Rotational motion of a rigid body around a fixed axis. Angular velocity and angular acceleration

The rotation of a rigid body around a fixed axis is such a movement in which two points of the body remain motionless during the entire time of movement. In this case, all points of the body located on a straight line passing through its fixed points also remain motionless. This line is called body rotation axis .

Let points A and B be stationary. Let's direct the axis along the axis of rotation. Through the axis of rotation we draw a stationary plane and a movable plane attached to a rotating body (at ).

The position of the plane and the body itself is determined by the dihedral angle between the planes and. Let's denote it . The angle is called body rotation angle .

The position of the body relative to the chosen reference system is uniquely determined at any time if the equation is given, where is any twice differentiable function of time. This equation is called equation of rotation of a rigid body around a fixed axis .

A body rotating around a fixed axis has one degree of freedom, since its position is determined by specifying only one parameter - angle.

An angle is considered positive if it is laid counterclockwise, and negative in the opposite direction. The trajectories of points of a body during its rotation around a fixed axis are circles located in planes perpendicular to the axis of rotation.

For characteristics rotational movement of a rigid body around a fixed axis, we introduce the concepts of angular velocity and angular acceleration.

Algebraic angular velocity of a body at any moment in time is called the first derivative with respect to time of the angle of rotation at this moment, that is.

Angular velocity is positive when the body rotates counterclockwise, since the angle of rotation increases with time, and negative when the body rotates clockwise, because the angle of rotation decreases.

The dimension of angular velocity by definition:

In engineering, angular velocity is the rotational speed expressed in revolutions per minute. In one minute the body will rotate through an angle , where n is the number of revolutions per minute. Dividing this angle by the number of seconds in a minute, we get

Algebraic angular acceleration of the body is called the first derivative with respect to time of the angular velocity, that is, the second derivative of the angle of rotation, i.e.

The dimension of angular acceleration by definition:

Let us introduce the concepts of vectors of angular velocity and angular acceleration of a body.

And , where is the unit vector of the rotation axis. Vectors and can be depicted at any point on the rotation axis; they are sliding vectors.

Algebraic angular velocity is the projection of the angular velocity vector onto the axis of rotation. Algebraic angular acceleration is the projection of the angular acceleration vector of velocity onto the axis of rotation.

If at , then the algebraic angular velocity increases with time and, therefore, the body rotates accelerated at the moment in time in the positive direction. The directions of the vectors and coincide, they are both directed in the positive direction of the axis of rotation.

When and the body rotates rapidly in the negative direction. The directions of the vectors and coincide, they are both directed in the negative direction of the axis of rotation.

Rotational they call such a movement in which two points associated with the body, therefore, the straight line passing through these points, remain motionless during movement (Fig. 2.16). Fixed straight line A B called axis of rotation.

Rice. 2.1V. Towards the definition of rotational motion of a body

The position of the body during rotational motion determines the angle of rotation φ, rad (see Fig. 2.16). When moving, the angle of rotation changes over time, i.e. the law of rotational motion of a body is defined as the law of change in time of the value of the dihedral angle Ф = Ф(/) between a fixed half-plane TO () , passing through the axis of rotation, and movable n 1 a half-plane connected to the body and also passing through the axis of rotation.

The trajectories of all points of the body during rotational motion are concentric circles located in parallel planes with centers on the axis of rotation.

Kinematic characteristics of the rotational motion of the body. In the same way that kinematic characteristics were introduced for a point, a kinematic concept is introduced that characterizes the rate of change of the function φ(c), which determines the position of the body during rotational motion, i.e. angular velocity co = f = s/f/s//, angular velocity dimension [co] = rad /With.

In technical calculations, the expression of angular velocity with a different dimension is often used - in terms of the number of revolutions per minute: [i] = rpm, and the relationship between n and co can be represented as: co = 27w/60 = 7w/30.

In general, angular velocity varies with time. The measure of the rate of change in angular velocity is angular acceleration e = c/co/c//= co = f, the dimension of angular acceleration [e] = rad/s 2 .

