List of exam questions

1. Technical mechanics, its definition. Mechanical motion and mechanical interaction. Material point, mechanical system, absolutely rigid body.

Technical mechanics - the science of mechanical motion and interaction of material bodies.

Mechanics is one of the most ancient sciences. The term "Mechanics" was introduced by the outstanding philosopher of antiquity Aristotle.

The achievements of scientists in the field of mechanics make it possible to solve complex practical problems in the field of technology, and in essence, not a single phenomenon of nature can be understood without understanding it from the mechanical side. And not a single creation of technology can be created without taking into account certain mechanical laws.

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

Mechanical interaction - these are the actions of material bodies on each other, as a result of which there is a change in the movement of these bodies or a change in their shape (deformation).

Basic concepts:

Material point is a body whose dimensions under given conditions can be neglected. It has mass and the ability to interact with other bodies.

mechanical system is a set of material points, the position and movement of each of which depend on the position and movement of other points in the system.

Absolutely rigid body (ATT) is a body, the distance between any two points of which always remains unchanged.

1. Theoretical mechanics and its sections. Problems of theoretical mechanics.

Theoretical mechanics is a branch of mechanics that studies the laws of motion of bodies and the general properties of these motions.

Theoretical mechanics consists of three sections: statics, kinematics and dynamics.

Statics considers the equilibrium of bodies and their systems under the action of forces.

Kinematics considers the general geometric properties of the motion of bodies.

Dynamics studies the motion of bodies under the action of forces.

Static tasks:

1. Transformation of systems of forces acting on ATT into systems equivalent to them, i.e. reduction of this system of forces to the simplest form.

2. Determination of the equilibrium conditions for the system of forces acting on the ATT.

To solve these problems, two methods are used: graphical and analytical.

1. Equilibrium. Force, system of forces. Resultant force, concentrated force and distributed forces.

Equilibrium is the state of rest of a body in relation to other bodies.

Force - this is the main measure of the mechanical interaction of material bodies. Is a vector quantity, i.e. Strength is characterized by three elements:

application point;

Line of action (direction);

Module (numerical value).

Force system is the totality of all forces acting on the considered absolutely rigid body (ATT)

The force system is called converging if the lines of action of all forces intersect at one point.

The system is called flat , if the lines of action of all forces lie in the same plane, otherwise spatial.

The force system is called parallel if the lines of action of all forces are parallel to each other.

The two systems of forces are called equivalent , if one system of forces acting on an absolutely rigid body can be replaced by another system of forces without changing the state of rest or motion of the body.

Balanced or equivalent to zero called a system of forces under the action of which a free ATT can be at rest.

resultant force is a force whose action on a body or material point is equivalent to the action of a system of forces on the same body.

Outside forces

The force applied to the body at any one point is called concentrated .

Forces acting on all points of a certain volume or surface are called distributed .

A body that is not prevented from moving in any direction by any other body is called a free body.

1. External and internal forces. Free and non-free body. The principle of release from bonds.

Outside forces called the forces with which the parts of a given body act on each other.

When solving most problems of statics, it is required to represent a non-free body as a free one, which is done using the principle of freeing the body, which is formulated as follows:

any non-free body can be considered as free, if we discard the connections, replacing them with reactions.

As a result of applying this principle, a body is obtained that is free from bonds and is under the action of a certain system of active and reactive forces.

1. Axioms of statics.

Conditions under which a body can be in equal Vesii, are derived from several basic provisions, accepted without evidence, but confirmed by experiments , and called axioms of statics. The basic axioms of statics were formulated by the English scientist Newton (1642-1727), and therefore they are named after him.

Axiom I (axiom of inertia or Newton's first law).

Any body retains its state of rest or rectilinear uniform motion, as long as some Forces will not bring him out of this state.

The ability of a body to maintain its state of rest or rectilinear uniform motion is called inertia. On the basis of this axiom, we consider the state of equilibrium to be such a state when the body is at rest or moves in a straight line and uniformly (i.e., the PO of inertia).

Axiom II (the axiom of interaction or Newton's third law).

If one body acts on the second with a certain force, then the second body simultaneously acts on the first with a force equal in magnitude to the opposite in direction.

