Methods of theoretical mechanics. Basic Mechanics for Dummies

In all its glory and elegance. With its help, Newton once derived his law based on Kepler's three empirical laws gravity. The subject, in general, is not so complicated, it is relatively easy to understand. But it’s difficult to pass, because teachers are often terribly picky (like Pavlova, for example). When solving problems, you need to be able to solve diffuses and calculate integrals.

Key Ideas

In fact, the theory of mechanics within this course is the application of the variational principle to calculate the "motion" of various physical systems. The calculus of variations is briefly covered in the course Integral Equations and the Calculus of Variations. Lagrange's equations are Euler's equations which are the solution to the problem with fixed ends.

One task can usually be solved by 3 different methods at once:

• Lagrange method (Lagrange function, Lagrange equations)
• Hamilton method (Hamilton function, Hamilton equations)
• Hamilton-Jacobi method (Hamilton-Jacobi equation)

It is important to choose the simplest of them for a particular task.

materials

First semester (test)

Basic formulas

Watch in big size!

Theory

Video recordings

Lectures V.R. Khalilova - Attention! not all lectures recorded

Second semester (exam)

You have to start with what different groups The exam is different. Usually Examination ticket consists of 2 theoretical questions and 1 task. Questions are obligatory for everyone, but you can both get rid of the task (for excellent work in the semester + written control ones), or grab an extra one (and more than one). Here you will be told about the rules of the game at seminars. In the groups of Pavlova and Pimenov, theormin is practiced, which is a kind of admission to the exam. It follows that this theory must be known perfectly.

Exam in Pavlova's groups goes something like this: To start a ticket with 2 questions of the term. There is little time to write, and the key here is to write it absolutely perfectly. Then Olga Serafimovna will be kind to you and the rest of the exam will be very pleasant. Next is a ticket with 2 theory questions + n tasks (depending on your work in the semester). Theory within a theory can be written off. Tasks to solve. There are many problems on the exam - it's not the end if you know how to solve them perfectly. This can be turned into an advantage - for each point of the exam you get +, + -, -+ or -. The rating is set "by the overall impression" => if in theory everything is not perfect for you, but then it goes 3 + for tasks, then the overall impression is good. But if you were without problems on the exam and the theory is not ideal, then there is nothing to smooth it over.

Theory

• Julia. Lecture notes (2014, pdf) - both semesters, 2nd stream
• Second stream tickets part 1 (lecture notes and part for tickets) (pdf)
• Second stream tickets and table of contents for all these parts (pdf)
• Answers to tickets of the 1st stream (2016, pdf) - in hard copy, very comfortably
• Recognized Theormin for the Pimenov Group Exam (2016, pdf) - both semesters
• Answers to theormin for Pimenov groups (2016, pdf) - accurate and apparently without errors

Tasks

• Pavlova's seminars 2nd semester (2015, pdf) - neat, beautifully and clearly written
• Tasks that may be on the exam (jpg) - once in some shaggy year they were on the 2nd stream, may also be relevant for V.R. groups. Khalilova (he gives similar problems on kr)
• Tasks for tickets (pdf)- for both streams (on the 2nd stream, these tasks were in the groups of A.B. Pimenov)

Introduction

Theoretical mechanics is one of the most important fundamental general scientific disciplines. It plays an essential role in the training of engineers of all specialties. General engineering disciplines are based on the results of theoretical mechanics: strength of materials, machine parts, theory of mechanisms and machines, and others.

The main task of theoretical mechanics is the study of the motion of material bodies under the action of forces. An important particular problem is the study of the equilibrium of bodies under the action of forces.

Lecture course. Theoretical mechanics

The structure of theoretical mechanics. Fundamentals of statics

Conditions for the equilibrium of an arbitrary system of forces.

Rigid Body Equilibrium Equations.

Flat system of forces.

Particular cases of equilibrium of a rigid body.

The problem of equilibrium of a beam.

