Mechanical movement. Material point

Trajectory description

It is customary to describe the trajectory of a material point using a radius vector, the direction, length and starting point of which depend on time. In this case, the curve described by the end of the radius vector in space can be represented in the form of conjugate arcs of varying curvature, generally located in intersecting planes. In this case, the curvature of each arc is determined by its radius of curvature, directed towards the arc from the instantaneous center of rotation, located in the same plane as the arc itself. Moreover, a straight line is considered as a limiting case of a curve, the radius of curvature of which can be considered equal to infinity. And therefore, in the general case, a trajectory can be represented as a set of conjugate arcs.

It is important that the shape of the trajectory depends on the reference system chosen to describe the movement of the material point. Thus, rectilinear motion in an inertial frame will generally be parabolic in a uniformly accelerating reference frame.

Relationship with speed and normal acceleration

The velocity of a material point is always directed tangent to the arc used to describe the point’s trajectory. In this case, there is a connection between the speed v, normal acceleration a n and the radius of curvature of the trajectory ρ at a given point:

Connection with equations of dynamics

Representation of a trajectory as a trace left by movement material point, connects the purely kinematic concept of trajectory, as a geometric problem, with the dynamics of the movement of a material point, that is, the problem of determining the causes of its movement. In fact, solving Newton's equations (in the presence of a complete set of initial data) gives the trajectory of a material point. And vice versa, knowing the trajectory of the material point in an inertial reference frame and its speed at each moment of time, you can determine the forces acting on it.

Trajectory of a free material point

In accordance with Newton's First Law, sometimes called the law of inertia, there must be a system in which a free body maintains (as a vector) its speed. Such a reference system is called inertial. The trajectory of such movement is a straight line, and the movement itself is called uniform and rectilinear.

Motion under the influence of external forces in an inertial reference frame

If in a known inertial system the speed of movement of an object with mass m changes in direction, even remaining the same in size, that is, the body turns and moves in an arc with a radius of curvature R, then the object experiences normal acceleration a n. The cause that causes this acceleration is a force directly proportional to this acceleration. This is the essence of Newton's Second Law:

Where is the vector sum of the forces acting on the body, its acceleration, and m- inertial mass.

In the general case, a body is not free in its movement, and its position, and in some cases its speed, are subject to restrictions - connections. If connections impose restrictions only on the coordinates of the body, then such connections are called geometric. If they also propagate at speed, then they are called kinematic. If the equation of a constraint can be integrated over time, then such a constraint is called holonomic.

The action of bonds on a system of moving bodies is described by forces called bond reactions. In this case, the force included in the left side of equation (1) is the vector sum of active (external) forces and the reaction of the connections.

It is significant that in the case of holonomic connections it becomes possible to describe the motion of mechanical systems in generalized coordinates included in the Lagrange equations. The number of these equations depends only on the number of degrees of freedom of the system and does not depend on the number of bodies included in the system, the position of which must be determined to fully describe the motion.

If the bonds operating in the system are ideal, that is, there is no transition of motion energy into other types of energy in them, then when solving the Lagrange equations, all unknown bond reactions are automatically eliminated.

Finally, if the acting forces belong to the class of potential forces, then with an appropriate generalization of concepts it becomes possible to use Lagrange’s equations not only in mechanics, but also in other areas of physics.

In this understanding, the forces acting on a material point unambiguously determine the shape of the trajectory of its movement (under known initial conditions). The converse statement is not true in the general case, since the same trajectory can take place with different combinations of active forces and coupling reactions.

Motion under the influence of external forces in a non-inertial reference frame

If the reference system is non-inertial (that is, it moves with a certain acceleration relative to the inertial reference system), then it is also possible to use expression (1), however, on the left side it is necessary to take into account the so-called inertial forces (including centrifugal force and Coriolis force associated with rotation of a non-inertial reference system).

Illustration

Trajectories of the same movement in different reference systems. At the top of the inertial frame, a leaky bucket of paint is carried in a straight line above a rotating stage. Below in non-inertial (paint trace for an observer standing on the stage)

As an example, consider a theater worker moving in the grate space above the stage in relation to the theater building evenly And straight forward and carrying over rotating stage with a leaky bucket of paint. It will leave a mark on it from falling paint in the form unwinding spiral(if moving from stage rotation center) and twisting- in the opposite case. At this time, his colleague, who is responsible for the cleanliness of the rotating stage and who is located on it, will therefore be forced to carry a leak-free bucket under the first one, constantly being under the first one. And its movement in relation to the building will also be uniform And straightforward, although in relation to the scene, which is non-inertial system, its movement will be twisted And uneven. Moreover, in order to counteract the drift in the direction of rotation, he must overcome the action of the Coriolis force by muscular effort, which his upper colleague above the stage does not experience, although the trajectories of both are in inertial system the theater buildings will represent straight lines.

But one can imagine that the task of the colleagues considered here is precisely to apply straight lines on rotating stage. In this case, the lower one must require the upper one to move along a curve that is a mirror image of the trace of the previously spilled paint. Hence, straight motion V non-inertial system countdown will not be such for the observer in an inertial frame.

Moreover, uniform body movement in one system, maybe uneven to another. So, two drops of paint that fell into different moments time from a leaky bucket, both in their own frame of reference and in the frame of the lower colleague stationary in relation to the building (on the stage that has already stopped rotating), will move in a straight line (towards the center of the Earth). The difference will be that for the lower observer this movement will be accelerated, and for his top colleague, if he stumbles, will fall, moving along with any of the drops, the distance between the drops will increase proportionally first degree time, that is, the mutual movement of drops and their observer in his accelerated the coordinate system will be uniform with speed v, determined by the delay Δ t between the moments of falling drops:

Where g- acceleration of gravity .

Therefore, the shape of the trajectory and the speed of movement of the body along it, considered in a certain frame of reference, about which nothing is known in advance, does not give an unambiguous idea of ​​the forces acting on the body. The question of whether this system is sufficiently inertial can be resolved only on the basis of an analysis of the causes of the appearance of the acting forces.

Thus, in a non-inertial frame:

• The curvature of the trajectory and/or the variability of the speed are insufficient arguments in favor of the statement that a body moving along it is acted upon by external forces, which in the final case can be explained by gravitational or electromagnetic fields.
• The straightness of the trajectory is an insufficient argument in favor of the statement that no forces act on a body moving along it.

Notes

Literature

• Newton I. Mathematical principles of natural philosophy. Per. and approx. A. N. Krylova. M.: Nauka, 1989
• Frisch S. A. and Timoreva A. V. Course of general physics, Textbook for physics-mathematics and physics-technical faculties of state universities, Volume I. M.: GITTL, 1957

Links

• http://av-physics.narod.ru/mechanics/trajectory.htm [ unreputable source?] Trajectory and displacement vector, section of a physics textbook

Section 1 MECHANICS

Chapter 1: BASIC KINEMATICS

Mechanical movement. Trajectory. Path and movement. Speed ​​addition

Mechanical body movement is called the change in its position in space relative to other bodies over time.

Mechanical movement of bodies studies Mechanics. The section of mechanics that describes the geometric properties of motion without taking into account the masses of bodies and acting forces is called kinematics .

Mechanical motion is relative. To determine the position of a body in space, you need to know its coordinates. To determine the coordinates of a material point, you must first of all select a reference body and associate a coordinate system with it.

Body of referencecalled a body relative to which the position of other bodies is determined. The reference body is chosen arbitrarily. It can be anything: Land, building, car, ship, etc.

The coordinate system, the reference body with which it is associated, and the indication of the time reference form frame of reference , relative to which the movement of the body is considered (Fig. 1.1).

A body whose dimensions, shape and structure can be neglected when studying a given mechanical movement is called material point . A material point can be considered a body whose dimensions are much smaller than the distances characteristic of the motion considered in the problem.

Trajectoryit is the line along which the body moves.

Depending on the type of trajectory, movements are divided into rectilinear and curvilinear

Pathis the length of the trajectory ℓ(m) ( fig.1.2)

The vector drawn from the initial position of the particle to its final position is called moving of this particle for a given time.

Unlike a path, displacement is not a scalar, but a vector quantity, since it shows not only how far, but also in what direction the body has moved during a given time.

Motion vector module(that is, the length of the segment that connects the starting and ending points of the movement) can be equal to the distance traveled or less than the distance traveled. But the displacement module can never be greater than the distance traveled. For example, if a car moves from point A to point B along a curved path, then the magnitude of the displacement vector is less than the distance traveled ℓ. The path and the module of displacement turn out to be equal only in one single case, when the body moves in a straight line.

Speedis a vector quantitative characteristic of body movement

average speed– this is a physical quantity equal to the ratio of the vector of movement of a point to the period of time

The direction of the average speed vector coincides with the direction of the displacement vector.

Instant speed, that is, the speed at a given moment in time is a vector physical quantity equal to the limit to which the average speed tends as the time interval Δt decreases infinitely.

The concept of a material point. Trajectory. Path and movement. Reference system. Speed ​​and acceleration during curved motion. Normal and tangential acceleration. Classification of mechanical movements.

Mechanics subject . Mechanics is a branch of physics devoted to the study of the laws of the simplest form of motion of matter - mechanical motion.

Mechanics consists of three subsections: kinematics, dynamics and statics.

Kinematics studies the movement of bodies without taking into account the reasons that cause it. It operates on such quantities as displacement, distance traveled, time, speed and acceleration.

Dynamics explores the laws and causes that cause the movement of bodies, i.e. studies the movement of material bodies under the influence of forces applied to them. The quantities force and mass are added to the kinematic quantities.

INstatics explore the conditions of equilibrium of a system of bodies.

Mechanical movement A body is called a change in its position in space relative to other bodies over time.

Material point - a body whose size and shape can be neglected under given conditions of motion, considering the mass of the body to be concentrated at a given point. The model of a material point is the simplest model of body motion in physics. A body can be considered a material point when its dimensions are much smaller than the characteristic distances in the problem.

To describe mechanical motion, it is necessary to indicate the body relative to which the motion is considered. An arbitrarily chosen stationary body in relation to which the movement of a given body is considered is called reference body .

Reference system - a reference body together with the coordinate system and clock associated with it.

Let us consider the movement of the material point M in a rectangular coordinate system, placing the origin of coordinates at point O.

The position of point M relative to the reference system can be specified not only using three Cartesian coordinates, but also using one vector quantity - the radius vector of point M drawn to this point from the origin of the coordinate system (Fig. 1.1). If are unit vectors (orts) of the axes of a rectangular Cartesian coordinate system, then

or the time dependence of the radius vector of this point

Three scalar equations (1.2) or their equivalent one vector equation (1.3) are called kinematic equations of motion of a material point .

Trajectory a material point is the line described in space by this point during its movement (the geometric location of the ends of the radius vector of the particle). Depending on the shape of the trajectory, rectilinear and curvilinear movements of the point are distinguished. If all parts of a point’s trajectory lie in the same plane, then the point’s movement is called flat.

