Portable relative and absolute motion. Addition of accelerations during translational motion

§ 20 . Relative, figurative and absolute

point movement

Complex point movement its movement is called such that it moves relative to a reference system moving relative to some other reference system taken as stationary. For example, we can assume that a passenger walking along the carriage of a moving train makes a complex movement in relation to the road surface, consisting of the movement of the passenger in relation to the carriage ( moving reference frame) and the movement of the passenger together with the carriage in relation to the road surface ( fixed frame of reference).

The movement of a point in relation to a moving coordinate system is called relative motion of a point. The speed and acceleration of this movement are called relative speed And relative acceleration and denote and .

The movement of a point due to the movement of a moving coordinate system is called portable movement of the point.

Portable speed And portable acceleration points call the speed and acceleration of the point rigidly connected with the moving coordinate system, with which the moving point coincides at a given moment in time, and denote And .

The movement of a point relative to a fixed coordinate system is called absolute or complex. The speed and acceleration of a point in this motion are called absolute speed And absolute acceleration and denote and .

In the above example, the movement of the passenger relative to the carriage will be relative, and the speed will be the relative speed of the passenger; the movement of the car in relation to the road surface will be portable motion for the passenger, and the speed of the car in which the passenger is located will be its portable speed at that moment; finally, the movement of the passenger in relation to the canvas will be his absolute movement, and the speed will be the absolute speed.

§ 21 .Determining the speed of a point with a complex

movement

Let there be a fixed reference system in relation to which the moving reference system moves . A point moves relative to the moving coordinate system (Fig. 2.26) . The equation of motion of a point in complex motion can be specified in a vector way

,(2.67)

where is the radius vector of a point, which determines its position relative to

fixed frame of reference;

Radius vector defining the position of the reference point of the moving

coordinate systems;

Radius vector of the point in question, defining it

position relative to the moving coordinate system.

Let the coordinates of the point be in the moving axes. Then

,(2.68)

where are unit vectors directed along the moving axes. Substituting (2.68) into equality (2.67), we obtain:

.(2.69)

With relative motion, the coordinates change over time. To find the speed of relative motion, it is necessary to differentiate the radius vector with respect to time, taking into account its change only due to relative motion, that is, only due to changes in coordinates, and assume the moving coordinate system to be stationary, that is, consider the vectors to be independent of time. Differentiating equality (2.68) with respect to time, taking into account the reservations made, we obtain the relative speed:

, (2.70)

where the dots above the quantities mean the derivatives of these quantities with respect to time:

, , .

If there is no relative motion, then the point will move along with the moving coordinate system and the speed of the point will be equal to the portable speed. Thus, the expression for the transfer speed can be obtained if we differentiate the radius vector with respect to time, considering it independent of time:

.(2.71)

We find the expression for absolute speed by differentiating with respect to time, taking into account that the relative coordinates and unit vectors of the moving coordinate system depend on time:

.(2.72)

In accordance with formulas (2.70), (2.71), the first bracket in (2.72) is the portable velocity of the point, and the second is the relative one. So,

.(2.73)

Equality (2.73) expresses velocity addition theorem : the absolute speed of a point is equal to the geometric sum of the portable and relative speeds.

Problem 2.9. The train is moving in a straight lineto himhorizontal path at constant speed . The passenger sees from the carriage window the trajectories of raindrops inclined to the vertical at an angle. Determine the absolute speed of falling raindrops from vertically falling rain, neglecting the friction of the drops on the glass.

Solution. Raindrops have absolute speed

where is the relative speed of the drop as it moves along the glass of the car;

Transferable drop speed, equal to speed train movement.

The resulting parallelogram of velocities (Fig. 2.27) is divided by the diagonal into two equal triangles. Considering any of these triangles, we find

.

We translate the resulting drop velocity into:

.

§ 22 .Determination of the acceleration of a point at a complex

movement

Expression for relative acceleration points can be obtained by differentiating the relative speed (2.70), taking into account it and the change only due to relative movement, that is, due to changes in the relative coordinates of the point , , . Vectors should be considered constant, since the movement of a fixed coordinate system is not taken into account when determining the relative speed and relative acceleration of a point. So we have

,(2.74)

Portable acceleration we obtain by differentiating equality (2.71) with respect to time, assuming that the point is at rest with respect to the moving coordinate system, i.e., that the relative coordinates of the point , , do not depend on time.