Determination of internal forces in bar structures.

Fundamentals of point kinematics.

natural coordinates.

Euler formula.

Distribution of accelerations of points of a rigid body.

Translational and rotational movements.

Plane-parallel motion.

Complicated point movement.

Fundamentals of point dynamics.

Differential equations of motion of a point.

Particular types of force fields.

Fundamentals of the dynamics of the system of points.

General theorems of the dynamics of a system of points.

Dynamics of rotational movement of the body.

Dobronravov V.V., Nikitin N.N. Course of theoretical mechanics. M., Higher School, 1983.

Butenin N.V., Lunts Ya.L., Merkin D.R. Course of Theoretical Mechanics, Parts 1 and 2. M., Higher School, 1971.

Petkevich V.V. Theoretical mechanics. M., Nauka, 1981.

Collection of tasks for term papers in theoretical mechanics. Ed. A.A. Yablonsky. M., Higher School, 1985.

Lecture 1 The structure of theoretical mechanics. Fundamentals of statics

AT theoretical mechanics the movement of bodies relative to other bodies, which are physical reference systems, is studied.

Mechanics allows not only to describe, but also to predict the movement of bodies, establishing causal relationships in a certain, very wide range of phenomena.

Basic abstract models of real bodies:

material point - has mass, but no dimensions;

absolutely rigid body - a volume of finite dimensions, completely filled with matter, and the distances between any two points of the medium filling the volume do not change during movement;

continuous deformable medium - fills a finite volume or unlimited space; the distances between the points of such a medium can vary.

Of these, systems:

System of free material points;

Systems with connections;

An absolutely solid body with a cavity filled with liquid, etc.

"Degenerate" models:

Infinitely thin rods;

Infinitely thin plates;

Weightless rods and threads connecting material points, etc.

From experience: mechanical phenomena run differently in different places physical reference system. This property is the inhomogeneity of space, determined by the physical reference system. Heterogeneity here is understood as the dependence of the nature of the occurrence of a phenomenon on the place in which we observe this phenomenon.

Another property is anisotropy (non-isotropy), the motion of a body relative to the physical reference frame can be different depending on the direction. Examples: the course of the river along the meridian (from north to south - the Volga); projectile flight, Foucault pendulum.

The properties of the reference system (heterogeneity and anisotropy) make it difficult to observe the motion of a body.

Practically free from this geocentric system: the center of the system is at the center of the Earth and the system does not rotate relative to the "fixed" stars). The geocentric system is convenient for calculating movements on the Earth.

For celestial mechanics(for solar system bodies): a heliocentric reference frame that moves with the center of mass solar system and does not rotate relative to "fixed" stars. For this system not found yet heterogeneity and anisotropy of space

in relation to the phenomena of mechanics.

So, we introduce an abstract inertial reference frame for which space is homogeneous and isotropic in relation to the phenomena of mechanics.

inertial frame of reference- one whose own movement cannot be detected by any mechanical experience. Thought experiment: "the point that is alone in the whole world" (isolated) is either at rest or moving in a straight line and uniformly.

All frames of reference moving relative to the original rectilinearly will be uniformly inertial. This allows you to introduce a single Cartesian coordinate system. Such a space is called Euclidean.

Conditional agreement - take the right coordinate system (Fig. 1).

AT time– in classical (non-relativistic) mechanics absolutely, which is the same for all reference systems, that is, the initial moment is arbitrary. In contrast to relativistic mechanics, where the principle of relativity is applied.

The state of motion of the system at time t is determined by the coordinates and velocities of the points at that moment.

Real bodies interact, and forces arise that change the state of motion of the system. This is the essence of theoretical mechanics.

How is theoretical mechanics studied?

The doctrine of the equilibrium of a set of bodies of a certain reference frame - section statics.

Chapter kinematics: a part of mechanics that studies the relationships between quantities that characterize the state of motion of systems, but does not consider the causes that cause a change in the state of motion.