The introduced angular kinematic characteristics are completely determined by specifying one function - the angle of rotation versus time.

Kinematic characteristics of body points during rotational motion. Consider the point M body located at a distance p from the axis of rotation. This point moves along a circle of radius p (Fig. 2.17).

Rice. 2.17.

points of the body during its rotation

Arc length M Q M circle of radius p is defined as s= ptp, where f is the angle of rotation, rad. If the law of motion of a body is given as φ = φ(g), then the law of motion of a point M along the trajectory is determined by the formula S= рф(7).

Using the expressions of kinematic characteristics with the natural method of specifying the motion of a point, we obtain kinematic characteristics for points of a rotating body: speed according to formula (2.6)

V= 5 = rf = rso; (2.22)

tangential acceleration according to expression (2.12)

i t = K = sor = er; (2.23)

normal acceleration according to formula (2.13)

a„ = And 2 /р = с 2 р 2 /р = ogr; (2.24)

total acceleration using expression (2.15)

A = -]A + a] = px/e 2 + co 4. (2.25)

The characteristic of the direction of total acceleration is taken to be p - the angle of deviation of the vector of total acceleration from the radius of the circle described by the point (Fig. 2.18).

From Fig. 2.18 we get

tgjLi = aja n=re/pco 2 =g/(o 2. (2.26)

Rice. 2.18.

Note that all kinematic characteristics of the points of a rotating body are proportional to the distances to the axis of rotation. Ve-

Their identities are determined through the derivatives of the same function - the angle of rotation.

Vector expressions for angular and linear kinematic characteristics. For an analytical description of the angular kinematic characteristics of a rotating body, together with the axis of rotation, the concept rotation angle vector(Fig. 2.19): φ = φ(/)A:, where To- eat

rotation axis vector

1; To=sop51 .

The vector f is directed along this axis so that it can be seen from the “end”

rotation occurring counterclockwise.

Rice. 2.19.

characteristics in vector form

If the vector φ(/) is known, then all other angular characteristics of rotational motion can be represented in vector form:

• angular velocity vector co = f = f To. The direction of the angular velocity vector determines the sign of the derivative of the rotation angle;
• angular acceleration vector є = сo = Ф To. The direction of this vector determines the sign of the derivative of the angular velocity.

The introduced vectors с and є allow us to obtain vector expressions for the kinematic characteristics of points (see Fig. 2.19).

Note that the modulus of the point’s velocity vector coincides with the modulus vector product vector of angular velocity and radius vector: |sokh G= sogvіpa = rubbish. Taking into account the directions of the vectors с and r and the rule for the direction of the vector product, we can write an expression for the velocity vector:

V= co xg.

Similarly, it is easy to show that

• ? X
• - egBіpa= єр = a t And

Sosor = co p = i.

(In addition, the vectors of these kinematic characteristics coincide in direction with the corresponding vector products.

Therefore, the tangent vectors and normal acceleration can be represented as vector products:

• (2.28)
• (2.29)

a x = g X G

A= co x V.

Absolutely rigid body - body relative position parts of which do not change during movement.

Translational motion of a rigid body - this is its movement in which any straight line rigidly connected to the body moves while remaining parallel to its original direction.

During the translational motion of a rigid body, all its points move equally in a short time dt, the radius vector of these points changes by the same amount. Accordingly, at each moment of time the velocities of all its points are the same and equal. Therefore, the kinematics of the considered translational motion of a rigid body comes down to the study of the movement of any of its points. Usually we consider the movement of the center of inertia of a rigid body moving freely in space.

Rotational motion of a rigid body - this is a movement in which all its points move in circles, the centers of which are located outside the body . The straight line is called the axis of rotation of the body.

Angular velocity– vector quantity, characterizing the speed of rotation of the body; the ratio of the angle of rotation to the time during which this rotation occurred; a vector determined by the first derivative of the angle of rotation of a body with respect to time. The angular velocity vector is directed along the axis of rotation according to the right screw rule. ω=φ/t=2π/T=2πn, where T is the rotation period, n is the rotation frequency. ω=lim Δt → 0 Δφ/Δt=dφ/dt.