Equations (1.2) and (1.3) define the trajectory of a point in the so-called parametric form. The role of the parameter is played by time t. Solving these equations together and excluding time t from them, we find the trajectory equation.

Length of the path of a material point is the sum of the lengths of all sections of the trajectory traversed by the point during the period of time under consideration.

Movement vector of a material point is a vector connecting the initial and final positions of the material point, i.e. increment of the radius vector of a point over the considered period of time

During rectilinear movement, the displacement vector coincides with the corresponding section of the trajectory. From the fact that movement is a vector, the law of independence of movements, confirmed by experience, follows: if a material point participates in several movements, then the resulting movement of the point is equal to the vector sum of its movements made by it during the same time in each of the movements separately

To characterize the motion of a material point, a vector physical quantity is introduced - speed , a quantity that determines both the speed of movement and the direction of movement at a given time.

Let a material point move along a curvilinear trajectory MN so that at time t it is in point M, and at time t in point N. The radius vectors of points M and N are respectively equal, and the arc length MN is equal (Fig. 1.3 ).

Average speed vector points in the time interval from t before t+Δ t is called the ratio of the increment of the radius vector of a point over this period of time to its value:

The average speed vector is directed in the same way as the displacement vector, i.e. along the chord MN.

Instantaneous speed or speed at a given time . If in expression (1.5) we go to the limit, tending to zero, then we obtain an expression for the speed vector of the m.t. at the moment of time t of its passage through the t.M trajectory.

In the process of decreasing the value, point N approaches t.M, and the chord MN, turning around t.M, in the limit coincides in the direction of the tangent to the trajectory at point M. Therefore the vectorand speedvmoving points are directed along a tangent trajectory in the direction of movement. The velocity vector v of a material point can be decomposed into three components directed along the axes of a rectangular Cartesian coordinate system.

From a comparison of expressions (1.7) and (1.8) it follows that the projection of the velocity of a material point on the axis of a rectangular Cartesian coordinate system is equal to the first time derivatives of the corresponding coordinates of the point:

Movement in which the direction of the velocity of a material point does not change is called rectilinear. If the numerical value of the instantaneous speed of a point remains unchanged during movement, then such movement is called uniform.

If, over arbitrary equal periods of time, a point traverses paths of different lengths, then the numerical value of its instantaneous speed changes over time. This type of movement is called uneven.

In this case, a scalar quantity is often used, called the average ground speed of uneven movement on a given section of the trajectory. It is equal to the numerical value of the speed of such a uniform movement, in which the same time is spent on traveling the path as for a given uneven movement:

Because only in the case of rectilinear motion with a constant speed in direction, then in the general case:

The distance traveled by a point can be represented graphically by the area of ​​the figure of the bounded curve v = f (t), straight t = t 1 And t = t 1 and the time axis on the speed graph.

Law of addition of speeds . If a material point simultaneously participates in several movements, then the resulting movements, in accordance with the law of independence of movement, are equal to the vector (geometric) sum of elementary movements caused by each of these movements separately:

According to definition (1.6):

Thus, the speed of the resulting movement is equal to the geometric sum of the speeds of all movements in which the material point participates (this position is called the law of addition of speeds).

When a point moves, the instantaneous speed can change both in magnitude and direction. Acceleration characterizes the speed of change in the magnitude and direction of the velocity vector, i.e. change in the magnitude of the velocity vector per unit time.

Average acceleration vector . The ratio of the speed increment to the time period during which this increment occurred expresses the average acceleration:

The vector of average acceleration coincides in direction with the vector.

Acceleration, or instantaneous acceleration equal to the limit of average acceleration as the time interval tends to zero:

In projections onto the corresponding axis coordinates:

During rectilinear motion, the velocity and acceleration vectors coincide with the direction of the trajectory. Let us consider the movement of a material point along a curvilinear flat trajectory. The velocity vector at any point of the trajectory is directed tangentially to it. Let us assume that in t.M of the trajectory the speed was , and in t.M 1 it became . At the same time, we believe that the time interval during the transition of a point on the path from M to M 1 is so small that the change in acceleration in magnitude and direction can be neglected. In order to find the velocity change vector, it is necessary to determine the vector difference:

To do this, let’s move it parallel to itself, combining its beginning with point M. The difference between the two vectors is equal to the vector connecting their ends and is equal to the side of the AS MAS, built on velocity vectors, as on the sides. Let us decompose the vector into two components AB and AD, and both respectively through and . Thus, the speed change vector is equal to the vector sum of two vectors:

Thus, the acceleration of a material point can be represented as the vector sum of the normal and tangential accelerations of this point

A-priory:

where is the ground speed along the trajectory, coinciding with the absolute value of the instantaneous speed at a given moment. The tangential acceleration vector is directed tangentially to the trajectory of the body.

Basic concepts of kinematics and kinematic characteristics

Human movement is mechanical, that is, it is a change in the body or its parts relative to other bodies. Relative movement is described by kinematics.

Kinematics – a branch of mechanics in which mechanical motion is studied, but the causes of this motion are not considered. The description of the movement of both the human body (its parts) in various sports and various sports equipment is an integral part of sports biomechanics and, in particular, kinematics.

Whatever material object or phenomenon we consider, it turns out that nothing exists outside of space and outside of time. Any object has spatial dimensions and shape, and is located in some place in space in relation to another object. Any process in which material objects participate has a beginning and an end in time, how long it lasts in time, and can occur earlier or later than another process. This is precisely why there is a need to measure spatial and temporal extent.

Basic units of measurement of kinematic characteristics in the international measurement system SI.

Space. One forty-millionth of the length of the earth's meridian passing through Paris was called a meter. Therefore, length is measured in meters (m) and its multiple units: kilometers (km), centimeters (cm), etc.

Time– one of the fundamental concepts. We can say that this is what separates two successive events. One way to measure time is to use any regularly repeated process. One eighty-six thousandth of an earthly day was chosen as a unit of time and was called the second (s) and its multiple units (minutes, hours, etc.).

In sports, special time characteristics are used:

Moment of time(t)- this is a temporary measure of the position of a material point, links of a body or system of bodies. Moments of time indicate the beginning and end of a movement or any part or phase of it.

Movement duration(∆t) – this is its temporary measure, which is measured by the difference between the moments of the end and the beginning of movement∆t = tcon. – tbeg.

Movement speed(N) – it is a temporal measure of the repetition of movements repeated per unit of time. N = 1/∆t; (1/s) or (cycle/s).

Rhythm of movements – this is a temporary measure of the relationship between parts (phases) of movements. It is determined by the ratio of the duration of the parts of the movement.

The position of a body in space is determined relative to a certain reference system, which includes a reference body (that is, relative to which the movement is considered) and a coordinate system necessary to describe at a qualitative level the position of the body in one or another part of space.

The beginning and direction of measurement are associated with the reference body. For example, in a number of competitions, the origin of coordinates can be chosen as the start position. Various competitive distances in all cyclic sports are already calculated from it. Thus, in the selected “start-finish” coordinate system, the distance in space that the athlete will move when moving is determined. Any intermediate position of the athlete’s body during movement is characterized by the current coordinate within the selected distance interval.

To accurately determine a sports result, the competition rules stipulate at what point (reference point) the count is taken: along the toe of a skater’s skate, at the protruding point of a sprinter’s chest, or along the back edge of the landing long jumper’s track.

In some cases, to accurately describe the movement of the laws of biomechanics, the concept of a material point is introduced.

Material point – this is a body whose dimensions and internal structure can be neglected under given conditions.

The movement of bodies can be different in nature and intensity. To characterize these differences, a number of terms are introduced in kinematics, presented below.

Trajectory – a line described in space by a moving point of a body. When biomechanical analysis of movements, first of all, the trajectories of movements of characteristic points of a person are considered. As a rule, such points are the joints of the body. Based on the type of movement trajectories, they are divided into rectilinear (straight line) and curvilinear (any line other than a straight line).

Moving – this is the vector difference between the final and initial position of the body. Therefore, displacement characterizes the final result of the movement.

Path – this is the length of the trajectory section traversed by a body or a point of the body during a selected period of time.

KINEMATICS OF A POINT

Introduction to Kinematics

Kinematics is a branch of theoretical mechanics that studies the motion of material bodies from a geometric point of view, regardless of the applied forces.

The position of a moving body in space is always determined in relation to any other unchanging body, called reference body. A coordinate system invariably associated with a reference body is called reference system. In Newtonian mechanics, time is considered absolute and not related to moving matter. In accordance with this, it proceeds identically in all reference systems, regardless of their motion. The basic unit of time is the second (s).

If the position of the body relative to the chosen frame of reference does not change over time, then it is said that body relative to a given frame of reference is at rest. If a body changes its position relative to the chosen reference system, then it is said to move relative to this system. A body can be at rest in relation to one reference system, but move (and in completely different ways) in relation to other reference systems. For example, a passenger sitting motionless on the bench of a moving train is at rest relative to the frame of reference associated with the car, but is moving with respect to the frame of reference associated with the Earth. A point lying on the rolling surface of the wheel moves in relation to the reference system associated with the car in a circle, and in relation to the reference system associated with the Earth, in a cycloid; the same point is at rest with respect to the coordinate system associated with the wheel pair.

Thus, the movement or rest of a body can be considered only in relation to any chosen frame of reference. Set the motion of a body relative to some reference system -means to give functional dependencies with the help of which one can determine the position of the body at any time relative to this system. Different points of the same body move differently in relation to the chosen reference system. For example, in relation to the system associated with the Earth, the tread point of the wheel moves along a cycloid, and the center of the wheel moves in a straight line. Therefore, the study of kinematics begins with the kinematics of a point.

§ 2. Methods for specifying the movement of a point

The movement of a point can be specified in three ways:natural, vector and coordinate.

With the natural way The movement assignment is given by a trajectory, i.e., a line along which the point moves (Fig. 2.1). On this trajectory, a certain point is selected, taken as the origin. The positive and negative directions of reference of the arc coordinate, which determines the position of the point on the trajectory, are selected. As the point moves, the distance will change. Therefore, to determine the position of a point at any time, it is enough to specify the arc coordinate as a function of time:

This equality is called equation of motion of a point along a given trajectory .

So, the movement of a point in the case under consideration is determined by a combination of the following data: the trajectory of the point, the position of the origin of the arc coordinate, the positive and negative directions of the reference and the function .

With the vector method of specifying the movement of a point, the position of the point is determined by the magnitude and direction of the radius vector drawn from the fixed center to a given point (Fig. 2.2). When a point moves, its radius vector changes in magnitude and direction. Therefore, to determine the position of a point at any time, it is enough to specify its radius vector as a function of time:

This equality is called vector equation of motion of a point .

With the coordinate method specifying the motion, the position of the point in relation to the selected reference system is determined using a rectangular Cartesian coordinate system (Fig. 2.3). When a point moves, its coordinates change over time. Therefore, to determine the position of a point at any time, it is enough to specify the coordinates , , as a function of time:

These equalities are called equations of motion of a point in rectangular Cartesian coordinates . The motion of a point in a plane is determined by two equations of system (2.3), rectilinear motion by one.