.(2.75)

Absolute acceleration we obtain by differentiating the expression for absolute speed (2.72), taking into account that over time they change as relative coordinates , , points and unit vectors of the moving coordinate system

.(2.76)

It can be seen that the first bracket in (2.76) is the portable acceleration, the third is the relative acceleration. The second bracket is an additional or Coriolis acceleration:

.(2.77)

So, equality (2.76) can be written in the form

.(2.78)

This formula expresses Coriolis theorem : in the case of non-translational translational motion, the absolute acceleration of a point is equal to the vector sum

portable, relative and rotational accelerations.

Let us transform formula (2.77) for Coriolis acceleration. For unit derivatives vectors of the mobile system coordinates the following take place Poisson formulas :

; ; .(2.79)

Here is the vector of instantaneous angular velocity moving coordinate system. The sign denotes the vector product of vectors.

Substituting formulas (2.79) into (2.77), we obtain:

The expression in brackets is nothing more than the relative speed (see (2.70)). Finally we get:

.(2.80)

So, Coriolis acceleration is equal to twice the vector product of the instantaneous angular velocity of the moving coordinate system and the relative velocity vector.

According to the general rule for determining the direction of the vector product, we have: the Coriolis acceleration is directed perpendicular to the plane passing through the vectors and in the direction from which the rotation of the vector to the vector at a smaller angle is visible counterclockwise (Fig. 2.28).

From formula (2.80) it also follows that the magnitude of the Coriolis acceleration

.(2.81)

It follows that Coriolis acceleration is zero in three cases:

1) if, i.e. in the case of translational portable motion or at moments when the angular velocity of non-translational portable motion vanishes;

2) if, i.e. in the case of relative rest of the point or at moments when the relative velocity of the point vanishes;

3) if, i.e. in the case when the vector of the relative velocity of the point is parallel to the vector of the angular velocity of the portable motion, as, for example, when the point moves along the generatrix of a cylinder rotating around its axis.

Problem 2.10. By railUti, laid along the parallel of northern latitude, a diesel locomotive moves at a speed from west to east. Find the Coriolis acceleration of the diesel locomotive.

Solution.Neglecting the size of the diesel locomotive, we will consider it as a certain point (point in Fig. 2.29). The point makes a complex movement. For portable motion we take the rotational motion of a point together with the Earth, and for relative motion we take the motion of this point in relation to the Earth at a constant speed.

The magnitude of the Coriolis acceleration according to (2.81) is equal to

,

where is the angular velocity of the Earth's rotation.

Let's find the angular velocity of the Earth's rotation. The Earth makes one revolution per day. The angle corresponding to one revolution is equal to and the number of seconds in a day is equal to , hence

.

The position and direction of the Coriolis acceleration vector are determined by the general rule for determining the direction of the vector product. The Coriolis acceleration vector is on a straight line, since it must be perpendicular to the vectors and , and is directed in the direction opposite to the direction of the vectors And .

So far we have studied the movement of a point or body in relation to one given system countdown. However, in a number of cases, when solving problems of mechanics, it turns out to be advisable (and sometimes necessary) to consider the movement of a point (or body) simultaneously in relation to two reference systems, of which one is considered the main or conditionally stationary, and the other moves in a certain way in relation to the first. The movement performed by the point (or body) is called composite or complex. For example, a ball rolling along the deck of a moving steamship can be considered to be performing a complex motion relative to the shore, consisting of rolling relative to the deck (moving frame of reference), and moving together with the deck of the steamship in relation to the shore (fixed frame of reference). In this way, the complex motion of the ball is decomposed into two simpler and more easily studied ones.

Fig.48

Consider the point M, moving relative to the moving reference system Oxyz, which in turn somehow moves relative to another reference system, which we call the main or conditionally stationary (Fig. 48). Each of these reference systems is associated, of course, with a specific body, not shown in the drawing. Let us introduce the following definitions.