Angular acceleration– vector determined by the first derivative of the angular velocity with respect to time. When a body rotates around a fixed axis, the angular acceleration vector is directed along the rotation axis towards the vector of the elementary increment of angular velocity. Second derivative of the rotation angle with respect to time. When a body rotates around a fixed axis, the angular acceleration vector is directed along the rotation axis towards the vector of the elementary increment of angular velocity. When the motion is accelerated, the vector ε is codirectional to the vector φ, and when it is slow, it is opposite to it. ε=dω/dt.

If dω/dt> 0, then εω

If dω/dt< 0, то ε ↓ω

4. The principle of inertia (Newton's first law). Inertial reference systems. The principle of relativity.

Newton's first law (law of inertia): every material point (body) maintains a state of rest or uniform rectilinear movement until influence from other bodies forces her to change this state

The desire of a body to maintain a state of rest or uniform rectilinear motion is called inertia. Therefore, Newton's first law is called the law of inertia.

Newton's first law states the existence of inertial frames of reference.

Inertial reference frame– this is a reference system relative to which a free material point, unaffected by other bodies, moves uniformly in a straight line; this is a system that is either at rest or moving uniformly and rectilinearly relative to some other inertial system.

The principle of relativity- fundamental physical law, according to which any process proceeds identically in an isolated material system at rest, and in the same system in a state of uniform rectilinear motion. States of motion or rest are defined with respect to an arbitrarily chosen inertial reference frame. The principle of relativity underlies special theory Einstein's relativity.

5. Galilean transformations.

Principle of relativity (Galilee): no experiments (mechanical, electrical, optical) carried out inside a given inertial reference system make it possible to detect whether this system is at rest or moving uniformly and rectilinearly; all laws of nature are invariant with respect to the transition from one inertial frame of reference to another.

Let us consider two reference systems: the inertial frame K (with coordinates x,y,z), which we will conventionally consider stationary and the system K’ (with coordinates x’, y’, z’), moving relative to K uniformly and rectilinearly with a speed U (U = const). Let's find the connection between the coordinates of an arbitrary point A in both systems. r = r’+r0=r’+Ut. (1.)

Equation (1.) can be written in projections on the coordinate axes:

y=y’+Uyt; (2.)

z=z’+Uzt; Equations (1.) and (2.) are called Galilean coordinate transformations.

Relationship between potential energy and force

Every point potential field corresponds, on the one hand, to a certain value of the force vector acting on the body, and, on the other hand, to a certain value of potential energy. Therefore, there must be a certain relationship between force and potential energy.

To establish this connection, let us calculate the elementary work performed by field forces during a small displacement of the body occurring along an arbitrarily chosen direction in space, which we denote by the letter . This work is equal to

where is the projection of the force onto the direction.

Since in this case the work is done due to the reserve of potential energy, it is equal to the loss of potential energy on the axis segment:

Of the two latest expressions we get

This formula determines the projection of the force vector onto coordinate axes. If these projections are known, the force vector itself turns out to be determined:

in mathematics vector ,

where a is a scalar function of x, y, z, called the gradient of this scalar and denoted by the symbol . Therefore, the force is equal to the potential energy gradient taken with the opposite sign

This article describes an important section of physics - “Kinematics and dynamics of rotational motion”.

Basic concepts of kinematics of rotational motion

Rotational motion of a material point around a fixed axis is called such motion, the trajectory of which is a circle located in a plane perpendicular to the axis, and its center lies on the axis of rotation.

Rotational motion of a rigid body is a motion in which all points of the body move along concentric (the centers of which lie on the same axis) circles in accordance with the rule for the rotational motion of a material point.

Let an arbitrary rigid body T rotate around the O axis, which is perpendicular to the plane of the drawing. Let us select point M on this body. When rotated, this point will describe a circle with radius around the O axis r.

After some time, the radius will rotate relative to its original position by an angle Δφ.