There is a mutual connection between the three described methods of specifying movement, which allows you to move from one method of specifying movement to another. This is easy to verify, for example, when considering the transition from the coordinate method of specifying movement to vector.

Let us assume that the motion of a point is given in the form of equations (2.3). Bearing in mind that

can be written down

And this is an equation of the form (2.2).

Task 2.1. Find the equation of motion and the trajectory of the middle point of the connecting rod, as well as the equation of motion of the slider of the crank-slider mechanism (Fig. 2.4), if ; .

Solution. The position of a point is determined by two coordinates and . From Fig. 2.4 it is clear that

, .

Then from and:

; ; .

Substituting values , and , we obtain the equations of motion of the point:

; .

To find the equation for the trajectory of a point in explicit form, it is necessary to exclude time from the equations of motion. For this purpose, we will carry out the necessary transformations in the equations of motion obtained above:

; .

By squaring and adding the left and right sides of these equations, we obtain the trajectory equation in the form

.

Therefore, the trajectory of the point is an ellipse.

The slider moves in a straight line. The coordinate , which determines the position of the point, can be written in the form

.

Speed ​​and acceleration

Point speed

In the previous article, the movement of a body or point is defined as a change in position in space over time. In order to more fully characterize the qualitative and quantitative aspects of motion, the concepts of speed and acceleration were introduced.

Velocity is a kinematic measure of the movement of a point, characterizing the speed of change of its position in space.
Velocity is a vector quantity, that is, it is characterized not only by its magnitude (scalar component), but also by its direction in space.

As is known from physics, with uniform motion, speed can be determined by the length of the path traveled per unit time: v = s/t = const (it is assumed that the origin of the path and time are the same).
During rectilinear motion, the speed is constant both in magnitude and direction, and its vector coincides with the trajectory.

Unit of speed in system SI is determined by the length/time ratio, i.e. m/s .

Obviously, with curvilinear movement, the speed of the point will change in direction.
In order to establish the direction of the velocity vector at each moment of time during curvilinear motion, we divide the trajectory into infinitesimal sections of the path, which can be considered (due to their smallness) rectilinear. Then at each section the conditional speed v p such a rectilinear motion will be directed along the chord, and the chord, in turn, with an infinite decrease in the length of the arc ( Δs tends to zero) will coincide with the tangent to this arc.
It follows from this that during curvilinear motion the velocity vector at each moment of time coincides with the tangent to the trajectory (Fig. 1a). Rectilinear motion can be represented as a special case of curvilinear motion along an arc whose radius tends to infinity (trajectory coincides with tangent).

When a point moves unevenly, the magnitude of its velocity changes over time.
Let's imagine a point whose movement is given in a natural way by the equation s = f(t) .

If in a short period of time Δt the point has passed the way Δs , then its average speed is:

vav = Δs/Δt.

Average speed does not give an idea of ​​the true speed at any given moment in time (true speed is also called instantaneous speed). Obviously, the shorter the time period for which the average speed is determined, the closer its value will be to the instantaneous speed.

True (instantaneous) speed is the limit to which the average speed tends as Δt tends to zero:

v = lim v av at t→0 or v = lim (Δs/Δt) = ds/dt.

Thus, the numerical value of the true speed is v = ds/dt .
The true (instantaneous) speed for any movement of a point is equal to the first derivative of the coordinate (i.e., the distance from the origin of the movement) with respect to time.

At Δt tending to zero, Δs also tends to zero, and, as we have already found out, the velocity vector will be directed tangentially (i.e., it coincides with the true velocity vector v ). It follows from this that the limit of the conditional speed vector v p , equal to the limit of the ratio of the point's displacement vector to an infinitesimal period of time, is equal to the vector of the point's true speed.

Fig.1

Let's look at an example. If a disk, without rotating, can slide along an axis fixed in a given reference system (Fig. 1, A), then in a given reference frame it obviously has only one degree of freedom - the position of the disk is uniquely determined, say, by the x coordinate of its center, measured along the axis. But if the disk, in addition, can also rotate (Fig. 1, b), then it acquires one more degree of freedom - to the coordinate x the rotation angle φ of the disk around the axis is added. If the axis with the disk is clamped in a frame that can rotate around a vertical axis (Fig. 1, V), then the number of degrees of freedom becomes equal to three - to x and φ the frame rotation angle is added ϕ .

A free material point in space has three degrees of freedom: for example, Cartesian coordinates x, y And z. The coordinates of a point can also be determined in cylindrical ( r, 𝜑, z) and spherical ( r, 𝜑, 𝜙) reference systems, but the number of parameters that uniquely determine the position of a point in space is always three.

A material point on a plane has two degrees of freedom. If we select a coordinate system in the plane xOy, then the coordinates x And y determine the position of a point on the plane, coordinate z is identically equal to zero.

A free material point on a surface of any kind has two degrees of freedom. For example: the position of a point on the Earth's surface is determined by two parameters: latitude and longitude.

A material point on a curve of any kind has one degree of freedom. The parameter that determines the position of a point on a curve can be, for example, the distance along the curve from the origin.

Consider two material points in space connected by a rigid rod of length l(Fig. 2). The position of each point is determined by three parameters, but a connection is imposed on them.

Fig.2

The equation l 2 =(x 2 -x 1) 2 +(y 2 -y 1) 2 +(z 2 -z 1) 2 is the coupling equation. From this equation, any one coordinate can be expressed in terms of the other five coordinates (five independent parameters). Therefore, these two points have (2∙3-1=5) five degrees of freedom.

Let us consider three material points in space that do not lie on the same straight line, connected by three rigid rods. The number of degrees of freedom of these points is (3∙3-3=6) six.

A free rigid body generally has 6 degrees of freedom. Indeed, the position of a body in space relative to any reference system is determined by specifying three of its points that do not lie on the same straight line, and the distances between points in a rigid body remain unchanged during any of its movements. According to the above, the number of degrees of freedom should be six.

Forward movement

In kinematics, as in statistics, we will consider all rigid bodies as absolutely rigid.

Absolutely solid body is a material body whose geometric shape and dimensions do not change under any mechanical influences from other bodies, and the distance between any two of its points remains constant.

Kinematics of a rigid body, as well as the dynamics of a rigid body, is one of the most difficult sections of the course in theoretical mechanics.

Rigid body kinematics problems fall into two parts:

1) setting the movement and determining the kinematic characteristics of the movement of the body as a whole;

2) determination of the kinematic characteristics of the movement of individual points of the body.

There are five types of rigid body motion:

1) forward movement;

2) rotation around a fixed axis;

3) flat movement;

4) rotation around a fixed point;

5) free movement.

The first two are called the simplest motions of a rigid body.

Let's start by considering the translational motion of a rigid body.

Progressive is the movement of a rigid body in which any straight line drawn in this body moves while remaining parallel to its initial direction.

Translational motion should not be confused with rectilinear motion. When a body moves forward, the trajectories of its points can be any curved lines. Let's give examples.

1. The car body on a straight horizontal section of the road moves forward. In this case, the trajectories of its points will be straight lines.

2. Sparnik AB(Fig. 3) when the cranks O 1 A and O 2 B rotate, they also move translationally (any straight line drawn in it remains parallel to its initial direction). The points of the partner move in circles.

Fig.3

The pedals of a bicycle move progressively relative to its frame during movement, the pistons in the cylinders of an internal combustion engine move relative to the cylinders, and the cabins of Ferris wheels in parks (Fig. 4) relative to the Earth.

Fig.4

The properties of translational motion are determined by the following theorem: during translational motion, all points of the body describe identical (overlapping, coinciding) trajectories and at each moment of time have the same magnitude and direction of velocity and acceleration.

To prove this, consider a rigid body undergoing translational motion relative to the reference frame Oxyz. Let's take two arbitrary points in the body A And IN, whose positions at the moment of time t are determined by radius vectors and (Fig. 5).

Fig.5

Let's draw a vector connecting these points.

In this case, the length AB constant, like the distance between points of a rigid body, and the direction AB remains unchanged as the body moves forward. So the vector AB remains constant throughout the body's movement ( AB=const). As a result, the trajectory of point B is obtained from the trajectory of point A by parallel displacement of all its points by a constant vector. Therefore, the trajectories of the points A And IN will actually be the same (when superimposed, coinciding) curves.

To find the velocities of points A And IN Let us differentiate both sides of the equality with respect to time. We get

But the derivative of a constant vector AB equal to zero. Derivatives of vectors and with respect to time give the velocities of points A And IN. As a result, we find that

those. what are the speeds of the points A And IN bodies at any moment of time are identical both in magnitude and direction. Taking derivatives with respect to time from both sides of the resulting equality:

Therefore, the accelerations of the points A And IN bodies at any moment of time are also identical in magnitude and direction.

Since the points A And IN were chosen arbitrarily, then from the results found it follows that for all points of the body their trajectories, as well as velocities and accelerations at any time, will be the same. Thus, the theorem is proven.

It follows from the theorem that the translational motion of a rigid body is determined by the movement of any one of its points. Consequently, the study of the translational motion of a body comes down to the problem of the kinematics of a point, which we have already considered.

During translational motion, the speed common to all points of the body is called the speed of translational motion of the body, and acceleration is called the acceleration of translational motion of the body. Vectors and can be depicted as applied at any point of the body.

Note that the concept of speed and acceleration of a body makes sense only in translational motion. In all other cases, the points of the body, as we will see, move with different speeds and accelerations, and the terms<<скорость тела>> or<<ускорение тела>> these movements lose their meaning.

Fig.6

During the time ∆t, the body, moving from point A to point B, makes a displacement equal to the chord AB and covers a path equal to the length of the arc l.

The radius vector rotates through an angle ∆φ. The angle is expressed in radians.

The speed of movement of a body along a trajectory (circle) is directed tangent to the trajectory. It is called linear speed. The modulus of linear velocity is equal to the ratio of the length of the circular arc l to the time interval ∆t during which this arc is passed:

A scalar physical quantity, numerically equal to the ratio of the angle of rotation of the radius vector to the period of time during which this rotation occurred, is called angular velocity:

The SI unit of angular velocity is radian per second.

With uniform motion in a circle, the angular velocity and the linear velocity module are constant values: ω=const; v=const.

The position of the body can be determined if the modulus of the radius vector and the angle φ it makes with the Ox axis (angular coordinate) are known. If at the initial moment of time t 0 =0 the angular coordinate is equal to φ 0, and at the moment of time t it is equal to φ, then the angle of rotation ∆φ of the radius vector during the time ∆t=t-t 0 is equal to ∆φ=φ-φ 0. Then from the last formula we can obtain the kinematic equation of motion of a material point in a circle:

It allows you to determine the position of the body at any time t.

Considering that , we get:

Formula for the relationship between linear and angular velocity.

The time period T during which the body makes one full revolution is called the period of rotation:

Where N is the number of revolutions made by the body during time Δt.