After that, consider the influence of forces [MAIN PART].

Chapter dynamics: part of mechanics, which considers the influence of forces on the state of motion of systems of material objects.

Principles of building the main course - dynamics:

1) based on a system of axioms (based on experience, observations);

Constantly - ruthless control of practice. Sign of exact science - the presence of internal logic (without it - set of unrelated recipes)!

static that part of mechanics is called, where the conditions that must be satisfied by the forces acting on a system of material points are studied in order for the system to be in equilibrium, and the conditions for the equivalence of systems of forces.

Problems of equilibrium in elementary statics will be considered using exclusively geometric methods based on the properties of vectors. This approach is applied in geometric statics(as opposed to analytic statics, which is not considered here).

The positions of various material bodies will be referred to the coordinate system, which we will take as fixed.

Ideal models of material bodies:

1) material point - a geometric point with mass.

2) absolutely rigid body - a set of material points, the distances between which cannot be changed by any actions.

By the forces we will call objective reasons, which are the result of the interaction of material objects, capable of causing the movement of bodies from a state of rest or changing the existing movement of the latter.

Since the force is determined by the motion it causes, it also has a relative character, depending on the choice of the frame of reference.

The question of the nature of forces is considered in physics.

A system of material points is in equilibrium if, being at rest, it does not receive any movement from the forces acting on it.

From everyday experience: forces are vector in nature, that is, magnitude, direction, line of action, point of application. The condition for the equilibrium of forces acting on a rigid body is reduced to the properties of systems of vectors.

Summarizing the experience of studying the physical laws of nature, Galileo and Newton formulated the basic laws of mechanics, which can be considered as axioms of mechanics, since they have based on experimental facts.

Axiom 1. The action of several forces on a point of a rigid body is equivalent to the action of one resultant force, constructed according to the rule of addition of vectors (Fig. 2).

Consequence. The forces applied to a point of a rigid body are added according to the parallelogram rule.

Axiom 2. Two forces applied to a rigid body mutually balanced if and only if they are equal in magnitude, directed in opposite directions and lie on the same straight line.

Axiom 3. The action of a system of forces on a rigid body will not change if add to this system or drop from it two forces of equal magnitude, directed in opposite directions and lying on the same straight line.

Consequence. The force acting on a point of a rigid body can be transferred along the line of action of the force without changing the balance (that is, the force is a sliding vector, Fig. 3)

1) Active - create or are able to create the movement of a rigid body. For example, the force of weight.

2) Passive - not creating movement, but limiting the movement of a rigid body, preventing movement. For example, the tension force of an inextensible thread (Fig. 4).

Axiom 4. The action of one body on the second is equal and opposite to the action of this second body on the first ( action equals reaction).

The geometric conditions that restrict the movement of points will be called connections.

Communication conditions: for example,

- rod of indirect length l.

- flexible inextensible thread of length l.

Forces due to bonds and preventing movement are called reaction forces.

Axiom 5. The bonds imposed on the system of material points can be replaced by reaction forces, the action of which is equivalent to the action of the bonds.

When passive forces cannot balance the action of active forces, movement begins.

Two particular problems of statics

1. System of converging forces acting on a rigid body

A system of converging forces such a system of forces is called, the lines of action of which intersect at one point, which can always be taken as the origin (Fig. 5).

Projections of the resultant:

;

;

.

If , then the force causes the motion of a rigid body.

Equilibrium condition for a convergent system of forces:

2. Balance of three forces

If three forces act on a rigid body, and the lines of action of two forces intersect at some point A, equilibrium is possible if and only if the line of action of the third force also passes through point A, and the force itself is equal in magnitude and oppositely directed to the sum (Fig. 6).

Examples:

Moment of force relative to point O define as a vector , in size equal to twice the area of ​​a triangle, the base of which is a force vector with a vertex at a given point O; direction- orthogonal to the plane of the considered triangle in the direction from where the rotation produced by the force around the point O is visible counterclockwise. is the moment of the sliding vector and is free vector(Fig. 9).

