Short course in theoretical mechanics. Targ S.M. Solving problems in theoretical mechanics Systems of bodies theoretical mechanics

Content

Kinematics

Kinematics of a material point

Determining the speed and acceleration of a point by given equations her movements

Given: Equations of motion of a point: x = 12 sin(πt/6), cm; y= 6 cos 2 (πt/6), cm.

Set the type of its trajectory for the moment of time t = 1 s find the position of a point on the trajectory, its speed, total, tangent and normal acceleration, as well as the radius of curvature of the trajectory.

Translational and rotational motion of a rigid body

Given:
t = 2 s; r 1 = 2 cm, R 1 = 4 cm; r 2 = 6 cm, R 2 = 8 cm; r 3 = 12 cm, R 3 = 16 cm; s 5 = t 3 - 6t (cm).

Determine at time t = 2 the velocities of points A, C; angular acceleration wheels 3; acceleration of point B and acceleration of rack 4.

Kinematic analysis of a flat mechanism

Given:
R 1, R 2, L, AB, ω 1.
Find: ω 2.

The flat mechanism consists of rods 1, 2, 3, 4 and a slider E. The rods are connected using cylindrical hinges. Point D is located in the middle of rod AB.
Given: ω 1, ε 1.
Find: velocities V A, V B, V D and V E; angular velocities ω 2, ω 3 and ω 4; acceleration a B ; angular acceleration ε AB of link AB; provisions instant centers speeds P 2 and P 3 of links 2 and 3 of the mechanism.

Determination of absolute speed and absolute acceleration of a point

A rectangular plate rotates around fixed axis according to the law φ = 6 t 2 - 3 t 3. The positive direction of the angle φ is shown in the figures by an arc arrow. Rotation axis OO 1 lies in the plane of the plate (the plate rotates in space).

Point M moves along the plate along straight line BD. The law of its relative motion is given, i.e. the dependence s = AM = 40(t - 2 t 3) - 40(s - in centimeters, t - in seconds). Distance b = 20 cm. In the figure, point M is shown in a position where s = AM > 0 (at s< 0 point M is on the other side of point A).

Find the absolute speed and absolute acceleration point M at time t 1 = 1 s.

Dynamics

Integration of differential equations of motion of a material point under the influence of variable forces

Load D with mass m, having received at point A initial speed V 0 moves in a curved pipe ABC located in a vertical plane. In a section AB, the length of which is l, the load is acted upon by a constant force T (its direction is shown in the figure) and a force R of the medium resistance (the modulus of this force R = μV 2, the vector R is directed opposite to the speed V of the load).

The load, having finished moving in section AB, at point B of the pipe, without changing the value of its speed module, moves to section BC. In section BC, the load is acted upon by a variable force F, the projection F x of which on the x axis is given.

Considering the load to be a material point, find the law of its motion in section BC, i.e. x = f(t), where x = BD. Neglect the friction of the load on the pipe.

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Theorem on the change in kinetic energy of a mechanical system

The mechanical system consists of weights 1 and 2, a cylindrical roller 3, two-stage pulleys 4 and 5. The bodies of the system are connected by threads wound on the pulleys; sections of threads are parallel to the corresponding planes. The roller (a solid homogeneous cylinder) rolls along the supporting plane without sliding. The radii of the stages of pulleys 4 and 5 are respectively equal to R 4 = 0.3 m, r 4 = 0.1 m, R 5 = 0.2 m, r 5 = 0.1 m. The mass of each pulley is considered to be uniformly distributed along its outer rim . The supporting planes of loads 1 and 2 are rough, the sliding friction coefficient for each load is f = 0.1.

Under the action of a force F, the modulus of which changes according to the law F = F(s), where s is the displacement of the point of its application, the system begins to move from a state of rest. When the system moves, pulley 5 is acted upon by resistance forces, the moment of which relative to the axis of rotation is constant and equal to M 5 .