The totality of forces applied to a given body (or system of bodies) is called force system. The force of action of a body on a given body and the force of reaction of a given body do not represent a system of forces, since they are applied to different bodies.

If some system of forces has such a property that, after being applied to a free body, it does not change its state of equilibrium, then such a system of forces is called balanced.

Axiom III (condition of balance of two forces).

For the equilibrium of a free rigid body under the action of two forces, it is necessary and sufficient that these forces be equal in absolute value and act in one straight line in opposite directions.

necessary to balance the two forces. This means that if the system of two forces is in equilibrium, then these forces must be equal in absolute value and act in one straight line in opposite directions.

The condition formulated in this axiom is sufficient to balance the two forces. This means that the reverse formulation of the axiom is true, namely: if two forces are equal in absolute value and act in the same straight line in opposite directions, then such a system of forces is necessarily in equilibrium.

In the following, we will get acquainted with the equilibrium condition, which will be necessary, but not sufficient for equilibrium.

Axiom IV.

The equilibrium of a rigid body will not be disturbed if a system of balanced forces is applied to it or removed.

Consequence from the axioms III and IV.

The equilibrium of a rigid body is not disturbed by the transfer of a force along its line of action.

Parallelogram axiom. This axiom is formulated as follows:

The resultant of two forces applied to body at one point, is equal in absolute value and coincides in direction with the diagonal of the parallelogram built on these forces, and is applied at the same point.

1. Connections, reactions of connections. Connection examples.

connections bodies that limit the movement of a given body in space are called. The force with which the body acts on the bond is called pressure; the force with which a bond acts on a body is called reaction. According to the axiom of interaction, the reaction and pressure modulo equal and act in the same straight line in opposite directions. Reaction and pressure are applied to different bodies. The external forces acting on the body are divided into active and reactive. Active forces tend to move the body to which they are applied, and reactive forces, through bonds, prevent this movement. The fundamental difference between active forces and reactive forces is that the magnitude of reactive forces, generally speaking, depends on the magnitude of active forces, but not vice versa. Active forces are often called

The direction of the reactions is determined by the direction in which this connection prevents the body from moving. The rule for determining the direction of reactions can be formulated as follows:

the direction of the reaction of the connection is opposite to the direction of the displacement destroyed by this connection.

1. Perfectly smooth plane

In this case, the reaction R directed perpendicular to the reference plane towards the body.

2. Ideally smooth surface (Fig. 16).

In this case, the reaction R is directed perpendicular to the tangent plane t - t, i.e., along the normal to the supporting surface towards the body.

3. Fixed point or corner edge (Fig. 17, edge B).

In this case, the reaction R in directed along the normal to the surface of an ideally smooth body towards the body.

4. Flexible connection (Fig. 17).

The reaction T of a flexible bond is directed along c to i s and. From fig. 17 it can be seen that the flexible connection, thrown over the block, changes the direction of the transmitted force.

5. Ideally smooth cylindrical hinge (Fig. 17, hinge BUT; rice. 18, bearing D).

In this case, it is only known in advance that the reaction R passes through the hinge axis and is perpendicular to this axis.

6. Perfectly smooth thrust bearing (Fig. 18, thrust bearing BUT).

The thrust bearing can be considered as a combination of a cylindrical hinge and a bearing plane. Therefore, we will

7. Perfectly smooth ball joint (Fig. 19).

In this case, it is only known in advance that the reaction R passes through the center of the hinge.

8. A rod fixed at both ends in ideally smooth hinges and loaded only at the ends (Fig. 18, rod BC).

In this case, the reaction of the rod is directed along the rod, since, according to axiom III, the reactions of the hinges B and C in equilibrium, the rod can only be directed along the line sun, i.e. along the rod.

1. System of converging forces. Addition of forces applied at one point.

converging called forces whose lines of action intersect at one point.

This chapter deals with systems of converging forces whose lines of action lie in the same plane (flat systems).

Imagine that a flat system of five forces acts on the body, the lines of action of which intersect at the point O (Fig. 10, a). In § 2 it was established that the force- sliding vector. Therefore, all forces can be transferred from the points of their application to the point O of the intersection of the lines of their action (Fig. 10, b).