So: or

,

where ;;.

Where F is the modulus of force, h is the shoulder (distance from the point to the direction of the force).

Moment of force about the axis is called the algebraic value of the projection onto this axis of the vector of the moment of force relative to an arbitrary point O, taken on the axis (Fig. 10).

This is a scalar independent of the choice of point. Indeed, we expand :|| and in the plane.

About moments: let О 1 be the point of intersection with the plane. Then:

a) from - moment => projection = 0.

b) from - moment along => is a projection.

So, the moment about the axis is the moment of the force component in the plane perpendicular to the axis about the point of intersection of the plane and the axis.

Varignon's theorem for a system of converging forces:

Moment of resultant force for a system of converging forces relative to an arbitrary point A is equal to the sum moments of all components of forces relative to the same point A (Fig. 11).

Proof in the theory of convergent vectors.

Explanation: addition of forces according to the parallelogram rule => the resulting force gives the total moment.

Test questions:

1. Name the main models of real bodies in theoretical mechanics.

2. Formulate the axioms of statics.

3. What is called the moment of force about a point?

Lecture 2 Equilibrium conditions for an arbitrary system of forces

From the basic axioms of statics, elementary operations on forces follow:

1) force can be transferred along the line of action;

2) forces whose lines of action intersect can be added according to the parallelogram rule (according to the rule of vector addition);

3) to the system of forces acting on a rigid body, one can always add two forces, equal in magnitude, lying on the same straight line and directed in opposite directions.

Elementary operations do not change the mechanical state of the system.

Let's name two systems of forces equivalent if one from the other can be obtained using elementary operations (as in the theory of sliding vectors).

A system of two parallel forces, equal in magnitude and directed in opposite directions, is called a couple of forces(Fig. 12).

Moment of a pair of forces- a vector equal in size to the area of ​​the parallelogram built on the vectors of the pair, and directed orthogonally to the plane of the pair in the direction from which the rotation reported by the vectors of the pair can be seen to occur counterclockwise.

, that is, the moment of force about point B.

A pair of forces is fully characterized by its moment.

A pair of forces can be transferred by elementary operations to any plane parallel to the plane of the pair; change the magnitude of the forces of the pair inversely proportional to the shoulders of the pair.

Pairs of forces can be added, while the moments of pairs of forces are added according to the rule of addition of (free) vectors.

Bringing the system of forces acting on a rigid body to an arbitrary point (reduction center)- means replacing the current system with a simpler one: system of three forces, one of which passes through a predetermined point, and the other two represent a pair.

It is proved with the help of elementary operations (fig.13).

The system of converging forces and the system of pairs of forces.

- resulting force.

The resulting pair

Which is what needed to be shown.

Two systems of forces will are equivalent if and only if both systems are reduced to one resultant force and one resultant pair, that is, under the following conditions:

General case of equilibrium of a system of forces acting on a rigid body

We bring the system of forces to (Fig. 14):

Resulting force through the origin;

The resulting pair, moreover, through the point O.

That is, they led to and - two forces, one of which passes through a given point O.

Equilibrium, if the other one straight line, are equal, directed oppositely (axiom 2).

Then passes through the point O, that is.

So, general terms and Conditions equilibrium of a rigid body:

These conditions are valid for an arbitrary point in space.

Test questions:

1. List elementary operations on forces.

2. What systems of forces are called equivalent?

3. Write the general conditions for the equilibrium of a rigid body.

Lecture 3 Rigid Body Equilibrium Equations

Let O be the origin of coordinates; is the resulting force; is the moment of the resulting pair. Let the point O1 be a new reduction center (Fig. 15).

New force system:

When the cast point changes, => changes only (in one direction with one sign, in the other with another). That is the point: match the lines

Analytically: (colinearity of vectors)

; point O1 coordinates.