During the time ∆t=T the body travels the path l=2πR. Hence,

At ∆t→0, the angle is ∆φ→0 and, therefore, β→90°. The perpendicular to the tangent to the circle is the radius. Therefore, it is directed radially towards the center and is therefore called centripetal acceleration:

Module , direction changes continuously (Fig. 8). Therefore, this movement is not uniformly accelerated.

Fig.8

Fig.9

Then the position of the body at any moment of time is uniquely determined by the angle φ between these half-planes taken with the appropriate sign, which we will call the angle of rotation of the body. We will consider the angle φ positive if it is plotted from the fixed plane in a counterclockwise direction (for an observer looking from the positive end of the Az axis), and negative if it is clockwise. We will always measure the angle φ in radians. To know the position of the body at any moment in time, you need to know the dependence of the angle φ on time t, i.e.

The equation expresses the law of rotational motion of a rigid body around a fixed axis.

During rotational motion of an absolutely rigid body around a fixed axis the angles of rotation of the radius vector of different points of the body are the same.

The main kinematic characteristics of the rotational motion of a rigid body are its angular velocity ω and angular acceleration ε.

If during a period of time ∆t=t 1 -t the body rotates through an angle ∆φ=φ 1 -φ, then the numerically average angular velocity of the body during this period of time will be . In the limit at ∆t→0 we find that

Thus, the numerical value of the angular velocity of a body at a given time is equal to the first derivative of the angle of rotation with respect to time. The sign of ω determines the direction of rotation of the body. It is easy to see that when rotation occurs counterclockwise, ω>0, and when clockwise, then ω<0.

The dimension of angular velocity is 1/T (i.e. 1/time); the unit of measurement is usually rad/s or, which is the same, 1/s (s -1), since the radian is a dimensionless quantity.

The angular velocity of a body can be represented as a vector whose modulus is equal to | | and which is directed along the axis of rotation of the body in the direction from which the rotation can be seen occurring counterclockwise (Fig. 10). Such a vector immediately determines the magnitude of the angular velocity, the axis of rotation, and the direction of rotation around this axis.

Fig.10

The angle of rotation and angular velocity characterize the motion of the entire absolutely rigid body as a whole. The linear velocity of any point of an absolutely rigid body is proportional to the distance of the point from the axis of rotation:

With uniform rotation of an absolutely rigid body, the angles of rotation of the body for any equal periods of time are the same, there are no tangential accelerations at various points of the body, and the normal acceleration of a point of the body depends on its distance to the axis of rotation:

The vector is directed along the radius of the point's trajectory towards the axis of rotation.

Angular acceleration characterizes the change in the angular velocity of a body over time. If over a period of time ∆t=t 1 -t the angular velocity of a body changes by the amount ∆ω=ω 1 -ω, then the numerical value of the average angular acceleration of the body over this period of time will be . In the limit at ∆t→0 we find,

Thus, the numerical value of the angular acceleration of a body at a given time is equal to the first derivative of the angular velocity or the second derivative of the angle of rotation of the body with respect to time.

The dimension of angular acceleration is 1/T 2 (1/time 2); the unit of measurement is usually rad/s 2 or, what is the same, 1/s 2 (s-2).

If the module of angular velocity increases with time, the rotation of the body is called accelerated, and if it decreases, it is called slow. It is easy to see that the rotation will be accelerated when the quantities ω and ε have the same signs, and slowed down when they are different.

The angular acceleration of a body (by analogy with angular velocity) can also be represented as a vector ε directed along the axis of rotation. Wherein

The direction of ε coincides with the direction of ω when the body rotates at an accelerated rate (Fig. 10, a), and is opposite to ω when the body rotates at a slow speed (Fig. 10, b).

Fig.11 Fig. 12

2. Acceleration of body points. To find the acceleration of a point M let's use the formulas

In our case ρ=h. Substituting the value v into the expressions a τ and a n, we get:

or finally:

The tangential component of acceleration a τ is directed tangentially to the trajectory (in the direction of motion during accelerated rotation of the body and in the opposite direction during slow rotation); the normal component a n is always directed along the radius MS to the axis of rotation (Fig. 12). Total point acceleration M will

The deviation of the total acceleration vector from the radius of the circle described by the point is determined by the angle μ, which is calculated by the formula

Substituting the values ​​of a τ and a n here, we get

Since ω and ε have the same value for all points of the body at a given moment in time, the accelerations of all points of a rotating rigid body are proportional to their distances from the axis of rotation and form at a given moment in time the same angle μ with the radii of the circles they describe . The acceleration field of points of a rotating rigid body has the form shown in Fig. 14.

Fig.13 Fig.14

3. Vectors of velocity and acceleration of body points. To find expressions directly for vectors v and a, let’s draw from an arbitrary point ABOUT axes AB radius vector of a point M(Fig. 13). Then h=r∙sinα and by the formula

So I can

Ticket 1.

Kinematics. Mechanical movement. Material point and absolutely rigid body. Kinematics of a material point and translational motion of a rigid body. Trajectory, path, displacement, speed, acceleration.

Ticket 2.

Kinematics of a material point. Speed, acceleration. Tangential, normal and total acceleration.

Kinematics- a branch of physics that studies the movement of bodies without being interested in the reasons that determine this movement.

Mechanicś logical movement́ nie - this is a change in body position in space relative to other bodies over time. (mechanical motion is characterized by three physical quantities: displacement, speed and acceleration)

The characteristics of mechanical motion are related to each other by basic kinematic equations:

Material point- a body whose dimensions, in the conditions of this problem, can be neglected.

Absolutely rigid body- a body whose deformation can be neglected under the conditions of a given problem.

Kinematics of a material point and translational motion of a rigid body: ?

movement in a rectangular, curvilinear coordinate system

how to write in different coordinate systems using a radius vector

Trajectory - some line described by the movement of the mat. points.

Path - scalar quantity characterizing the length of the trajectory of the body.

Moving - a straight line segment drawn from the initial position of a moving point to its final position (vector quantity)

Speed:

A vector quantity that characterizes the speed of movement of a particle along the trajectory in which this particle moves at each moment of time.

Derivative of the particle vector radius with respect to time.

Derivative of displacement with respect to time.

Acceleration:

A vector quantity characterizing the rate of change of the velocity vector.

Derivative of speed with respect to time.

Tangential acceleration - directed tangentially to the trajectory. Is a component of the acceleration vector a. Characterizes the change in speed modulo.

Centripetal or Normal acceleration - occurs when a point moves in a circle. Is a component of the acceleration vector a. The normal acceleration vector is always directed towards the center of the circle.

The total acceleration is the square root of the sum of the squares of the normal and tangential accelerations.

Ticket 3

Kinematics of rotational motion of a material point. Angular values. Relationship between angular and linear quantities.

Kinematics of rotational motion of a material point.

Rotational movement is a movement in which all points of the body describe circles, the centers of which lie on the same straight line, called the axis of rotation.

The axis of rotation passes through the center of the body, through the body, or may be located outside it.

Rotational motion of a material point is the movement of a material point in a circle.

Main characteristics of the kinematics of rotational motion: angular velocity, angular acceleration.

Angular displacement is a vector quantity that characterizes the change in angular coordinates during its movement.

Angular velocity is the ratio of the angle of rotation of the radius vector of a point to the period of time during which this rotation occurred. (direction along the axis around which the body rotates)

Rotation frequency is a physical quantity measured by the number of full revolutions made by a point per unit time with uniform movement in one direction (n)

Rotation period - the period of time during which a point makes a full revolution,

moving in a circle (T)

N is the number of revolutions made by the body during time t.

Angular acceleration is a quantity characterizing the change in the angular velocity vector over time.

Relationship between angular and linear quantities:

Relationship between linear and angular speed.

Relationship between tangential and angular acceleration.

the relationship between normal (centripetal) acceleration, angular velocity and linear velocity.

Ticket 4.

Dynamics of a material point. Classical mechanics, the limits of its applicability. Newton's laws. Inertial reference systems.

Dynamics of a material point:

Newton's laws

Laws of conservation (momentum, angular momentum, energy)

Classical mechanics is a branch of physics that studies the laws of changes in the positions of bodies and the causes that cause them, based on Newton's laws and Galileo's principle of relativity.

Classical mechanics is divided into:

statics (which considers the balance of bodies)

kinematics (which studies the geometric property of motion without considering its causes)

dynamics (which considers the movement of bodies).

Limits of applicability of classical mechanics:

At speeds close to the speed of light, classical mechanics stops working

The properties of the microcosm (atoms and subatomic particles) cannot be understood within the framework of classical mechanics

Classical mechanics becomes ineffective when considering systems with very large numbers of particles

Newton's first law (law of inertia):

There are reference systems relative to which a material point, in the absence of external influences, is at rest or moves uniformly and rectilinearly.

Newton's second law:

In an inertial frame of reference, the product of the mass of a body and its acceleration is equal to the force acting on the body.

Newton's third law:

The forces with which interacting bodies act on each other are equal in magnitude and opposite in direction.

A reference system is a set of bodies that are not elevated relative to each other, in relation to which movements are considered (includes a reference body, a coordinate system, a clock)

An inertial reference system is a reference system in which the law of inertia is valid: any body that is not acted upon by external forces or the action of these forces is compensated is in a state of rest or uniform linear motion.

Inertia is a property inherent in bodies (it takes time to change the speed of a body).

Mass is a quantitative characteristic of inertia.

Ticket 5.

Center of mass (inertia) of the body. Momentum of a material point and a rigid body. Law of conservation of momentum. Movement of the center of mass.

The center of mass of a system of material points is a point whose position characterizes the distribution of the mass of the system in space.

distribution of masses in the coordinate system.

The position of the center of mass of a body depends on how its mass is distributed throughout the volume of the body.

The movement of the center of mass is determined only by external forces acting on the system. Internal forces of the system do not affect the position of the center of mass.

position of the center of mass.

The center of mass of a closed system moves in a straight line and uniformly or remains stationary.

The momentum of a material point is a vector quantity equal to the product of the mass of the point and its speed.

The momentum of a body is equal to the sum of the impulses of its individual elements.

Change in momentum mat. point is proportional to the applied force and has the same direction as the force.

Impulse of the mat system. points can only be changed by external forces, and the change in the momentum of the system is proportional to the sum of the external forces and coincides with it in direction. Internal forces, changing the impulses of individual bodies of the system, do not change the total impulse of the system.

Law of conservation of momentum:

if the sum of external forces acting on the body of the system is equal to zero, then the momentum of the system is conserved.

Ticket 6.

Work of force. Energy. Power. Kinetic and potential energy.Forces in nature.

Work is a physical quantity that characterizes the result of the action of a force and is numerically equal to the scalar product of the force vector and the displacement vector, completely under the influence of this force.

A = F S cosа (a-angle between the direction of force and the direction of movement)

No work is done if:

The force acts, but the body does not move

The body moves but the force is zero

The angle m/d by the force and displacement vectors is 90 degrees

Power is a physical quantity that characterizes the speed of work and is numerically equal to the ratio of work to the interval during which the work is performed.

Average power; instant power.

Power shows how much work is done per unit of time.