The direction of the right screw (clockwise) is taken as the positive direction of rotation. The change in the angle of rotation over time is called the equation of rotational motion of a rigid body:

φ = φ(t).

If φ is measured in radians (1 rad is the angle corresponding to an arc of length equal to its radius), then the length of the circular arc ΔS, which the material point M will pass in time Δt, is equal to:

ΔS = Δφr.

Basic elements of the kinematics of uniform rotational motion

A measure of the movement of a material point over a short period of time dt serves as an elementary rotation vector dφ.

The angular velocity of a material point or body is physical quantity, which is determined by the ratio of the vector of an elementary rotation to the duration of this rotation. The direction of the vector can be determined by the rule of the right screw along the O axis. In scalar form:

ω = dφ/dt.

If ω = dφ/dt = const, then such motion is called uniform rotational motion. With it, the angular velocity is determined by the formula

ω = φ/t.

According to the preliminary formula, the dimension of angular velocity

[ω] = 1 rad/s.

The uniform rotational motion of a body can be described by the period of rotation. The period of rotation T is a physical quantity that determines the time during which a body makes one full revolution around the axis of rotation ([T] = 1 s). If in the formula for angular velocity we take t = T, φ = 2 π (one full revolution of radius r), then

ω = 2π/T,

Therefore, we define the rotation period as follows:

T = 2π/ω.

The number of revolutions that a body makes per unit time is called the rotation frequency ν, which is equal to:

ν = 1/T.

Frequency units: [ν]= 1/s = 1 s -1 = 1 Hz.

Comparing the formulas for angular velocity and rotation frequency, we obtain an expression connecting these quantities:

ω = 2πν.

Basic elements of the kinematics of uneven rotational motion

The uneven rotational motion of a rigid body or material point around a fixed axis is characterized by its angular velocity, which changes with time.

Vector ε , characterizing the rate of change of angular velocity, is called the angular acceleration vector:

ε = dω/dt.

If a body rotates, accelerating, that is dω/dt > 0, the vector has a direction along the axis in the same direction as ω.

If the rotational movement is slow - dω/dt< 0 , then the vectors ε and ω are oppositely directed.

Comment. When uneven rotational motion occurs, the vector ω can change not only in magnitude, but also in direction (when the axis of rotation is rotated).

Relationship between quantities characterizing translational and rotational motion

It is known that the arc length with the angle of rotation of the radius and its value are related by the relation

ΔS = Δφ r.

Then the linear speed of a material point performing rotational motion

υ = ΔS/Δt = Δφr/Δt = ωr.

The normal acceleration of a material point that performs rotational translational motion is defined as follows:

a = υ 2 /r = ω 2 r 2 /r.

So, in scalar form

a = ω 2 r.

Tangential accelerated material point that performs rotational motion

a = ε r.

Momentum of a material point

The vector product of the radius vector of the trajectory of a material point of mass m i and its momentum is called the angular momentum of this point about the axis of rotation. The direction of the vector can be determined using the right screw rule.

Momentum of a material point ( L i) is directed perpendicular to the plane drawn through r i and υ i, and forms a right-hand triple of vectors with them (that is, when moving from the end of the vector r i To υ i the right screw will show the direction of the vector L i).

In scalar form

L = m i υ i r i sin(υ i , r i).

Considering that when moving in a circle, the radius vector and the linear velocity vector for i-th material mutually perpendicular points,

sin(υ i , r i) = 1.

So the angular momentum of a material point for rotational motion will take the form

L = m i υ i r i .

The moment of force that acts on the i-th material point

The vector product of the radius vector, which is drawn to the point of application of the force, and this force is called the moment of the force acting on i-th material point relative to the axis of rotation.

In scalar form

M i = r i F i sin(r i , F i).

Considering that r i sinα = l i ,M i = l i F i .

Magnitude l i, equal to the length of the perpendicular lowered from the point of rotation to the direction of action of the force, is called the arm of the force F i.

Dynamics of rotational motion

The equation for the dynamics of rotational motion is written as follows:

M = dL/dt.