1. Movement made by a point M in relation to the moving reference system (to the axes Oxyz), called relative movement(such movement will be seen by an observer associated with these axes and moving with them). Trajectory AB described by a point in relative motion is called a relative trajectory. Point speed M in relation to the axes Oxyz is called relative speed (denoted by ), and acceleration is called relative acceleration (denoted by ). From the definition it follows that when calculating and it is possible to move the axes Oxyz do not take into account (consider them as motionless).

2. Movement performed by a moving frame of reference Oxyz(and all points of space invariably associated with it) in relation to the fixed system, is for the point M portable movement.

The speed is invariably associated with the moving axes Oxyz points m, with which the moving point coincides at a given moment in time M, is called the transfer speed of the point M at this moment (denoted by ), and the acceleration of this point m- portable acceleration of a point M(denoted by ). Thus,

If we imagine that the relative movement of a point occurs on the surface (or inside) of a solid body, to which the movable axes are rigidly connected Oxyz, then the portable speed (or acceleration) of the point M at a given moment of time there will be the speed (or acceleration) of that point m of the body with which the point coincides at this moment M.

3. The movement made by a point in relation to a fixed frame of reference is called absolute or complex. Trajectory CD of this movement is called the absolute trajectory, the speed is called absolute speed (denoted by ) and acceleration is called absolute acceleration (denoted by ).

In the above example, the motion of the ball relative to the deck of the steamship will be relative, and the speed will be the relative speed of the ball; the movement of the steamer in relation to the shore will be a portable motion for the ball, and the speed of that point on the deck that the ball touches at a given moment in time will be its portable speed at that moment; finally, the motion of the ball relative to the shore will be its absolute motion, and the speed will be the absolute speed of the ball.

When studying the complex movement of a point, it is useful to apply the “Stopping Rule”. In order for a stationary observer to see the relative movement of a point, the portable movement must be stopped.

Then only relative motion will occur. Relative motion will become absolute. And vice versa, if you stop the relative movement, the portable one will become absolute and a stationary observer will see only this portable movement.

In the latter case, when determining the portable movement of a point, one very important circumstance is revealed. The portable movement of a point depends on the moment at which the relative movement is stopped, on where the point is on the medium at that moment. Since, generally speaking, all points of the medium move differently. Therefore, it is more logical to determine portable movement of a point as the absolute movement of that point in the environment with which the moving point currently coincides.

Complex point movement

The movement of a body is judged by the movement of each of its points. Previously, we considered the movement of a point in a certain coordinate system, which was conventionally taken to be stationary. However, in practice, you have to solve problems in which you know how a point moves relative to one coordinate system and you need to find out how it moves relative to another coordinate system, if you know how these coordinate systems move relative to each other. To describe the movement of a point, moving from one coordinate system to another, it is necessary to establish how the quantities characterizing the movement of a point in these systems are related to each other. For this purpose, one coordinate system is conventionally accepted as stationary, and the other as moving, and the concepts of absolute, relative and portable motion of a point are introduced.

Absolute motion– movement of a point in a fixed coordinate system.

Relative motion– movement of a point in a moving coordinate system.

Portable movement– movement of moving space relative to fixed space.

Problems in which translational motion is given and absolute motion needs to be found are called problems on addition movements.

In some cases it is necessary to solve the inverse problem.

By rational choice of a moving coordinate system, it is often possible to reduce the complex absolute movement of a point to two simple ones: relative and figurative. Such problems are called problems on decomposition of movements.

fixed system coordinates are called absolute speed And absolute acceleration.

Speed ​​and acceleration of a point relative to mobile system coordinates are called relative speed And relative acceleration.

Portable speed And portable acceleration of a moving point are called the absolute speed and absolute acceleration of that moving space points, with which the moving point coincides at a given time.

All previously obtained results for velocity and acceleration are fully applicable to relative motion, because when deriving them we do not impose any restrictions on the choice of coordinate system.