This is the equation of a straight line, for all points of which the direction of the resulting vector coincides with the direction of the moment of the resulting pair - the straight line is called dynamo.

If on the axis of the dynamas => , then the system is equivalent to one resultant force, which is called the resultant force of the system. In this case, always, that is.

Four cases of bringing forces:

1.) ;- dynamo.

2.) ; - resultant.

3.) ;- pair.

4.) ;- balance.

Two vector equilibrium equations: the main vector and the main moment are equal to zero,.

Or six scalar equations in projections onto Cartesian coordinate axes:

Here:

The complexity of the type of equations depends on the choice of the reduction point => the art of the calculator.

Finding the equilibrium conditions for a system of rigid bodies in interaction<=>the problem of the balance of each body separately, and the body is affected by external forces and internal forces (the interaction of bodies at points of contact with equal and oppositely directed forces - axiom IV, Fig. 17).

We choose for all bodies of the system one referral center. Then for each body with the equilibrium condition number:

, , (= 1, 2, …, k)

where , - the resulting force and the moment of the resulting pair of all forces, except for internal reactions.

The resulting force and moment of the resulting pair of forces of internal reactions.

Formally summing up and taking into account the IV axiom

we get necessary conditions for the equilibrium of a rigid body:

,

Example.

Equilibrium: = ?

Test questions:

1. Name all cases of bringing the system of forces to one point.

2. What is a dynamo?

3. Formulate the necessary conditions for the equilibrium of a system of rigid bodies.

Lecture 4 Flat system of forces

A special case of the general task delivery.

Let all active forces lie in the same plane - for example, a sheet. Let us choose the point O as the center of reduction - in the same plane. We get the resulting force and the resulting pair in the same plane, that is (Fig. 19)

Comment.

The system can be reduced to one resultant force.

Equilibrium conditions:

or scalars:

Very common in applications such as strength of materials.

Example.

With the friction of the ball on the board and on the plane. Equilibrium condition: = ?

The problem of the equilibrium of a non-free rigid body.

A rigid body is called non-free, the movement of which is constrained by constraints. For example, other bodies, hinged fastenings.

When determining the conditions of equilibrium: a non-free body can be considered as free, replacing the bonds with unknown reaction forces.

Example.

Test questions:

1. What is called a flat system of forces?

2. Write the equilibrium conditions for a flat system of forces.

3. What kind of solid body is called non-free?

Lecture 5 Special cases of rigid body equilibrium

Theorem. Three forces balance a rigid body only if they all lie in the same plane.

Proof.

We choose a point on the line of action of the third force as the point of reduction. Then (fig.22)

That is, the planes S1 and S2 coincide, and for any point on the axis of force, etc. (Easier: in the plane just for balance).

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, 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.

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 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 special case when the dimensions of the body are large and the speed is small. You can learn more about it from our article.

Generally speaking, quantum and relativistic effects never disappear, 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 turn to, which individually shed light on dark spot the most difficult task.

Examples of problem solving in theoretical mechanics

Statics

Task Conditions

Kinematics

Kinematics of a material point

The task

Determination of the speed and acceleration of a point according to the given equations of its motion.
According to the given equations of motion of the point, establish the type of its trajectory and for the moment of time t = 1 s find the position of a point on the trajectory, its velocity, full, tangential and normal accelerations, as well as the radius of curvature of the trajectory.
Point motion equations:
x= 12 sin(πt/6), cm;
y= 6 cos 2 (πt/6), cm.

Kinematic analysis of a flat mechanism

The task

The flat mechanism consists of rods 1, 2, 3, 4 and slider E. The rods are interconnected, with sliders and fixed supports connected with cylindrical hinges. Point D is located in the middle of bar AB. The lengths of the rods are equal, respectively
l 1 \u003d 0.4 m; l 2 = 1.2 m; l 3 \u003d 1.6 m; l 4 \u003d 0.6 m.