Energy is a scalar physical quantity, which is a single measure of various forms of motion of matter and a measure of the transition of the motion of matter from one form to another.

Mechanical energy is a quantity that characterizes the movement and interaction of bodies and is a function of the speeds and relative positions of bodies. It is equal to the sum of kinetic and potential energies.

A physical quantity equal to half the product of the mass of a body by the square of its speed is called the kinetic energy of the body.

Kinetic energy is the energy of motion.

A physical quantity equal to the product of the mass of a body by the acceleration modulus of gravity and the height to which the body is raised above the surface of the Earth is called the potential energy of interaction between the body and the Earth.

Potential energy is the energy of interaction.

A= – (Er2 – Er1).

1.Friction force.

Friction is one of the types of interaction between bodies. It occurs when two bodies come into contact. They arise due to the interaction between atoms and molecules of the bodies in contact. (Dry friction forces are the forces that arise when two solid bodies come into contact in the absence of a liquid or gaseous layer between them. The static friction force is always equal in magnitude to the external force and directed in the opposite direction. If the external force is greater than (Ftr)max, sliding friction occurs.)

μ is called the sliding friction coefficient.

2.Elasticity force. Hooke's law.

When a body is deformed, a force arises that strives to restore the previous size and shape of the body - the force of simplification.

(proportional to the deformation of the body and directed in the direction opposite to the direction of movement of body particles during deformation)

Fcontrol = –kx.

The coefficient k is called the rigidity of the body.

Tensile (x > 0) and compressive (x< 0).

Hooke's law: relative strain ε is proportional to stress σ, where E is Young's modulus.

3. Ground reaction force.

The elastic force acting on the body from the side of the support (or suspension) is called the support reaction force. When bodies come into contact, the support reaction force is directed perpendicular to the contact surface.

The weight of a body is the force with which the body, due to its attraction to the Earth, acts on a support or suspension.

4.Gravity. One of the manifestations of the force of universal gravity is the force of gravity.

5.Gravitational force (gravitational force)

All bodies are attracted to each other with a force directly proportional to their masses and inversely proportional to the square of the distance between them.

Ticket 7.

Conservative and dissipative forces. Law of conservation of mechanical energy. Equilibrium condition for a mechanical system.

Conservative forces (potential forces) - forces whose work does not depend on the shape of the trajectory (depends only on the starting and ending points of application of forces)

Conservative forces are those forces whose work along any closed trajectory is equal to 0.

The work done by conservative forces along an arbitrary closed contour is 0;

A force acting on a material point is called conservative or potential if the work done by this force when moving this point from an arbitrary position 1 to another 2 does not depend on the trajectory along which this movement occurred:

Changing the direction of movement of a point along a trajectory to the opposite causes a change in the sign of the conservative force, since the quantity changes sign. Therefore, when a material point moves along a closed trajectory, for example, the work done by the conservative force is zero.

Examples of conservative forces are the forces of universal gravitation, the forces of elasticity, and the forces of electrostatic interaction of charged bodies. A field whose work of forces in moving a material point along an arbitrary closed trajectory is equal to zero is called potential.

Dissipative forces are forces, under the action of which on a moving mechanical system, its total mechanical energy decreases, turning into other, non-mechanical forms of energy, for example into heat.

example of dissipative forces: the force of viscous or dry friction.

Law of conservation of mechanical energy:

The sum of kinetic and potential energy of bodies that make up a closed system and interact with each other through gravitational and elastic forces remains unchanged.

Ek1 + Ep1 = Ek2 + Ep2

A closed system is a system that is not affected by external forces or is compensated for.

Equilibrium condition for a mechanical system:

Statics is a branch of mechanics that studies the conditions of equilibrium of bodies.

For a non-rotating body to be in equilibrium, it is necessary that the resultant of all forces applied to the body be equal to zero.

If a body can rotate about a certain axis, then for its equilibrium it is not enough for the resultant of all forces to be zero.

Rule of moments: a body having a fixed axis of rotation is in equilibrium if the algebraic sum of the moments of all forces applied to the body relative to this axis is equal to zero: M1 + M2 + ... = 0.

The length of the perpendicular drawn from the axis of rotation to the line of action of the force is called the arm of the force.

The product of the force modulus F and the arm d is called the moment of force M. The moments of those forces that tend to rotate the body counterclockwise are considered positive.

Ticket 8.

Kinematics of rotational motion of a rigid body. Angular displacement, angular velocity, angular acceleration. Relationship between linear and angular characteristics. Kinetic energy of rotational motion.

For a kinematic description of the rotation of a rigid body, it is convenient to use angular quantities: angular displacement Δφ, angular velocity ω

In these formulas, angles are expressed in radians. When a rigid body rotates relative to a fixed axis, all its points move with the same angular velocities and the same angular accelerations. The positive direction of rotation is usually taken to be counterclockwise.

Rotational motion of a rigid body:

1) around an axis - movement in which all points of the body lying on the axis of rotation are motionless, and the remaining points of the body describe circles with centers on the axis;

2) around a point - the movement of a body in which one of its points O is stationary, and all others move along the surfaces of spheres with a center at point O.

Kinetic energy of rotational motion.

Kinetic energy of rotational motion is the energy of a body associated with its rotation.

Let us divide the rotating body into small elements Δmi. Let us denote the distances to the axis of rotation by ri, and the linear velocity modules by υi. Then the kinetic energy of the rotating body can be written as:

The physical quantity depends on the distribution of masses of the rotating body relative to the axis of rotation. It is called the moment of inertia I of the body relative to a given axis:

In the limit as Δm → 0, this sum goes into an integral.

Thus, the kinetic energy of a rigid body rotating about a fixed axis can be represented as:

The kinetic energy of rotational motion is determined by the moment of inertia of the body relative to the axis of rotation and its angular velocity.

Ticket 9.

Dynamics of rotational motion. Moment of power. Moment of inertia. Steiner's theorem.

The moment of force is a quantity that characterizes the rotational effect of a force when it acts on a solid body. A distinction is made between the moment of force relative to the center (point) and relative to the axis.

1. The moment of force relative to the center O is a vector quantity. Its modulus Mo = Fh, where F is the modulus of the force, and h is the arm (the length of the perpendicular lowered from O to the line of action of the force)

Using the vector product, the moment of force is expressed by the equality Mo =, where r is the radius vector drawn from O to the point of application of the force.

2. The moment of force relative to an axis is an algebraic quantity equal to the projection onto this axis.

Moment of force (torque; rotational moment; torque) is a vector physical quantity equal to the product of the radius vector drawn from the axis of rotation to the point of application of the force and the vector of this force.

this expression is Newton's second law for rotational motion.

It is only true then:

a) if by moment M we mean part of the moment of an external force, under the influence of which the body rotates around an axis - this is the tangential component.

b) the normal component of the moment of force does not participate in the rotational motion, since Mn tries to displace the point from the trajectory, and by definition is identically equal to 0, with r- const Mn=0, and Mz determines the pressure force on the bearings.

The moment of inertia is a scalar physical quantity, a measure of the inertia of a body in rotational motion around an axis, just as the mass of a body is a measure of its inertia in translational motion.

The moment of inertia depends on the mass of the body and on the location of the particles of the body relative to the axis of rotation.

Thin hoop Rod (fixed in the middle) Rod See

Homogeneous cylinder Disc Ball.

(on the right is the picture for point 2 in Steiner’s volume)

Steiner's theorem.

The moment of inertia of a given body relative to any given axis depends not only on the mass, shape and size of the body, but also on the position of the body relative to this axis.

According to the Huygens-Steiner theorem, the moment of inertia of a body J relative to an arbitrary axis is equal to the sum:

1) the moment of inertia of this body Jо, relative to the axis passing through the center of mass of this body, and parallel to the axis under consideration,

2) the product of body mass by the square of the distance between the axes.

Ticket 10.

Moment of impulse. Basic equation for the dynamics of rotational motion (equation of moments). Law of conservation of angular momentum.

Momentum is a physical quantity that depends on how much mass is rotating and how it is distributed relative to the axis of rotation and at what speed the rotation occurs.

The angular momentum relative to a point is a pseudovector.

Momentum about an axis is a scalar quantity.

The angular momentum L of a particle relative to a certain reference point is determined by the vector product of its radius vector and momentum: L=

r is the radius vector of the particle relative to the selected reference point that is stationary in a given reference frame.

P is the momentum of the particle.

L = rp sin A = p l;

For systems rotating around one of the axes of symmetry (generally speaking, around the so-called principal axes of inertia), the following relation is valid:

moment of momentum of a body relative to the axis of rotation.

The angular momentum of a rigid body relative to the axis is the sum of the angular momentum of the individual parts.

Equation of moments.

The time derivative of the angular momentum of a material point relative to a fixed axis is equal to the moment of force acting on the point relative to the same axis:

M=JE=J dw/dt=dL/dt

The law of conservation of angular momentum (the law of conservation of angular momentum) - the vector sum of all angular momentum relative to any axis for a closed system remains constant in the case of equilibrium of the system. In accordance with this, the angular momentum of a closed system relative to any fixed point does not change with time.

=> dL/dt=0 i.e. L=const

Work and kinetic energy during rotational motion. Kinetic energy in plane motion.

External force applied to a point of mass

The distance traveled by the mass in time dt

But is equal to the modulus of the moment of force relative to the axis of rotation.

hence

taking into account that

we get the expression for work:

The work of rotational motion is equal to the work expended on turning the entire body.

Work during rotational motion occurs by increasing kinetic energy:

Plane (plane-parallel) motion is a motion in which all its points move parallel to some fixed plane.

Kinetic energy during plane motion is equal to the sum of the kinetic energies of translational and rotational motion:

Ticket 12.

Harmonic vibrations. Free undamped oscillations. Harmonic oscillator. Differential equation of a harmonic oscillator and its solution. Characteristics of undamped oscillations. Velocity and acceleration in undamped oscillations.

Mechanical vibrations are movements of bodies that repeat exactly (or approximately) at equal intervals of time. The law of motion of a body oscillating is specified using a certain periodic function of time x = f (t).

Mechanical vibrations, like oscillatory processes of any other physical nature, can be free and forced.

Free vibrations are carried out under the influence of the internal forces of the system, after the system has been brought out of equilibrium. Oscillations of a weight on a spring or oscillations of a pendulum are free oscillations. Oscillations that occur under the influence of external periodically changing forces are called forced.

Harmonic oscillation is a phenomenon of periodic change of any quantity, in which the dependence on the argument has the character of a sine or cosine function.

Oscillations are called harmonic if the following conditions are met:

1) the pendulum oscillations continue indefinitely (since there are no irreversible energy transformations);

2) its maximum deviation to the right from the equilibrium position is equal to the maximum deviation to the left;

3) the time of deviation to the right is equal to the time of deviation to the left;

4) the nature of the movement to the right and to the left from the equilibrium position is the same.

X = Xm cos (ωt + φ0).