The formulation of the law is as follows: the rate of change of angular momentum of a body that rotates around a fixed axis is equal to the resulting moment relative to this axis of all external forces applied to the body.

Moment of impulse and moment of inertia

It is known that for the i-th material point the angular momentum in scalar form is given by the formula

L i = m i υ i r i .

If instead of linear speed we substitute its expression through angular speed:

υ i = ωr i ,

then the expression for the angular momentum will take the form

L i = m i r i 2 ω.

Magnitude I i = m i r i 2 is called the moment of inertia relative to the axis of the i-th material point of an absolutely rigid body passing through its center of mass. Then we write the angular momentum of the material point:

L i = I i ω.

We write the angular momentum of an absolutely rigid body as the sum of the angular momentum of the material points that make up this body:

L = Iω.

Moment of force and moment of inertia

The law of rotational motion states:

M = dL/dt.

It is known that the angular momentum of a body can be represented through the moment of inertia:

L = Iω.

M = Idω/dt.

Considering that the angular acceleration is determined by the expression

ε = dω/dt,

we obtain a formula for the moment of force, represented through the moment of inertia:

M = Iε.

Comment. A moment of force is considered positive if the angular acceleration that causes it is greater than zero, and vice versa.

Steiner's theorem. Law of addition of moments of inertia

If the axis of rotation of a body does not pass through its center of mass, then relative to this axis one can find its moment of inertia using Steiner’s theorem:
I = I 0 + ma 2,

Where I 0- initial moment of inertia of the body; m- body weight; a- distance between axles.

If a system that rotates around a fixed axis consists of n bodies, then the total moment of inertia of this type of system will be equal to the sum of the moments of its components (the law of addition of moments of inertia).

Angle of rotation, angular velocity and angular acceleration

Rotation of a rigid body around a fixed axis It is called such a movement in which two points of the body remain motionless during the entire time of movement. In this case, all points of the body located on a straight line passing through its fixed points also remain motionless. This line is called axis of rotation of the body.

If A And IN- fixed points of the body (Fig. 15 ), then the axis of rotation is the axis Oz, which can have any direction in space, not necessarily vertical. One axis direction Oz is taken as positive.

We draw a fixed plane through the axis of rotation By and mobile P, attached to a rotating body. Let at the initial moment of time both planes coincide. Then at a moment in time t the position of the moving plane and the rotating body itself can be determined by the dihedral angle between the planes and the corresponding linear angle φ between straight lines located in these planes and perpendicular to the axis of rotation. Corner φ called body rotation angle.

The position of the body relative to the chosen reference system is completely determined in any

moment in time, if given the equation φ =f(t) (5)

Where f(t)- any twice differentiable function of time. This equation is called equation for the rotation of a rigid body around a fixed axis.

A body rotating around a fixed axis has one degree of freedom, since its position is determined by specifying only one parameter - the angle φ .

Corner φ is considered positive if it is plotted counterclockwise, and negative in the opposite direction when viewed from the positive direction of the axis Oz. The trajectories of points of a body during its rotation around a fixed axis are circles located in planes perpendicular to the axis of rotation.

To characterize the rotational motion of a rigid body around a fixed axis, we introduce the concepts of angular velocity and angular acceleration. Algebraic angular velocity of the body at any moment in time is called the first derivative with respect to time of the angle of rotation at this moment, i.e. dφ/dt = φ. It is a positive quantity when the body rotates counterclockwise, since the angle of rotation increases with time, and negative when the body rotates clockwise, because the angle of rotation decreases.

The angular velocity module is denoted by ω. Then ω= ׀dφ/dt׀= ׀φ ׀ (6)

The dimension of angular velocity is set in accordance with (6)

[ω] = angle/time = rad/s = s -1.