The mutual arrangement of the elements of the mechanism in a particular version of the problem is determined by the angles α, β, γ, φ, ϑ. Rod 1 (rod O 1 A) rotates around a fixed point O 1 counterclockwise with a constant angular velocity ω 1 .

For a given position of the mechanism, it is necessary to determine:

• linear speeds V A , V B , V D and V E points A, B, D, E;
• angular speeds ω 2 , ω 3 and ω 4 links 2, 3 and 4;
• linear acceleration a B point B;
• angular acceleration ε AB of link AB;
• positions of instantaneous centers of speeds C 2 and C 3 of links 2 and 3 of the mechanism.

Determining the absolute speed and absolute acceleration of a point

The task

The diagram below considers the movement of point M in the chute of a rotating body. According to the given equations of translational motion φ = φ(t) and relative motion OM = OM(t), determine the absolute speed and absolute acceleration of a point in this moment time.

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Dynamics

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

The task

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

The load, having completed its movement in section AB, at point B of the pipe, without changing the value of its velocity modulus, passes to section BC. On the section BC, a variable force F acts on the load, the projection F x of which on the x axis is given.

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

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Theorem on the change in the kinetic energy of a mechanical system

The task

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 pulleys; sections of threads are parallel to the corresponding planes. The roller (solid homogeneous cylinder) rolls along the reference plane without slipping. The radii of the steps of the pulleys 4 and 5 are respectively 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 uniformly distributed along its outer rim . The supporting planes of weights 1 and 2 are rough, the coefficient of sliding friction for each weight is f = 0.1.

Under the action of 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, resistance forces act on the pulley 5, 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 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

The task

For a mechanical system, determine the linear acceleration a 1 . Consider that for blocks and rollers the masses are distributed along the outer radius. Cables and belts are considered weightless and inextensible; there is no slippage. Ignore rolling and sliding friction.

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

The task

vertical shaft AK, rotating uniformly with an angular velocity ω = 10 s -1 , is fixed with a thrust bearing at point A and a cylindrical bearing at point D.

A weightless rod 1 with a length of l 1 = 0.3 m is rigidly attached to the shaft, at the free end of which there is a load of mass 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 the bearing.

Point kinematics.

1. The subject of theoretical mechanics. Basic abstractions.

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

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

Mechanical interaction is called such an interaction of material bodies, which changes the nature of their mechanical movement.

Statics - This is a branch of theoretical mechanics, which studies methods for converting systems of forces into equivalent systems and establishes the conditions for the equilibrium of forces applied to a solid body.

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

Dynamics - This 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 rigid body.

Absolute space and absolute time are independent of each other. 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. The subject of kinematics.

Kinematics - this is a branch of mechanics that studies the geometric properties of the motion 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, rigidly, some coordinate system is connected, 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), to determine all the kinematic quantities that characterize its motion (velocity and acceleration).

3. Methods for specifying the movement of a point

· natural way

Should be known:

Point movement trajectory;

Start and direction of counting;

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 the point M.

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

· Vector way

Relationship between coordinate and natural ways point movement assignments

Determine the trajectory of the point, excluding time from equations (1.2);

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

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 movement of a point is determined by equation (1.4)

4. Determining the speed of a point with the vector method of specifying the movement.

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

The speed of a point at a given time

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

(1.6)

(1.7)

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

(unit¾ m/s, km/h)

Mean 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 is equal to the first derivative of the velocity vector or the second derivative of the point's radius vector with respect to time.

(unit - )

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

At rectilinear motion the vector is directed along the straight line along which the point moves. If the trajectory of the point is a flat curve, then the acceleration vector , as well as the vector cp, lies in the plane of this curve and is directed towards its concavity. If the trajectory is not a plane curve, then the vector cp 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 . AT limit when the pointM 1 tends to M this plane occupies the position of the so-called contiguous plane. Therefore, in the general case, the acceleration vector lies in a contiguous plane and is directed towards the concavity of the curve.