V= -A w o sin(w o + φ)=A w o cos(w o t+ φ+P/2)

a= -A w o *2 cos(w o t+ φ)= A w o *2 cos(w o t+ φ+P)

x – displacement of the body from the equilibrium position,

xm – amplitude of oscillations, i.e. maximum displacement from the equilibrium position,

ω – cyclic or circular vibration frequency,

t – time.

φ = ωt + φ0 is called the phase of the harmonic process

φ0 is called the initial phase.

The minimum time interval through which the movement of the body is repeated is called the period of oscillation T

The oscillation frequency f shows how many oscillations occur in 1 s.

Undamped oscillations are oscillations with a constant amplitude.

Damped oscillations are oscillations whose energy decreases over time.

Free undamped oscillations:

Let's consider the simplest mechanical oscillatory system - a pendulum in a non-viscous medium.

Let's write the equation of motion according to Newton's second law:

Let's write this equation in projections onto the x-axis. Let's represent the acceleration projection onto the x-axis as the second derivative of the x-coordinate with respect to time.

Let's denote k/m by w2, and give the equation the form:

Where

The solution to our equation is a function of the form:

A harmonic oscillator is a system that, when displaced from an equilibrium position, experiences a restoring force F proportional to the displacement x (according to Hooke's law):

k is a positive constant describing the rigidity of the system.

1.If F is the only force acting on the system, then the system is called a simple or conservative harmonic oscillator.

2. If there is also a frictional force (damping) proportional to the speed of movement (viscous friction), then such a system is called a damped or dissipative oscillator.

Differential equation of a harmonic oscillator and its solution:

As a model of a conservative harmonic oscillator, we take a load of mass m attached to a spring with stiffness k. Let x be the displacement of the load relative to the equilibrium position. Then, according to Hooke's law, a restoring force will act on it:

Using Newton's second law, we write:

Denoting and replacing acceleration with the second derivative of the coordinate with respect to time, we write:

This differential equation describes the behavior of a conservative harmonic oscillator. The coefficient ω0 is called the cyclic frequency of the oscillator.

We will look for a solution to this equation in the form:

Here is the amplitude, is the oscillation frequency (not necessarily equal to the natural frequency), and is the initial phase.

Substitute into the differential equation.

The amplitude is reduced. This means that it can have any value (including zero - this means that the load is at rest in the equilibrium position). You can also reduce by sine, since equality must be true at any time t. And the condition for the oscillation frequency remains:

The negative frequency can be discarded, since the arbitrariness in the choice of this sign is covered by the arbitrariness of the choice of the initial phase.

The general solution to the equation is written as:

where the amplitude A and the initial phase are arbitrary constants.

Kinetic energy is written as:

and there is potential energy

Characteristics of continuous oscillations:

Amplitude does not change

Frequency depends on stiffness and mass (spring)

Continuous oscillation speed:

Acceleration of continuous oscillations:

Ticket 13.

Free damped oscillations. Differential equation and its solution. Decrement, logarithmic decrement, damping coefficient. Relaxation time.

Free damped oscillations

If the forces of resistance to motion and friction can be neglected, then when the system is removed from the equilibrium position, only the elastic force of the spring will act on the load.

Let us write the equation of motion of the load, compiled according to Newton’s 2nd law:

Let's project the equation of motion onto the X axis.

transform:

because

this is a differential equation of free harmonic undamped oscillations.

The solution to the equation is:

Differential equation and its solution:

In any oscillatory system there are resistance forces, the action of which leads to a decrease in the energy of the system. If the loss of energy is not replenished by the work of external forces, the oscillations will die out.

The resistance force is proportional to the speed:

r is a constant value called the resistance coefficient. The minus sign is due to the fact that force and velocity have opposite directions.

The equation of Newton's second law in the presence of resistance forces has the form:

Using the notation , , we rewrite the equation of motion as follows:

This equation describes the damped oscillations of the system

The solution to the equation is:

The attenuation coefficient is a value inversely proportional to the time during which the amplitude decreased by e times.

The time after which the amplitude of oscillations decreases by a factor of e is called the damping time

During this time, the system oscillates.

The damping decrement, a quantitative characteristic of the speed of damping of oscillations, is the natural logarithm of the ratio of two subsequent maximum deviations of the oscillating value in the same direction.

The logarithmic damping decrement is the logarithm of the ratio of amplitudes at the moments of successive passages of an oscillating quantity through a maximum or minimum (the attenuation of oscillations is usually characterized by a logarithmic damping decrement):

It is related to the number of oscillations N by the relation:

Relaxation time is the time during which the amplitude of a damped oscillation decreases by a factor of e.

Ticket 14.

Forced vibrations. Complete differential equation of forced oscillations and its solution. Period and amplitude of forced oscillations.

Forced oscillations are oscillations that occur under the influence of external forces that change over time.

Newton's second law for the oscillator (pendulum) will be written as:

If

and replace the acceleration with the second derivative of the coordinate with respect to time, we obtain the following differential equation:

General solution of the homogeneous equation:

where A,φ are arbitrary constants

Let's find a particular solution. Let's substitute a solution of the form: into the equation and get the value for the constant:

Then the final solution will be written as:

The nature of forced oscillations depends on the nature of the action of the external force, on its magnitude, direction, frequency of action and does not depend on the size and properties of the oscillating body.

Dependence of the amplitude of forced oscillations on the frequency of the external force.

Period and amplitude of forced oscillations:

The amplitude depends on the frequency of forced oscillations; if the frequency is equal to the resonant frequency, then the amplitude is maximum. It also depends on the attenuation coefficient; if it is equal to 0, then the amplitude is infinite.

The period is related to the frequency; forced oscillations can have any period.

Ticket 15.

Forced vibrations. Period and amplitude of forced oscillations. Oscillation frequency. Resonance, resonant frequency. Family of resonance curves.

Ticket 14.

When the frequency of the external force and the frequency of the body’s own vibrations coincide, the amplitude of the forced vibrations increases sharply. This phenomenon is called mechanical resonance.

Resonance is the phenomenon of a sharp increase in the amplitude of forced oscillations.

An increase in amplitude is only a consequence of resonance, and the reason is the coincidence of the external frequency with the internal frequency of the oscillatory system.

Resonant frequency – the frequency at which the amplitude is maximum (slightly less than the natural frequency)

The graph of the amplitude of forced oscillations versus the frequency of the driving force is called a resonance curve.

Depending on the damping coefficient, we obtain a family of resonance curves; the lower the coefficient, the smaller the curve, the larger and higher it is.

Ticket 16.

Addition of oscillations of one direction. Vector diagram. Beating.

The addition of several harmonic oscillations of the same direction and the same frequency becomes clear if the oscillations are depicted graphically as vectors on a plane. The diagram obtained in this way is called a vector diagram.

Consider the addition of two harmonic oscillations of the same direction and the same frequency:

Let's represent both vibrations using vectors A1 and A2. Using the rules of vector addition, we construct the resulting vector A; the projection of this vector onto the x-axis is equal to the sum of the projections of the vectors being added:

Therefore, vector A represents the resulting oscillation. This vector rotates with the same angular velocity as vectors A1 and A2, so the sum of x1 and x2 is a harmonic oscillation with the same frequency, amplitude and phase. Using the cosine theorem, we find that

Representing harmonic oscillations using vectors allows you to replace the addition of functions with the addition of vectors, which is much simpler.

Beats are oscillations with periodically changing amplitude, resulting from the superposition of two harmonic oscillations with slightly different, but similar frequencies.

Ticket 17.

Addition of mutually perpendicular vibrations. Relationship between angular velocity of rotational motion and cyclic frequency. Lissajous figures.

Addition of mutually perpendicular vibrations:

Oscillations in two mutually perpendicular directions occur independently of each other:

Here the natural frequencies of harmonic oscillations are equal:

Let's consider the trajectory of cargo movement:

during the transformations we get:

Thus, the load will make periodic movements along an elliptical path. The direction of movement along the trajectory and the orientation of the ellipse relative to the axes depend on the initial phase difference

If the frequencies of two mutually perpendicular oscillations do not coincide, but are multiples, then the trajectories of motion are closed curves called Lissajous figures. Note that the ratio of oscillation frequencies is equal to the ratio of the numbers of points of contact of the Lissajous figure to the sides of the rectangle in which it is inscribed.

Ticket 18.

Oscillations of a load on a spring. Mathematical and physical pendulum. Characteristics of vibrations.

In order for free vibrations to occur according to the harmonic law, it is necessary that the force tending to return the body to the equilibrium position be proportional to the displacement of the body from the equilibrium position and directed in the direction opposite to the displacement.

F (t) = ma (t) = –m ω2 x (t)

Fpr = –kx Hooke’s law.

The circular frequency ω0 of free oscillations of a load on a spring is found from Newton’s second law:

The frequency ω0 is called the natural frequency of the oscillatory system.

Therefore, Newton’s second law for a load on a spring can be written as:

The solution to this equation is harmonic functions of the form:

x = xm cos (ωt + φ0).

If the load, which was in the equilibrium position, was given an initial speed with the help of a sharp push

A mathematical pendulum is an oscillator, which is a mechanical system consisting of a material point suspended on a weightless inextensible thread or on a weightless rod in a gravitational field. The period of small oscillations of a mathematical pendulum of length l in a gravitational field with free fall acceleration g is equal to

and depends little on the amplitude and mass of the pendulum.

A physical pendulum is an oscillator, which is a solid body that oscillates in a field of any forces relative to a point that is not the center of mass of this body, or a fixed axis perpendicular to the direction of action of the forces and not passing through the center of mass of this body

Ticket 19.

Wave process. Elastic waves. Longitudinal and transverse waves. Plane wave equation. Phase speed. Wave equation and its solution.

A wave is a phenomenon of disturbance of a physical quantity propagating in space over time.

Depending on the physical medium in which the waves propagate, there are:

Waves on the surface of a liquid;

Elastic waves (sound, seismic waves);

Body waves (propagating through the medium);

Electromagnetic waves (radio waves, light, x-rays);

Gravitational waves;

Waves in plasma.

In relation to the direction of vibration of the particles of the medium:

Longitudinal waves (compression waves, P-waves) - particles of the medium oscillate parallel (along) the direction of wave propagation (as, for example, in the case of sound propagation);

Transverse waves (shear waves, S-waves) - particles of the medium oscillate perpendicular to the direction of propagation of the wave (electromagnetic waves, waves on the separation surfaces of media);

Mixed waves.

According to the type of wave front (surface of equal phases):

Plane wave - phase planes are perpendicular to the direction of wave propagation and parallel to each other;

Spherical wave - the surface of the phases is a sphere;

Cylindrical wave - the surface of the phases resembles a cylinder.

Elastic waves (sound waves) are waves propagating in liquid, solid and gaseous media due to the action of elastic forces.

Transverse waves are waves propagating in a direction perpendicular to the plane in which the displacements and vibrational velocities of particles are oriented.

Longitudinal waves, waves whose direction of propagation coincides with the direction of displacement of particles of the medium.