Law of addition of speeds

The law of addition of velocities determines the relationship between the velocities of point M in a fixed coordinate system XYZ and mobile coordinate system https://pandia.ru/text/78/244/images/image002_52.jpg" width="588" height="243">

– law of addition of velocities.

KINEMATICS OF AN ABSOLUTELY RIGID BODY

Let's move on to considering the movement absolutely solid(ATT). A rigid body consists of an infinite number of points, however, as will be shown later, to describe the movement of the ATT there is no need to specify the movement of each of its points.

The constancy of the distance between points of a rigid body leads to a dependence between the velocities of individual points. This dependence is expressed by the following basic theorem of rigid body kinematics: the projections of the velocities of any two points of a rigid body onto the segment connecting them are equal.

To prove this, consider arbitrary points A and B of a rigid body.

The positions of points A and B in space will be specified by radius vectors and https://pandia.ru/text/78/244/images/image007_36.gif" width="29" height="24 src=">, the direction of which is in progress the movement of the body changes, but the modulus remains constant (due to the constant distance between the points of the rigid body). This vector can be represented in the form . Differentiating this equality with respect to time, we obtain

. (2.1)

To determine the vector, note that , where AB vector modulus. Because AB does not change over time, then, differentiating this equality with respect to t, we get:

,

i.e..gif" width="29" height="24 src="> is directed perpendicular to the vector itself:

Now designing each part of the equality (2..gif" width="37" height="24"> – ex.=0

,

which proves the formulated theorem.

Translational motion of a rigid body

Let us first consider simple cases of motion - translational motion of a rigid body and rotation of a rigid body.

The simplest type of motion of a rigid body is one in which the velocity vectors of its three points that do not lie on the same straight line are equal to each other at each moment of time. Let us determine the position of these points at some point in time using radius vectors:

https://pandia.ru/text/78/244/images/image020_14.gif" width="263 height=43" height="43">

Therefore, vectors are independent of time and therefore move in space while remaining parallel to themselves. Three points of a rigid body define a coordinate system clearly related to the rigid body. In the case under consideration, the movement will be such that the axes will move while remaining parallel to themselves. But this means that any straight line drawn in solid body, remains parallel to itself during the movement. Such movement is called translational (for example, the movement of a cabin in a Ferris wheel attraction).

Let us choose two arbitrary points A and B in a rigid body moving translationally.

During forward movement of the ATT

(2.2)

Since then (2.2) will take the form:

Points A and B are chosen randomly. Consequently: during translational motion, all points of a rigid body have identical velocity vectors at any given moment of time.

Differentiating the equation with respect to time (2..gif" width="56" height="24"> (2.4)

Points A and B are chosen randomly. Hence: points of a rigid body moving translationally have identical accelerations at each given moment of time.

Since the trajectories of points A and B are congruent, i.e. their. can be combined with each other when overlaying. Thus, the trajectories described by the points of a rigid body moving translationally are identical and equally located.

From the results obtained we can conclude: for description forward motion for a rigid body, it is enough to specify the movement of only one of its points.

Rigid body rotation

Rotation of a rigid body is a type of motion in which at least one point of the rigid body remains motionless. Let's consider, however, a simpler case - rotation of the ATT around a fixed axis.

Rotation of an absolutely rigid body around a fixed axis

Let's fix two ATT points:. Let's consider how all the points of a rigid body will move and learn how to determine the velocities and accelerations of these points. It is clear that points of a rigid body lying on a straight line passing through two fixed points will not move: this straight line is called stationary axis of rotation. The motion of a rigid body, in which at least two of its points are motionless, is called rotation of the ATT around a motionless axis.

It is clear that points not lying on the axis of rotation describe circles whose centers lie on the axis of rotation. The planes in which such circles lie are perpendicular to the axis of rotation. Consequently: we know the trajectories of all points of the body. This allows you to start finding the speed of any point on a rigid body.

With the natural way of specifying the movement of a point:

Let us choose a fixed reference system, the axis 0 Z which coincides with the axis of rotation. Angle between fixed plane X0Z, passing through the axis of rotation and a plane rigidly connected to the rigid body and passing through the axis of rotation, we denote it by https://pandia.ru/text/78/244/images/image036_12.gif" width="73" height="31 ">. Consider the movement of point M along a circle of radius R.

