motion is callednatural ... Thus, the equation of motion can be represented as S \u003d f (t).

Figure 2.2

The position of a point can also be determined if its coordinates are known as a function of time (Figure 2.2). Then, in the case of motion on a plane, two equations must be given:

In the case of spatial movement, a third coordinate is also addedz= f 3 ( t)

This way of setting movement is calledcoordinate .

Travel speed Is a vector quantity that characterizes at the moment the speed and direction of movement along the trajectory.

Velocity is a vector at any moment directed tangentially to the trajectory in the direction of the direction of movement (Figure 2.3).

Figure 2.3

If a point travels equal distances in equal intervals of time, then the movement is calleduniform .

Average speed on the path ΔS determined:

whereΔS- distance traveled in time Δt; Δ t- time interval.

If a point travels unequal paths in equal periods of time, then the movement is calleduneven ... In this case, speed is a variable quantity and depends on timev= f( t)

The speed is currently defined as

Point acceleration is a vector quantity that characterizes the rate of change in speed in magnitude and direction.

The speed of a point when moving from point M1 to point Mg changes in magnitude and direction. Average acceleration value for this period of time

Acceleration at the moment:

Usually, for convenience, two mutually perpendicular acceleration components are considered: normal and tangential (Figure 2.4)

Normal acceleration a n , characterizes the change in speed along

direction and is defined as

Normal acceleration is always perpendicular to the velocity toward the center of the arc.

Figure 2.4

Tangential acceleration a t , characterizes the change in speed in magnitude and is always directed tangentially to the trajectory; when accelerating, its direction coincides with the direction of the velocity, and when decelerating, it is directed opposite to the direction of the velocity vector.

The full acceleration value is defined as:

Analysis of types and kinematic parameters of movements

Uniform movement - this movement at a constant speed:

For straight, even motion:

For curved, uniform motion:

The law of uniform motion :

Equivalent motion – this motion with constant tangential acceleration:

For rectilinear equal motion

For curvilinear equal-variable motion:

The law of equal motion:

Kinematic graphs

Kinematic graphs - these are graphs of changes in path, speed and accelerations depending on time.

Uniform movement (figure 2.5)

Figure 2.5

Equivalent motion (Figure 2.6)

Figure 2.6

The simplest movements of a rigid body

Translational motion is called the movement of a rigid body, in which any straight line on the body during movement remains parallel to its initial position (Figure 2.7)

Figure 2.7

In translational motion, all points of the body move in the same way: the speeds and accelerations at each moment are the same.

Whenrotary motion all points of the body describe a circle around a common fixed axis.

The fixed axis around which all points of the body rotate is calledaxis of rotation.

To describe the rotational motion of a body around a fixed axis, onlyangular parameters. (figure 2.8)

φ - body rotation angle;

ω – angular velocity, determines the change in the angle of rotation per unit of time;

The change in angular velocity over time is determined by the angular acceleration:

2.2. Examples of solving practical problems

Example 1: The equation of motion of a point is given. Determine the speed of the point at the end of the third second of movement and the average speed for the first three seconds.

Decision:

1. Equation of speed

2. Speed \u200b\u200bat the end of the third second (t=3 c)

3. Average speed

Example 2: According to the given law of motion, determine the type of motion, the initial speed and tangential acceleration of the point, the time to stop.

Decision:

1. Type of movement: equal-variable ()
2. When comparing the equations, it is obvious that

- the initial path, traversed before the start of counting 10m;

- initial speed 20m / s

- constant tangential acceleration

- the acceleration is negative, therefore, the movement is slowed down, the acceleration is directed in the direction opposite to the speed of movement.

3. You can define the time at which the point speed will be zero.

3.Dynamics: basic concepts and axioms

Dynamics - a section of theoretical mechanics, in which a connection is established between the motion of bodies and the forces acting on them.

Two types of problems are solved in dynamics:

determine the parameters of movement for given forces;

determine the forces acting on the body, according to the given kinematic parameters of motion.

Undermaterial point imply a certain body that has a certain mass (i.e., containing a certain amount of matter), but does not have linear dimensions (an infinitely small volume of space).
Isolated a material point is considered, which is not affected by other material points. In the real world, isolated material points, like isolated bodies, do not exist, this concept is conditional.

When moving forward, all points of the body move in the same way, so the body can be taken as a material point.

If the dimensions of the body are small compared to the trajectory, it can also be considered as a material point, while the point coincides with the center of gravity of the body.

During the rotational motion of the body, the points may not move in the same way, in this case, some provisions of the dynamics can be applied only to individual points, and the material object can be considered as a set of material points.

Therefore, the dynamics are divided into the dynamics of the point and the dynamics of the material system.

Dynamics axioms

The first axiom ( principle of inertia): in any isolated material point is in a state of rest or uniform and rectilinear motion until the applied forces bring it out of this state.

This state is called the stateinertia. Remove the point from this state, i.e. to give it some acceleration, an external force can.