Determine the value of the angular velocity of pulley 4 at the moment in time when the displacement s of the point of application of force F becomes equal to s 1 = 1.2 m.

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Application of the general equation of dynamics to the study of the motion of a mechanical system

For mechanical system determine linear acceleration a 1. Assume that the masses of blocks and rollers are distributed along the outer radius. Cables and belts should be considered weightless and inextensible; there is no slippage. Neglect rolling and sliding friction.

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Application of d'Alembert's principle to determining the reactions of the supports of a rotating body

Vertical shaft AK rotating uniformly with angular velocityω = 10 s -1, secured by a thrust bearing at point A and a cylindrical bearing at point D.

Rigidly attached to the shaft are a weightless rod 1 with a length of l 1 = 0.3 m, at the free end of which there is a load with a mass of m 1 = 4 kg, and a homogeneous rod 2 with a length of l 2 = 0.6 m, having a mass of m 2 = 8 kg. Both rods lie in the same vertical plane. The points of attachment of the rods to the shaft, as well as the angles α and β are indicated in the table. Dimensions AB=BD=DE=EK=b, where b = 0.4 m. Take the load as a material point.

Neglecting the mass of the shaft, determine the reactions of the thrust bearing and bearing.

Within any training course The study of physics begins with mechanics. Not from theoretical, not from applied or 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 falling, and it was this phenomenon that prompted him to discover the law universal 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 fundamentals, 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 we build machines, we are still like the Moon, so let’s follow in the footsteps of our ancestors and study the movement of stones thrown at an angle to the horizon, and apples falling on our heads from a height h.

Why does the study of physics begin with mechanics? Because this is completely natural, shouldn’t we start with thermodynamic equilibrium?!

Mechanics is one of the oldest sciences, and historically the study of physics began with the foundations of mechanics. Placed within the framework of time and space, people, in fact, could not start with something else, no matter how much they wanted. 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 the person standing on the side of the road at a certain speed, and is at rest relative to his neighbor in the seat next to him, and moves at some other speed relative to the passenger in the car that is overtaking 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 heliocentric system countdown. In everyday life, we carry out almost all our measurements in geocentric system reference associated with the Earth. The earth is a body of reference relative to which cars, planes, people, and 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 builds a mathematical description of motion and finds connections between physical quantities, which characterize it.

In order to move further, we need the concept “ material point " They say physics - 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 smelled an ideal gas, but they exist! They are simply 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, it studies exactly how a body moves. In other words, this section deals with the quantitative characteristics of movement. Find speed, path - typical kinematics problems

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 balance of bodies under the influence of forces, that is, answers the question: why doesn’t it 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 framework of applicability. In general, the laws of classical mechanics are valid in the world we are accustomed to in size (macroworld). They stop working in the case of the particle world, when quantum mechanics replaces classical mechanics. Also, classical mechanics is not applicable 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 is special case, when the body size is large and the speed is low.

Generally speaking, quantum and relativistic effects never go away; they also occur during the ordinary motion of macroscopic bodies at a speed much lower 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. 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 to our authors, which will individually shed light on the dark spot of the most difficult task.

As part of any educational course, the study of physics begins with mechanics. Not from theoretical, not from applied or computational, but from good old classical mechanics. This mechanics is also called Newtonian mechanics. According to legend, a scientist was walking in the garden and saw an apple falling, 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 fundamentals, 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 this is completely natural, shouldn’t we start with thermodynamic equilibrium?!

What is movement?

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

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 body of reference relative to which cars, planes, people, and animals move.

In order to move further, we need the concept “ 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 smelled an ideal gas, but they exist! They are simply 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

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 balance of bodies under the influence of forces, that is, answers the question: why doesn’t it 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 framework of applicability. In general, the laws of classical mechanics are valid in the world we are accustomed to in size (macroworld). They stop working in the case of the particle world, when quantum mechanics replaces classical mechanics. Also, classical mechanics is not applicable 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.