; ; https://pandia.ru/text/78/244/images/image040_13.gif" width="20" height="26 src="> are constant:

Substituting (2.6) into (2.5) we get:

This formula is inconvenient because it includes the unit vector https://pandia.ru/text/78/244/images/image044_12.gif" width="14" height="18 src=">. It must be included in the formula for speed. To do this, we will carry out the following transformations:

using that , we rewrite relation (2.7) in the form

(2.8)

Let's denote:

– does not depend on the choice of the considered point M; (2.9)

– vector drawn from the center of the circle to point M. (2.10)

It is clear that the modulus is equal to the radius of the circle.

Let's substitute (2.9) and (2.10) into (2.8):

https://pandia.ru/text/78/244/images/image051_11.gif" width="91" height="27"> (2.12)

The directions coincide with the direction of the unit touch vector https://pandia.ru/text/78/244/images/image054_10.gif" width="64" height="29"> – linear speed of point M. (2.13)

– angular velocity. (2.14)

Angular velocity is the same value for all points of a rigid body.

The linear speed of any point of a rigid body rotating around a fixed axis is equal to vector product We expand the angular velocity of the ATT into a radius vector drawn from an arbitrary point on the axis of rotation https://pandia.ru/text/78/244/images/image057_9.gif" width="145" height="29">. (2.15 )

Comparing (2.15) and (2.14) we get:

;

The modulus of angular velocity is related to the rotation frequency of an absolutely rigid body:

When a body rotates, its angular velocity can change; it is necessary to be able to determine the angular velocity of the body at any time. For this purpose, a value has been introduced that characterizes the change in angular velocity over time. This quantity is called angular acceleration.

Let's give the definition of angular acceleration.

Let at a moment in time t angular velocity. And at a moment in time t+∆t angular velocity is equal to . Let us make up the ratio of the change in angular velocity to the time period during which this change occurs, and find the limit of this ratio at ∆ t→ 0. In mechanics this limit is called angular acceleration of the body and therefore denote:

.

Angular acceleration is the same value for all points of a rigid body.

The unit of measurement for angular acceleration is https://pandia.ru/text/78/244/images/image068_7.gif" width="273" height="48">.

For angular acceleration, its projection onto the axis 0 Z, modulus of angular acceleration, the following relations are valid:

(2.16)

Let's rewrite the expression for the acceleration of a point:

(2.17)

The tangential acceleration of any point of a rigid body rotating around a fixed axis is equal to the vector product of the angular acceleration of the body and the radius - the vector of this point drawn from an arbitrary point on the axis of rotation.

Rotation of a rigid body with constant angular acceleration

Let's see how the kinematic equation of body motion is written during this movement. First, we obtain a formula by which in this case we can find the angular velocity of the body. Let's direct the axis 0 Z along the axis of rotation of the body.

Since, then https://pandia.ru/text/78/244/images/image078_5.gif" width="98" height="54"> (since) Rotational movements (physics)" href=" /text/category/vrashatelmznie_dvizheniya__fizika_/" rel="bookmark">rotational motion around a pole with angular velocity independent of the choice of pole.

It can be shown that the speed of any point of the body relative to a fixed coordinate system is equal to:

– angular acceleration of rotation of the body relative to the pole.

Law of addition of accelerations

The formula expressing the law of addition of accelerations in complex motion is called the Coriolis formula, and the fact it expresses is the Coriolis theorem. According to this theorem, the absolute acceleration of a point is equal to the sum of three vectors: the relative acceleration vector, the transfer acceleration vector and the vector representing the rotational or Coriolis acceleration:

(2.21)

It appears due to two reasons that are not taken into account by relative and portable accelerations: it does not take into account the change in the direction of relative speed in a stationary space due to the rotation of the moving coordinate system in portable motion. does not take into account the change in portable speed resulting from the transition of a moving point from one point in moving space to another (this transition is caused by relative motion).

In the following cases:

The portable movement of a point is its movement at the considered moment in time together with the moving coordinate system relative to a fixed coordinate system.