Thus, any system of converging forces applied to different points of the body can be replaced by an equivalent system of forces applied to one point. This system of forces is often called bundle of forces.

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

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

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 constructs a mathematical description of motion and finds connections between the physical quantities that characterize 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 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 away; they also take place during the usual motion of macroscopic bodies at a speed much lower than the speed of light. Another thing is that the action of these effects is so small that it does not go beyond the most accurate measurements. Classical mechanics will thus never lose its fundamental importance.

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

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

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

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

Application of statics in dynamics

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

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

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

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

Material point statics

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

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

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

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

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

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

Rigid body statics

Moment of force about a point

Determining the moment of force

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

Geometric interpretation

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

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

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

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

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

Moment components in rectangular coordinates

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

Properties of the moment of force about the center

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

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

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

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

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

Power couple

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

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

Moment of force about a given axis

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

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

Properties of the moment of force about the axis

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

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

Calculation of the moment of force about an axis

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

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

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

Equilibrium conditions for a rigid body

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

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

The equilibrium conditions can be formulated in another way.

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

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

Center of gravity of the body

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

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

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

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

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

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

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

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

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

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

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

Friction forces

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

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

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

The course covers: kinematics of a point and a rigid body (and from different points of view it is proposed to consider the problem of orientation of a rigid body), classical problems of the dynamics of mechanical systems and the dynamics of a rigid body, elements of celestial mechanics, motion of systems of variable composition, impact theory, differential equations of analytical dynamics.

The course covers all the traditional sections of theoretical mechanics, but special attention is paid to the most meaningful and valuable for theory and applications sections of dynamics and methods of analytical mechanics; statics is studied as a section of dynamics, and in the section of kinematics, the concepts necessary for the section of dynamics and the mathematical apparatus are introduced in detail.

Informational resources

Gantmakher F.R. Lectures on Analytical Mechanics. - 3rd ed. – M.: Fizmatlit, 2001.
Zhuravlev V.F. Fundamentals of theoretical mechanics. - 2nd ed. - M.: Fizmatlit, 2001; 3rd ed. – M.: Fizmatlit, 2008.
Markeev A.P. Theoretical mechanics. - Moscow - Izhevsk: Research Center "Regular and Chaotic Dynamics", 2007.

Requirements

The course is designed for students who own the apparatus of analytical geometry and linear algebra in the scope of the first-year program of a technical university.

Course program

1. Kinematics of a point
1.1. Problems of kinematics. Cartesian coordinate system. Decomposition of a vector in an orthonormal basis. Radius vector and point coordinates. Point speed and acceleration. Trajectory of movement.
1.2. Natural triangular. Expansion of velocity and acceleration in the axes of a natural trihedron (Huygens' theorem).
1.3. Curvilinear point coordinates, examples: polar, cylindrical and spherical coordinate systems. Velocity components and projections of acceleration on the axes of a curvilinear coordinate system.

2. Methods for specifying the orientation of a rigid body
2.1. Solid. Fixed and body-bound coordinate systems.
2.2. Orthogonal rotation matrices and their properties. Euler's finite turn theorem.
2.3. Active and passive points of view on orthogonal transformation. Addition of turns.
2.4. Finite rotation angles: Euler angles and "airplane" angles. Expression of an orthogonal matrix in terms of finite rotation angles.

3. Spatial motion of a rigid body
3.1. Translational and rotational motion of a rigid body. Angular velocity and angular acceleration.
3.2. Distribution of velocities (Euler's formula) and accelerations (Rivals' formula) of points of a rigid body.
3.3. Kinematic invariants. Kinematic screw. Instant screw axle.

4. Plane-parallel motion
4.1. The concept of plane-parallel motion of the body. Angular velocity and angular acceleration in the case of plane-parallel motion. Instantaneous center of speed.

5. Complex motion of a point and a rigid body
5.1. Fixed and moving coordinate systems. Absolute, relative and figurative movement of a point.
5.2. The theorem on the addition of velocities in the case of a complex motion of a point, relative and figurative velocities of a point. The Coriolis theorem on the addition of accelerations for a complex motion of a point, relative, translational and Coriolis accelerations of a point.
5.3. Absolute, relative and portable angular velocity and angular acceleration of a body.