In engineering, angular velocity is the rotational speed expressed in revolutions per minute. In 1 minute the body will rotate through an angle 2πп, If n- number of revolutions per minute. Dividing this angle by the number of seconds in a minute, we get: (7)

Algebraic angular acceleration of the body is called the first derivative with respect to time of the algebraic speed, i.e. second derivative of the rotation angle d 2 φ/dt 2 = ω. Let us denote the angular acceleration module ε , Then ε=|φ| (8)

The dimension of angular acceleration is obtained from (8):

[ε ] = angular velocity/time = rad/s 2 = s -2

If φ’’>0 at φ’>0 , then the algebraic angular velocity increases with time and, therefore, the body rotates accelerated at the moment in time in the positive direction (counterclockwise). At φ’’<0 And φ’<0 the body rotates rapidly in a negative direction. If φ’’<0 at φ’>0 , then we have slow rotation in a positive direction. At φ’’>0 And φ’<0 , i.e. slow rotation occurs in the negative direction. Angular velocity and angular acceleration in the figures are depicted by arc arrows around the axis of rotation. The arc arrow for angular velocity indicates the direction of rotation of the bodies;

For accelerated rotation, the arc arrows for angular velocity and angular acceleration have the same directions; for slow rotation, their directions are opposite.

Special cases of rotation of a rigid body

Rotation is said to be uniform if ω=const, φ= φ’t

The rotation will be uniform if ε=const. φ’= φ’ 0 + φ’’t and

In general, if φ’’ not all the time

Velocities and accelerations of body points

The equation for the rotation of a rigid body around a fixed axis is known φ= f(t)(Fig. 16). Distance s points M in a moving plane P along a circular arc (point trajectory), measured from the point M o, located in a fixed plane, expressed through the angle φ addiction s=hφ, Where h-radius of the circle along which the point moves. It is the shortest distance from a point M to the axis of rotation. This is sometimes called the radius of rotation of a point. At each point of the body, the radius of rotation remains unchanged when the body rotates around a fixed axis.

Algebraic speed of a point M determined by the formula v τ =s’=hφ Point speed module: v=hω(9)

The velocities of body points when rotating around a fixed axis are proportional to their shortest distances to this axis. The proportionality coefficient is the angular velocity. The velocities of the points are directed along tangents to the trajectories and, therefore, are perpendicular to the radii of rotation. Velocities of body points located on a straight line segment OM, in accordance with (9) are distributed according to a linear law. They are mutually parallel, and their ends are located on the same straight line passing through the axis of rotation. We decompose the acceleration of a point into tangential and normal components, i.e. a=a τ +a nτ Tangential and normal accelerations are calculated using formulas (10)

since for a circle the radius of curvature is p=h(Fig. 17 ). Thus,

Tangent, normal and total accelerations of points, as well as velocities, are also distributed according to a linear law. They depend linearly on the distances of the points to the axis of rotation. Normal acceleration is directed along the radius of the circle towards the axis of rotation. The direction of the tangential acceleration depends on the sign of the algebraic angular acceleration. At φ’>0 And φ’’>0 or φ’<0 And φ’<0 we have accelerated rotation of the body and directions of vectors a τ And v match. If φ’ And φ’" have different signs (slow rotation), then a τ And v directed opposite to each other.

Having designated α the angle between the total acceleration of a point and its radius of rotation, we have

tgα = | a τ |/a n = ε/ω 2 (11)

since normal acceleration a p always positive. Corner A the same for all points of the body. It should be postponed from acceleration to the radius of rotation in the direction of the arc arrow of angular acceleration, regardless of the direction of rotation of the rigid body.

Vectors of angular velocity and angular acceleration

Let us introduce the concepts of vectors of angular velocity and angular acceleration of a body. If TO is the unit vector of the rotation axis directed in its positive direction, then the angular velocity vectors ώ and angular acceleration ε determined by expressions (12)

Because k is a vector constant in magnitude and direction, then from (12) it follows that

ε=dώ/dt(13)

At φ’>0 And φ’’>0 vector directions ώ And ε match. They are both directed towards the positive side of the rotation axis Oz(Fig. 18.a)If φ’>0 And φ’’<0 , then they are directed in opposite directions (Fig. 18.b ). The angular acceleration vector coincides in direction with the angular velocity vector during accelerated rotation and is opposite to it during slow rotation. Vectors ώ And ε can be depicted at any point on the rotation axis. They are moving vectors. This property follows from the vector formulas for the velocities and accelerations of body points.