Plane wave, a wave in which all points lying in any plane perpendicular to the direction of its propagation at each moment correspond to the same displacements and velocities of particles of the medium

Plane wave equation:

Phase velocity is the speed of movement of a point with a constant phase of oscillatory motion in space along a given direction.

The geometric location of the points to which the oscillations reach at time t is called the wave front.

The geometric location of points oscillating in the same phase is called a wave surface.

Wave equation and its solution:

The propagation of waves in a homogeneous isotropic medium is generally described by the wave equation - a partial differential equation.

Where

The solution to the equation is the equation of any wave, which has the form:

Ticket 20.

Transfer of energy by a traveling wave. Vector Umov. Addition of waves. Superposition principle. Standing wave.

A wave is a change in the state of a medium, propagating in this medium and carrying energy with it. (a wave is a spatial alternation of maxima and minima of any physical quantity that changes over time, for example, the density of a substance, electric field strength, temperature)

A traveling wave is a wave disturbance that changes in time t and space z according to the expression:

where is the amplitude envelope of the wave, K is the wave number and is the oscillation phase. The phase speed of this wave is given by

where is the wavelength.

Energy transfer - the elastic medium in which the wave propagates has both the kinetic energy of the vibrational motion of particles and the potential energy caused by the deformation of the medium.

A traveling wave, when propagating through a medium, transfers energy (unlike a standing wave).

Standing wave - oscillations in distributed oscillatory systems with a characteristic arrangement of alternating maxima (antinodes) and minima (nodes) of amplitude. In practice, such a wave occurs when reflected from obstacles and inhomogeneities as a result of the superposition of the reflected wave on the incident one. In this case, the frequency, phase and attenuation coefficient of the wave at the place of reflection are extremely important. Examples of a standing wave include vibrations of a string, vibrations of air in an organ pipe

The Umov (Umov-Poynting) vector is the vector of the energy flux density of the physical field; is numerically equal to the energy transferred per unit time through a unit area perpendicular to the direction of energy flow at a given point.

The principle of superposition is one of the most general laws in many branches of physics.

In its simplest formulation, the principle of superposition states: the result of the action of several external forces on a particle is simply the sum of the results of the influence of each of the forces.

The principle of superposition can also take other formulations, which, we emphasize, are completely equivalent to the one given above:

The interaction between two particles does not change when a third particle is introduced, which also interacts with the first two.

The interaction energy of all particles in a many-particle system is simply the sum of the energies of pairwise interactions between all possible pairs of particles. There are no many-particle interactions in the system.

The equations describing the behavior of a many-particle system are linear in the number of particles.

Addition of waves - addition of oscillations at each point.

The addition of standing waves is the addition of two identical waves propagating in different directions.

Ticket 21.

Inertial and non-inertial reference systems. Galileo's principle of relativity.

Inertial- such reference systems in which the body, which is not acted upon by forces, or they are balanced, is at rest or moves uniformly and rectilinearly

Non-inertial reference frame- an arbitrary reference system that is not inertial. Examples of non-inertial reference systems: a system moving rectilinearly with constant acceleration, as well as a rotating system

The principle of relativity Galilee- a fundamental physical principle according to which all physical processes in inertial reference systems proceed in the same way, regardless of whether the system is stationary or in a state of uniform and rectilinear motion.

It follows that all laws of nature are the same in all inertial frames of reference.

Ticket 22.

Physical foundations of molecular kinetic theory. Basic gas laws. Equation of state of an ideal gas. Basic equation of molecular kinetic theory.

Molecular kinetic theory (abbreviated as MKT) is a theory that considers the structure of matter, mainly gases, from the point of view of three main approximately correct provisions:

all bodies consist of particles whose size can be neglected: atoms, molecules and ions;

particles are in continuous chaotic motion (thermal);

particles interact with each other through absolutely elastic collisions.

The main evidence for these provisions was considered:

Diffusion

Brownian motion

Changes in aggregate states of matter

Clapeyron-Mendeleev equation - a formula establishing the relationship between pressure, molar volume and absolute temperature of an ideal gas.

PV = υRT υ = m/μ

The Boyle-Mariotte law states:

At constant temperature and mass of an ideal gas, the product of its pressure and volume is constant

pV= const,

Where p- gas pressure; V- gas volume

Gay Lussac - V / T= const

Charles - P / T= const

Boyle - Mariotta – PV= const

Avogadro's law is one of the important fundamental principles of chemistry, which states that “equal volumes of different gases, taken at the same temperature and pressure, contain the same number of molecules.”

Corollary from Avogadro's law: one mole of any gas under the same conditions occupies the same volume.

In particular, under normal conditions, i.e. at 0 ° C (273 K) and 101.3 kPa, the volume of 1 mole of gas is 22.4 l/mol. This volume is called the molar volume of gas V m

Dalton's laws:

Law on the total pressure of a mixture of gases - The pressure of a mixture of chemically non-interacting ideal gases is equal to the sum of the partial pressures

Ptot = P1 + P2 + … + Pn

Law on the solubility of gas mixture components - At a constant temperature, the solubility in a given liquid of each of the components of the gas mixture located above the liquid is proportional to their partial pressure

Both Dalton's laws are strictly satisfied for ideal gases. For real gases, these laws are applicable provided that their solubility is low and their behavior is close to that of an ideal gas.

Equation of states of an ideal gas - see Clapeyron - Mendeleev equation PV = υRT υ = m/μ

The basic equation of molecular kinetic theory (MKT) is

= (i/2) * kT where k is the Boltzmann constant - the ratio of the gas constant R to Avogadro's number, and i- number of degrees of freedom of molecules.

Basic equation of molecular kinetic theory. Gas pressure on the wall. Average energy of molecules. Law of equidistribution. Number of degrees of freedom.

Gas pressure on the wall - During their movement, molecules collide with each other, as well as with the walls of the vessel in which the gas is located. There are many molecules in a gas, so the number of their impacts is very large. Although the impact force of an individual molecule is small, the effect of all molecules on the walls of the vessel is significant, and it creates gas pressure

Average energy of a molecule –

The average kinetic energy of gas molecules (per one molecule) is determined by the expression

Ek= ½ m

The kinetic energy of the translational motion of atoms and molecules, averaged over a huge number of randomly moving particles, is a measure of what is called temperature. If the temperature T is measured in degrees Kelvin (K), then its relationship with E k is given by the relation

The law of equipartition is a law of classical statistical physics, which states that for a statistical system in a state of thermodynamic equilibrium, for each translational and rotational degree of freedom there is an average kinetic energy kT/2, and for each vibrational degree of freedom - the average energy kT(Where T - absolute temperature of the system, k - Boltzmann constant).

The equipartition theorem states that in thermal equilibrium, energy is divided equally between its different forms

The number of degrees of freedom is the smallest number of independent coordinates that determine the position and configuration of the molecule in space.

The number of degrees of freedom for a monatomic molecule is 3 (translational motion in the direction of three coordinate axes), for diatomic - 5 (three translational and two rotational, since rotation around the X axis is possible only at very high temperatures), for triatomic - 6 (three translational and three rotational).

Ticket 24.

Elements of classical statistics. Distribution functions. Maxwell distribution by absolute value of velocities.

Ticket 25.

Maxwell distribution by absolute value of velocity. Finding the characteristic velocities of molecules.

Elements of classical statistics:

A random variable is a quantity that, as a result of experiment, takes on one of many values, and the appearance of one or another value of this quantity cannot be accurately predicted before its measurement.

A continuous random variable (CRV) is a random variable that can take all values ​​from some finite or infinite interval. The set of possible values ​​of a continuous random variable is infinite and uncountable.

The distribution function is the function F(x), which determines the probability that the random variable X as a result of the test will take a value less than x.

Distribution function is the probability density of the distribution of particles of a macroscopic system over coordinates, momenta or quantum states. The distribution function is the main characteristic of a wide variety of (not only physical) systems that are characterized by random behavior, i.e. random change in the state of the system and, accordingly, its parameters.

Maxwell distribution by absolute value of velocities:

Gas molecules constantly collide as they move. The speed of each molecule upon collision changes. It can increase and decrease. However, the RMS speed remains unchanged. This is explained by the fact that in a gas at a certain temperature, a certain stationary velocity distribution of molecules that does not change over time is established, which obeys a certain statistical law. The speed of an individual molecule may change over time, but the proportion of molecules with speeds in a certain speed range remains unchanged.

Graph of the ratio of the fraction of molecules to the speed interval Δv i.e. .

In practice, the graph is described by the velocity distribution function of molecules or Maxwell’s law:

Derived formula:

When the gas temperature changes, the speed of movement of all molecules will change, and, consequently, the most probable speed. Therefore, the maximum of the curve will shift to the right as the temperature increases and to the left as the temperature decreases.

The height of the maximum changes with temperature changes. The fact that the distribution curve begins at the origin means that there are no stationary molecules in the gas. From the fact that the curve asymptotically approaches the x-axis at infinitely high speeds, it follows that there are few molecules with very high speeds.

Ticket 26.

Boltzmann distribution. Maxwell-Boltzmann distribution. Boltzmann's barometric formula.

Boltzmann distribution is the energy distribution of particles (atoms, molecules) of an ideal gas under conditions of thermodynamic equilibrium.

Boltzmann distribution law:

where n is the concentration of molecules at height h,

n0 – concentration of molecules at the initial level h = 0,

m – mass of particles,

g – free fall acceleration,

k – Boltzmann constant,

T – temperature.

Maxwell-Boltzmann distribution:

equilibrium distribution of ideal gas particles by energy (E) in an external force field (for example, in a gravitational field); determined by the distribution function:

where E is the sum of the kinetic and potential energies of the particle,

T - absolute temperature,

k - Boltzmann constant

The barometric formula is the dependence of the pressure or density of a gas on the height in the gravitational field. For an ideal gas that has a constant temperature T and is located in a uniform gravitational field (at all points of its volume the acceleration of gravity g is the same), the barometric formula has the following form:

where p is the gas pressure in the layer located at height h,

p0 - pressure at zero level (h = h0),

M is the molar mass of the gas,

R - gas constant,

T - absolute temperature.

From the barometric formula it follows that the concentration of molecules n (or gas density) decreases with altitude according to the same law:

where m is the mass of a gas molecule, k is Boltzmann’s constant.

Ticket 27.

The first law of thermodynamics. Work and warmth. Processes. Work done by gas in various isoprocesses. The first law of thermodynamics in various processes. Formulations of the first principle.

Ticket 28.

Internal energy of an ideal gas. Heat capacity of an ideal gas at constant volume and constant pressure. Mayer's equation.