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 to our authors, which will individually shed light on the dark spot of the most difficult task.

Kinematics of a point.

1. Subject of theoretical mechanics. Basic abstractions.

Theoretical mechanicsis a science in which general laws are studied mechanical movement and mechanical interaction of material bodies

Mechanical movementis the movement of a body in relation to another body, occurring in space and time.

Mechanical interaction is the interaction of material bodies that changes the nature of their mechanical movement.

Statics is a branch of theoretical mechanics in which methods of transforming systems of forces into equivalent systems are studied and conditions for the equilibrium of forces applied to a solid body are established.

Kinematics - is a branch of theoretical mechanics that studies movement of material bodies in space with geometric point vision, regardless of the forces acting on them.

Dynamics is a branch of mechanics that studies the movement of material bodies in space depending on the forces acting on them.

Objects of study in theoretical mechanics:

material point,

system of material points,

Absolutely solid body.

Absolute space and absolute time are independent of one another. Absolute space - three-dimensional, homogeneous, motionless Euclidean space. Absolute time - flows from the past to the future continuously, it is homogeneous, the same at all points in space and does not depend on the movement of matter.

2. Subject of kinematics.

Kinematics - is a branch of mechanics that studies geometric properties movement of bodies without taking into account their inertia (i.e. mass) and the forces acting on them

To determine the position of a moving body (or point) with the body in relation to which the movement of this body is being studied, some coordinate system is rigidly associated, which together with the body forms reference system.

The main task of kinematics is to, knowing the law of motion of a given body (point), determine all the kinematic quantities that characterize its movement (speed and acceleration).

3. Methods for specifying the movement of a point

· The natural way

It should be known:

The trajectory of the point;

Origin and direction of reference;

The law of motion of a point along a given trajectory in the form (1.1)

· Coordinate method

Equations (1.2) are the equations of motion of point M.

The equation for the trajectory of point M can be obtained by eliminating the time parameter « t » from equations (1.2)

· Vector method

(1.3)

Relationship between coordinate and vector methods of specifying the movement of a point

(1.4)

Relationship between coordinate and natural methods of specifying the movement of a point

Determine the trajectory of the point by eliminating time from equations (1.2);

-- find the law of motion of a point along a trajectory (use the expression for the differential of the arc)

After integration, we obtain the law of motion of a point along a given trajectory:

The connection between the coordinate and vector methods of specifying the motion of a point is determined by equation (1.4)

4. Determining the speed of a point using the vector method of specifying motion.

Let at a moment in timetthe position of the point is determined by the radius vector, and at the moment of timet 1 – radius vector, then over a period of time the point will move.

(1.5)

average point speed,

the direction of the vector is the same as that of the vector

Point speed in at the moment time

To obtain the speed of a point at a given time, it is necessary to make a passage to the limit

(1.6)

(1.7)

Velocity vector of a point at a given time equal to the first derivative of the radius vector with respect to time and directed tangentially to the trajectory at a given point.

(unit¾ m/s, km/h)

Average acceleration vector has the same direction as the vectorΔ v , that is, directed towards the concavity of the trajectory.

Acceleration vector of a point at a given time equal to the first derivative of the velocity vector or the second derivative of the radius vector of the point with respect to time.

(unit - )

How is the vector located in relation to the trajectory of the point?

At straight motion the vector is directed along the straight line along which the point moves. If the trajectory of a point is a flat curve, then the acceleration vector, as well as the vector ср, lies in the plane of this curve and is directed towards its concavity. If the trajectory is not a plane curve, then the vector ср will be directed towards the concavity of the trajectory and will lie in the plane passing through the tangent to the trajectory at the pointM and a line parallel to the tangent at an adjacent pointM 1 . IN limit when pointM 1 strives for M this plane occupies the position of the so-called osculating plane. Therefore, in the general case, the acceleration vector lies in the contacting plane and is directed towards the concavity of the curve.

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