Complex point movement

Basic Concepts

To study some more complex types of motion of a rigid body, it is advisable to consider the simplest complex motion of a point. In many problems, the motion of a point must be considered relative to two (or more) reference systems moving relative to each other. Thus, the movement of a spacecraft moving towards the Moon must be considered simultaneously both relative to the Earth and relative to the Moon, which is moving relative to the Earth. Any movement of a point can be considered complex, consisting of several movements. For example, the movement of a ship along a river relative to the Earth can be considered complex, consisting of movement through the water and together with the flowing water.

In the simplest case, the complex movement of a point consists of relative and translational movements. Let's define these movements. Let us have two reference systems moving relative to each other. If one of these systems O l x 1 y 1 z 1(Fig. 19 ) taken as the main or stationary one (its movement relative to other reference systems is not considered), then the second reference system Oxyz will move relative to the first one. Motion of a point relative to a moving reference frame Oxyz called relative. The characteristics of this movement, such as trajectory, speed and acceleration, are called relative. They are designated by the index r; for speed and acceleration v r , a r . Motion of a point relative to the main or fixed system reference frame O 1 x 1 y 1 z 1 called absolute(or complex ). It is also sometimes called composite movement. The trajectory, speed and acceleration of this movement are called absolute. The speed and acceleration of absolute motion are indicated by the letters v, a no indexes.

The portable movement of a point is the movement that it makes together with a moving frame of reference, as a point rigidly attached to this system at the moment in time under consideration. Due to relative motion, a moving point at different times coincides with different points of the body S, with which the moving reference system is attached. The portable speed and the portable acceleration are the speed and acceleration of that point of the body S, with which the moving point currently coincides. Portable speed and acceleration denote v e , a e.

If the trajectories of all points of the body S, attached to the moving reference system, depicted in the figure (Fig. 20), then we obtain a family of lines - a family of trajectories of the portable movement of a point M. Due to the relative motion of the point M at each moment of time it is on one of the trajectories of portable movement. Dot M can coincide with only one point on each of the trajectories of this family of portable trajectories. In this regard, it is sometimes believed that there are no trajectories of portable movement, since it is necessary to consider lines as trajectories of portable movement, for which only one point is actually a point of the trajectory.

In the kinematics of a point, the movement of a point relative to any reference system was studied, regardless of whether this reference system moves relative to other systems or not. Let us supplement this study by considering complex motion, in the simplest case consisting of relative and figurative motion. One and the same absolute motion, choosing different moving frames of reference, can be considered to consist of different portable and, accordingly, relative motions.

Speed ​​addition

Let us determine the speed of the absolute movement of a point if the speeds of the relative and portable movements of this point are known. Let the point make only one, relative movement with respect to the moving frame of reference Oxyz and at the moment of time t occupy position M on the trajectory of the relative movement (Fig. 20). At time t+ t, due to relative motion, the point will be in position M 1, having moved MM 1 along the trajectory of relative motion. Let's assume that the point is involved Oxyz and with a relative trajectory it will move along some curve on MM 2. If a point participates simultaneously in both relative and portable movements, then in time A; she will move to MM" along the trajectory of absolute motion and at the moment of time t+At will take the position M". If time At little and then go to the limit at At, tending to zero, then small displacements along curves can be replaced by segments of chords and taken as displacement vectors. Adding the vector displacements, we get

In this respect, small quantities of a higher order are discarded, tending to zero at At, tending to zero. Passing to the limit, we have (14)

Therefore, (14) will take the form (15)

The so-called velocity addition theorem was obtained: the speed of the absolute movement of a point is equal to the vector sum of the speeds of the portable and relative movements of this point. Since in the general case the velocities of the portable and relative movements are not perpendicular, then (15’)

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