The portable speed and portable acceleration of a point are indicated by the index e: ,.

Portable speed (acceleration ) point M at a given time is called a vector equal to the speed
(acceleration
) that pointmmoving coordinate system with which the moving point M currently coincides(Fig. 8.1).

Let's draw the radius vector of the origin of coordinates (Fig. 8.1). From the figure it is clear that

To find the portable speed of a point at a given point in time, it is necessary to differentiate the radius vector provided that the coordinates of the point x, y, z do not change at a given time:

The transfer acceleration is correspondingly equal to

Thus, to determine the transfer speed and portable acceleration at a given moment in time it is necessary to mentally stop at this moment in time the relative movement of the point, determine the point m a body invariably associated with a moving coordinate system where the point is located at a stopped moment M, and calculate the speed and acceleration of the point m a body undergoing portable motion relative to a fixed coordinate system.

Setting tasks for complex point motion

1.Direct task:

Based on the given portable and relative movements of the point, find the kinematic characteristics of the absolute movement of the point.

2. Inverse problem:

To represent some given motion of a point in a complex manner, decomposing it into relative and portable, and determine the kinematic characteristics of these movements. To solve this problem unambiguously, additional conditions are required.

Velocity addition theorem

Absolute point speed is determined by the theorem on the addition of velocities, according to which the absolute speed of a point performing a complex movement is equal to the geometric sum of the portable and relative speeds:

Proof:

To determine the absolute speed of a point, we differentiate the expression on the right (8.4) with respect to time, using the properties of the derivative of a vector with respect to a scalar argument:

(8.8)

In the last expression on the left, the first four terms in formula (8.5) represent the transfer speed , the last three terms in formula (8.1) are the relative speed . The theorem has been proven.

Theorem for addition of accelerations in portable translational motion

The absolute acceleration of a point performing a complex movement during portable translational motion is equal to the geometric sum of the relative and portable acceleration:

. (8.9)

Proof:

Let's return to Fig. 8.1. With portable translational movement of the orta
do not change not only in size, but also in direction, i.e. these are constant vectors, and since derivatives of constant vectors, and since derivatives of constant vectors are equal to zero, then according to formula (8.6)

. (8.10)

To determine the absolute acceleration of a point, we differentiate the radius vector twice (8.4) in time, taking into account the constancy of the unit vectors
:

In the last expression, the first term in formula (8.10) represents the portable acceleration , and the last three according to formula (8.2) are the relative acceleration . The theorem has been proven.

Theorem for addition of accelerations during arbitrary translational motion (Coriolis theorem)

The absolute acceleration of a point is determined by Coriolis theorem, according to which the absolute acceleration of a point performing a complex movement is equal to the geometric sum of the portable, relative and Coriolis accelerations:

. (8.11)

Coriolis acceleration calculated by the formula:

, (8.12)

where is the vector of the angular velocity of the portable movement, is the vector of the relative speed of the point. The direction of the Coriolis acceleration vector is determined by the vector product rule: the Coriolis acceleration will be directed perpendicular to the plane in which the vectors lie (Fig. 8.2), in the direction from which the shortest turn from the vector to the vector is seen to occur counterclockwise.

The modulus of Coriolis acceleration is equal to .

Let us prove the validity of the theorem for portable rotational motion.

Let the moving coordinate system Oxyz rotates around an axis l with angular velocity
(Fig. 8.3). During the entire movement, the radius vectors of the point are still connected by the dependence

Since by definition
, let us differentiate expression (8.8) with respect to time, taking into account the properties of the derivative of a vector with respect to the scalar argument:

In the last expression, the first four terms represent the portable acceleration , the next three terms represent the relative speed . We denote the remaining terms (*). In expression (*), the derivative of each unit vector with respect to time represents the linear velocity of the point for which this unit unit is a radius vector. For example, for Orta (Fig. 8.3) speed
points A its end is equal

.

But since ort rotates around an axis l, then the speed of its end can be determined using Euler’s vector formula:

.

Hence

. (8.14)

Similarly for orts And :

,
. (8.15)

Substituting formulas (8.14) and (8.15) into expression (*), we obtain

Using the combinatory property of a vector product with respect to numerical factors, which are
, we have

Thus,

.