The first law of thermodynamics - one of the three basic laws of thermodynamics, is the law of conservation of energy for thermodynamic systems

There are several equivalent formulations of the first law of thermodynamics:

1) The amount of heat received by the system goes to change its internal energy and perform work against external forces

2) The change in the internal energy of a system during its transition from one state to another is equal to the sum of the work of external forces and the amount of heat transferred to the system and does not depend on the method in which this transition is carried out

3) The change in the total energy of the system in a quasi-static process is equal to the amount of heat Q, communicated to the system, in sum with the change in energy associated with the amount of matter N at chemical potential μ, and work A"performed on the system by external forces and fields, minus the work A committed by the system itself against external forces

ΔU = Q - A + μΔΝ + A`

An ideal gas is a gas in which it is assumed that the potential energy of the molecules can be neglected compared to their kinetic energy. There are no forces of attraction or repulsion between molecules, collisions of particles with each other and with the walls of the vessel are absolutely elastic, and the interaction time between molecules is negligible compared to the average time between collisions.

Work - When expanding, the work of a gas is positive. When compressed, it is negative. Thus:

A" = pDV - gas work (A" - gas expansion work)

A= - pDV - work of external forces (A - work of external forces on gas compression)

Heat-kinetic part of the internal energy of a substance, determined by the intense chaotic movement of the molecules and atoms of which this substance consists.

The heat capacity of an ideal gas is the ratio of the heat imparted to the gas to the temperature change δT that occurred.

The internal energy of an ideal gas is a quantity that depends only on its temperature and does not depend on volume.

Mayer's equation shows that the difference in the heat capacities of a gas is equal to the work done by one mole of an ideal gas when its temperature changes by 1 K, and explains the meaning of the universal gas constant R.

For any ideal gas, Mayer's relation is valid:

,

Processes:

An isobaric process is a thermodynamic process occurring in a system at constant pressure.

The work done by the gas during expansion or compression of the gas is equal to

Work done by gas during expansion or compression of gas:

The amount of heat received or given off by the gas:

at a constant temperature dU = 0, therefore the entire amount of heat imparted to the system is spent on doing work against external forces.

Heat capacity:

Ticket 29.

Adiabatic process. Adiabatic equation. Poisson's equation. Work in an adiabatic process.

An adiabatic process is a thermodynamic process in a macroscopic system in which the system neither receives nor releases thermal energy.

For an adiabatic process, the first law of thermodynamics, due to the absence of heat exchange between the system and the environment, has the form:

In an adiabatic process, heat exchange with the environment does not occur, i.e. δQ=0. Consequently, the heat capacity of an ideal gas in an adiabatic process is also zero: Sadiab=0.

Work is done by the gas due to changes in internal energy Q=0, A=-DU

In an adiabatic process, the gas pressure and its volume are related by the relation:

pV*g=const, where g= Cp/Cv.

In this case, the following relations are valid:

p2/p1=(V1/V2)*g, *g-degree

T2/T1=(V1/V2)*(g-1), *(g-1)-degree

T2/T1=(p2/p1)*(g-1)/g. *(g-1)/g -degree

The given relations are called Poisson’s equations

equation of the adiabatic process. (Poisson equation) g - adiabatic exponent

Ticket 30.

Second law of thermodynamics. Carnot cycle. Efficiency of an ideal heat engine. Entropy and thermodynamic probability. Various formulations of the second law of thermodynamics.

The second law of thermodynamics is a physical principle that imposes restrictions on the direction of heat transfer processes between bodies.

The second law of thermodynamics states that spontaneous transfer of heat from a less heated body to a more heated body is impossible.

The second law of thermodynamics prohibits the so-called perpetual motion machines of the second kind, showing the impossibility of converting all the internal energy of the system into useful work.

The second law of thermodynamics is a postulate that cannot be proven within the framework of thermodynamics. It was created on the basis of a generalization of experimental facts and received numerous experimental confirmations.

Clausius's postulate: “A process is impossible, the only result of which would be the transfer of heat from a colder body to a hotter one”(this process is called Clausius process).

Thomson's postulate: “A circular process is impossible, the only result of which would be the production of work by cooling the thermal reservoir”(this process is called Thomson process).

The Carnot cycle is an ideal thermodynamic cycle.

A Carnot heat engine operating in this cycle has the highest efficiency of all machines in which the maximum and minimum temperatures of the cycle being carried out coincide, respectively, with the maximum and minimum temperatures of the Carnot cycle.

The Carnot cycle consists of four stages:

1.Isothermal expansion (in the figure - process A→B). At the beginning of the process, the working fluid has a temperature Tn, that is, the temperature of the heater. The body is then brought into contact with a heater, which transfers an amount of heat QH to it isothermally (at a constant temperature). At the same time, the volume of the working fluid increases.

2. Adiabatic (isentropic) expansion (in the figure - process B→C). The working fluid is disconnected from the heater and continues to expand without heat exchange with the environment. At the same time, its temperature decreases to the temperature of the refrigerator.

3.Isothermal compression (in the figure - process B→G). The working fluid, which by that time has a temperature TX, is brought into contact with the refrigerator and begins to compress isothermally, giving the amount of heat QX to the refrigerator.

4. Adiabatic (isentropic) compression (in the figure - process G→A). The working fluid is disconnected from the refrigerator and compressed without heat exchange with the environment. At the same time, its temperature increases to the temperature of the heater.

Entropy- an indicator of randomness or disorder in the structure of a physical system. In thermodynamics, entropy expresses the amount of thermal energy available to do work: the less energy, the less entropy. On the scale of the Universe, entropy increases. Energy can be extracted from a system only by transforming it into a less ordered state. According to the second law of thermodynamics, entropy in an isolated system either does not increase or increases during any process.

Thermodynamic probability, the number of ways in which the state of a physical system can be realized. In thermodynamics, the state of a physical system is characterized by certain values ​​of density, pressure, temperature and other measurable quantities.

Ticket 31.

Micro- and macrostates. Statistical weight. Reversible and irreversible processes. Entropy. Law of increasing entropy. Nernst's theorem.

Ticket 30.

Statistical weight is the number of ways in which a given system state can be realized. The statistical weights of all possible states of the system determine its entropy.

Reversible and irreversible processes.

A reversible process (that is, equilibrium) is a thermodynamic process that can occur in both the forward and reverse directions, passing through the same intermediate states, and the system returns to its original state without energy expenditure, and no macroscopic changes remain in the environment.

(A reversible process can be made to flow in the opposite direction at any time by changing any independent variable by an infinitesimal amount.

Reversible processes produce the most work.

In practice, a reversible process cannot be realized. It flows infinitely slowly, and you can only get closer to it.)

An irreversible process is a process that cannot be carried out in the opposite direction through all the same intermediate states. All real processes are irreversible.

In an adiabatically isolated thermodynamic system, entropy cannot decrease: it is either preserved if only reversible processes occur in the system, or increases if at least one irreversible process occurs in the system.

The written statement is another formulation of the second law of thermodynamics.

Nernst's theorem (Third law of thermodynamics) is a physical principle that determines the behavior of entropy as temperature approaches absolute zero. It is one of the postulates of thermodynamics, accepted on the basis of a generalization of a significant amount of experimental data.

The third law of thermodynamics can be formulated as follows:

“The increase in entropy at absolute zero temperature tends to a finite limit, independent of the equilibrium state the system is in.”

Where x is any thermodynamic parameter.

(The third law of thermodynamics applies only to equilibrium states.

Since, based on the second law of thermodynamics, entropy can only be determined up to an arbitrary additive constant (that is, it is not the entropy itself that is determined, but only its change):

The third law of thermodynamics can be used to accurately determine entropy. In this case, the entropy of the equilibrium system at absolute zero temperature is considered equal to zero.

According to the third law of thermodynamics, at value .)

Ticket 32.

Real gases. Van de Waals equation. The internal energy is really gas.

A real gas is a gas that is not described by the Clapeyron-Mendeleev equation of state for an ideal gas.

Molecules in a real gas interact with each other and occupy a certain volume.

In practice, it is often described by the generalized Mendeleev-Clapeyron equation:

The van der Waals gas equation of state is an equation that relates the basic thermodynamic quantities in the van der Waals gas model.

(To more accurately describe the behavior of real gases at low temperatures, a van der Waals gas model was created that takes into account the forces of intermolecular interaction. In this model, the internal energy U becomes a function of not only temperature, but also volume.)

The thermal equation of state (or, often, simply the equation of state) is the relationship between pressure, volume and temperature.

For n moles of van der Waals gas, the equation of state looks like this:

p - pressure,

• T - absolute temperature,

  R is the universal gas constant.

The internal energy of a real gas consists of the kinetic energy of the thermal motion of molecules and the potential energy of intermolecular interaction

Ticket 33.

Physical kinetics. The phenomenon of transport in gases. Number of collisions and mean free path of molecules.

Physical kinetics is a microscopic theory of processes in nonequilibrium media. In kinetics, the methods of quantum or classical statistical physics are used to study the processes of transfer of energy, momentum, charge and matter in various physical systems (gases, plasma, liquids, solids) and the influence of external fields on them.

Transport phenomena in gases are observed only if the system is in a nonequilibrium state.

Diffusion is the process of transferring matter or energy from an area of ​​high concentration to an area of ​​low concentration.

Thermal conductivity is the transfer of internal energy from one part of the body to another or from one body to another upon their direct contact.

Number (Frequency) of collisions and mean free path of molecules.

Moving at medium speed On average, in time τ the particle travels a distance equal to the mean free path< l >:

< l > = τ

τ is the time that a molecule moves between two successive collisions (analogous to a period)

Then the average number of collisions per unit time (average collision frequency) is the reciprocal of the period:

v= 1 / τ = / = σn

Path length< l>, at which the probability of collision with target particles becomes equal to one, is called the mean free path.

= 1/σn

Ticket 34.

Diffusion in gases. Diffusion coefficient. Viscosity of gases. Viscosity coefficient. Thermal conductivity. Coefficient of thermal conductivity.

Diffusion is the process of transferring matter or energy from an area of ​​high concentration to an area of ​​low concentration.

Diffusion in gases occurs much faster than in other states of aggregation, which is due to the nature of the thermal movement of particles in these media.

Diffusion coefficient - the amount of a substance passing per unit time through a section of unit area with a concentration gradient equal to unity.

The diffusion coefficient reflects the rate of diffusion and is determined by the properties of the medium and the type of diffusing particles.

Viscosity (internal friction) is one of the transfer phenomena, the property of fluid bodies (liquids and gases) to resist the movement of one part relative to another.

When talking about viscosity, the number that is usually considered is viscosity coefficient. There are several different viscosity coefficients, depending on the acting forces and the nature of the fluid:

Dynamic viscosity (or absolute viscosity) determines the behavior of an incompressible Newtonian fluid.

Kinematic viscosity is the dynamic viscosity divided by the density for Newtonian fluids.

Bulk viscosity determines the behavior of a compressible Newtonian fluid.

Shear Viscosity (Shear Viscosity) - coefficient of viscosity under shear loads (for non-Newtonian fluids)

Bulk viscosity - compression viscosity coefficient (for non-Newtonian fluids)

Thermal conduction is the process of heat transfer, leading to equalization of temperature throughout the entire volume of the system.

Thermal conductivity coefficient is a numerical characteristic of the thermal conductivity of a material, equal to the amount of heat passing through a material with a thickness of 1 m and an area of ​​1 sq.m per hour when the temperature difference on two opposite surfaces is 1 degree C.