Every body (point) hasinertia. The measure of inertia is body weight.

Mass calledthe amount of substance in the volume of the body, in classical mechanics, it is considered a constant value. The unit of measure for mass is kilogram (kg).

Second axiom (Newton's second law is the basic law of dynamics)

F \u003d ma

wheret - point mass, kg;a - point acceleration, m / s 2 .

The acceleration imparted to a material point by force is proportional to the magnitude of the force and coincides with the direction of the force.

The force of gravity acts on all bodies on the Earth, it imparts to the body the acceleration of free fall directed towards the center of the Earth:

G \u003d mg,

whereg - 9.81 m / s², acceleration of gravity.

Third axiom (Newton's third law): csilts of interaction of two bodies are equal in size and directed along one straight line in different directions.

When interacting, accelerations are inversely proportional to masses.

Fourth axiom (the law of independence of the action of forces): toeach force of a system of forces acts as it would act alone.

The acceleration imparted to the point by the system of forces is equal to the geometric sum of the accelerations imparted to the point by each force separately (Figure 3.1):

Figure 3.1

Friction concept. Types of friction.

Friction- resistance arising from the movement of one rough body on the surface of another. When bodies slide, sliding friction occurs, while rolling - swing friction.

Sliding friction

Figure 3.2.

The reason is the mechanical engagement of the protrusions. The force of resistance to movement during sliding is called the sliding friction force (Figure 3.2)

Sliding friction laws:

1. The sliding friction force is directly proportional to the normal pressure force:

whereR- the force of normal pressure, directed perpendicular to the supporting surface;f- coefficient of sliding friction.

Figure 3.3.

In the case of body movement along an inclined plane (Figure 3.3)

Rolling friction

Rolling resistance is associated with the mutual deformation of the soil and the wheel and is significantly less sliding friction.

For uniform wheel rolling, force must be appliedF dv (Figure 3.4)

The rolling condition of the wheel is that the moving moment must not be less than the moment of resistance:

Figure 3.4.

Example 1: Example 2: To two material points with massm 1 \u003d 2kg andm 2 \u003d 5 kg the same forces are applied. Compare the values \u200b\u200bfaster.

Decision:

According to the third axiom, acceleration dynamics are inversely proportional to masses:

Example 3: Determine the work of gravity when moving the load from point A to point C along an inclined plane (Figure 3. 7). The gravity of the body is 1500N. AB \u003d 6 m, BC \u003d 4m. Example 3: Determine the work of the cutting force in 3 min. Workpiece rotation speed 120 rpm, workpiece diameter 40mm, cutting force 1kN. (Figure 3.8)

Decision:

1. Work with rotary motion:

2. Angular speed 120 rpm

Figure 3.8.

3. The number of revolutions for a given time isz\u003d 120 * 3 \u003d 360 rev.

The angle of rotation during this time is φ \u003d 2πz\u003d 2 * 3.14 * 360 \u003d 2261rad

4. Work in 3 turns:W\u003d 1 * 0.02 * 2261 \u003d 45.2 kJ

Bibliography

Olofinskaya, V.P. "Technical Mechanics", Moscow "Forum" 2011

Erdedi A.A. Erdedi N.A. Theoretical mechanics. Resistance of materials.- Rn-D; Phoenix, 2010

In any academic 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, the scientist was walking in the garden, saw an apple falling, and it was this phenomenon that pushed him to the discovery of the law of universal gravitation. Of course, the law has always existed, and Newton only gave it a form that people can understand, 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 is translated as "the art of building machines." But before constructing machines, we are still like the Moon, so we will 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 of 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 from the foundations of mechanics. Placed within the framework of time and space, people, in fact, could not start from something else, with all their desire. Moving bodies are the first thing we turn our attention to.

What is movement?

Mechanical movement 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 on the seat next to him, and moves at a different 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 of our measurements in a geocentric frame of reference associated with the Earth. The earth is a reference body relative to which cars, airplanes, people, animals move.

Mechanics, as a science, has its own task. The task of mechanics is to know the position of a 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.

In order to move further, we need the concept “ material point ”. They say 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. Nobody has ever seen a material point or smelled ideal gas, but they are! It's just much easier to live with them.

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

Sections of classical mechanics

Mechanics consists of several sections

• Kinematics
• Dynamics
• Statics

Kinematicsfrom a physical point of view, it studies exactly how the body moves. In other words, this section deals with the quantitative characteristics of movement. Find speed, path - typical kinematic problems

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

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

The 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 framework of applicability. In general, the laws of classical mechanics are true for the world we are accustomed to in terms of size (macrocosm). They stop working in the case of the particle world, when quantum mechanics replaces the classical one. Also, classical mechanics is inapplicable to cases when 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.

Generally speaking, quantum and relativistic effects never go anywhere, they also take place during the ordinary motion of macroscopic bodies with a speed much less than the speed of light. Another thing is that the effect of these effects is so small that it does not go beyond the most accurate measurements. Thus, classical mechanics will never lose their fundamental importance.

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