The theorem for portable rotational motion has been proven.

§ 2. 5. Movement: absolute, relative, figurative. Euler's theorem. Angular velocity.

In addition to the fixed axes Oxyz (system S), we introduce into consideration some moving rigid body and the system of rectangular coordinate axes O’x’y’z’ (system S’) invariably associated with it.

The movement of a point relative to the moving system of axes S’ is called relative movement.

The movement of a point relative to the fixed axes S is called absolute movement.

Portable movement point over a time interval (t,t+Dt) is the movement relative to the axes S that this point would have if at time t and for the interval (t,t+Dt) it was invariably connected with a moving system of axes and, therefore, would move along with this system.

Trajectory, speed and acceleration are called absolute, relative or portable, depending on whether they relate to absolute, relative or portable motion.

Euler's theorem: If, relative to the system S, the system S" has one fixed point, then the movement of S" from one arbitrary position to any other can be accomplished by one rotation through a certain angle relative to the axis passing through this fixed point.

To prove it, it is enough to show the possibility of translation with one turn of the arc, for example, .

		Let's draw two equators: a, perpendicular to the midpoint x 1 "x 2", and b, perpendicular to the midpoint z 1 "z 2". We get two points of intersection of these equators - c and d. Dx 1"z 1"d = Dz 2"x 2"d (since x 1 "z 1 " = x 2 "z 2 ", and x 1 "d = x 2 "d due to the fact that point d lies on the equator perpendicular to the midpoint of x 1 "x 2", z 1 "d = z 2 "d for the same reason) Thus, Ðx 1 "dz 1" = Ðz 2 "dx 2" and the angle between the arcs x 1 "d and x 2 "d equal to angle between the arcs z 1 "d and z 2 "d, that is, you need to rotate x 1 "z 1 " relative to the dO "c axis by an angle x 1 "dz 1 " (or equal to it z 2 "dx 2 ")

Euler's theorem is valid for both finite and infinitesimal rotations. Although there can be any sequence of infinitesimal rotations, the result will be the same, the same finite rotations do not commute. This is all the more true for infinitesimal rotations, the closer the arcs described by any point are to the chords connecting the ends of the arcs.

When considering problems about the motion of a body with one fixed point, which have a large practical significance, to determine (fix) the position of the system S" relative to S, three Euler angles are widely used.

The intersection of the planes O"xy and O"x"y" gives a straight line, which is called the line of nodes (ort of the line of nodes - ). The first Euler angle j is the angle between the O"x axis and the line of nodes. The second angle y is the angle between the line of nodes and the O"x axis. The third angle q is the angle between the O"z and O"z axes.

These three angles uniquely determine the position of the system S" relative to S

Thus, with an infinitesimal rotation of the system S" relative to S by angles dj, dy, dq (some of them may be equal to zero), they can be replaced by one rotation by an angle dc around a certain axis passing through the point O".

Let us introduce the vector of infinitesimal rotation into consideration:

(Here directed along the axis of rotation according to the rule of the right screw)

The magnitude and direction of the dc vector can change during complex motion. The axis is called the axis of instantaneous rotation. Let's see what happens to the unit vectors of the system S" when it is rotated by an angle

§ 2. 6. Complex point movement.

Differentiating this relationship with respect to time, we obtain:

Absolute speed of a point (relative to system S),

Speed ​​of origin S" relative to S,

It is not the velocity of point M relative to the system S", since the unit vectors of this system are functions of time.

,

using formulas (2.5.1) we will have:

The last term means that the derivative is taken with constant unit vectors of the system O’x’y’z’, .

Now for speeds we have:

here v h is portable, v is absolute, v’ is the relative speed of the point, that is, the connection between these speeds is obtained. The transfer speed consists of two terms: the first is present if the moving frame of reference moves translationally, the second appears if the moving frame of reference rotates.

To obtain the connection between accelerations, we differentiate the relation for speeds with respect to time:

Absolute acceleration is the acceleration of the origin of coordinates S’ relative to S.