6. Motion of a rigid body with a fixed point (quaternion presentation)
6.1. The concept of complex and hypercomplex numbers. Algebra of quaternions. Quaternion product. Conjugate and inverse quaternion, norm and modulus.
6.2. Trigonometric representation of the unit quaternion. Quaternion method of specifying body rotation. Euler's finite turn theorem.
6.3. Relationship between quaternion components in different bases. Addition of turns. Rodrigues-Hamilton parameters.

7. Exam work

8. Basic concepts of dynamics.
8.1 Momentum, angular momentum (kinetic moment), kinetic energy.
8.2 Power of forces, work of forces, potential and total energy.
8.3 Center of mass (center of inertia) of the system. The moment of inertia of the system about the axis.
8.4 Moments of inertia about parallel axes; the Huygens–Steiner theorem.
8.5 Tensor and ellipsoid of inertia. Principal axes of inertia. Properties of axial moments of inertia.
8.6 Calculation of the angular momentum and kinetic energy of the body using the inertia tensor.

9. Basic theorems of dynamics in inertial and non-inertial frames of reference.
9.1 Theorem on the change in the momentum of the system in an inertial frame of reference. The theorem on the motion of the center of mass.
9.2 Theorem on the change in the angular momentum of the system in an inertial frame of reference.
9.3 Theorem on the change in the kinetic energy of the system in an inertial frame of reference.
9.4 Potential, gyroscopic and dissipative forces.
9.5 Basic theorems of dynamics in non-inertial frames of reference.

10. Movement of a rigid body with a fixed point by inertia.
10.1 Euler dynamic equations.
10.2 Euler case, first integrals of dynamical equations; permanent rotations.
10.3 Interpretations of Poinsot and Macculag.
10.4 Regular precession in the case of dynamic symmetry of the body.

11. Motion of a heavy rigid body with a fixed point.
11.1 General formulation of the problem of the motion of a heavy rigid body around.
fixed point. Euler dynamic equations and their first integrals.
11.2 Qualitative analysis of the motion of a rigid body in the case of Lagrange.
11.3 Forced regular precession of a dynamically symmetric rigid body.
11.4 The basic formula of gyroscopy.
11.5 The concept of the elementary theory of gyroscopes.

12. Dynamics of a point in the central field.
12.1 Binet's equation.
12.2 Orbit equation. Kepler's laws.
12.3 The scattering problem.
12.4 The problem of two bodies. Equations of motion. Area integral, energy integral, Laplace integral.

13. Dynamics of systems of variable composition.
13.1 Basic concepts and theorems on the change of basic dynamic quantities in systems of variable composition.
13.2 Movement of a material point of variable mass.
13.3 Equations of motion of a body of variable composition.

14. Theory of impulsive movements.
14.1 Basic concepts and axioms of the theory of impulsive movements.
14.2 Theorems about changing the basic dynamic quantities during impulsive motion.
14.3 Impulsive motion of a rigid body.
14.4 Collision of two rigid bodies.
14.5 Carnot's theorems.

15. Control work

Learning Outcomes

As a result of mastering the discipline, the student must:

• Know:
  • basic concepts and theorems of mechanics and the methods of studying the motion of mechanical systems arising from them;
• Be able to:
  • correctly formulate problems in terms of theoretical mechanics;
  • develop mechanical and mathematical models that adequately reflect the main properties of the phenomena under consideration;
  • apply the acquired knowledge to solve relevant specific problems;
• Own:
  • skills in solving classical problems of theoretical mechanics and mathematics;
  • the skills of studying the problems of mechanics and building mechanical and mathematical models that adequately describe a variety of mechanical phenomena;
  • skills in the practical use of methods and principles of theoretical mechanics in solving problems: force calculation, determining the kinematic characteristics of bodies with various methods of setting motion, determining the law of motion of material bodies and mechanical systems under the action of forces;
  • skills to independently master new information in the process of production and scientific activities, using modern educational and information technologies;