Rotational motion of a rigid body: equation, formulas. Rotational motion of a rigid body around a fixed axis

Rotational motion of a rigid body around a fixed axis is such a motion in which any two points belonging to the body (or invariably associated with it) remain motionless throughout the movement(Fig. 2.2) .

Figure 2.2

Passing through fixed points A And IN the straight line is called axis of rotation. Since the distance between the points of a rigid body must remain unchanged, it is obvious that during rotational motion all points belonging to the axis will be motionless, and all others will describe circles, the planes of which are perpendicular to the axis of rotation, and the centers lie on this axis. To determine the position of a rotating body, we draw through the axis of rotation along which the axis is directed Az, half-plane І – fixed and half-plane ІІ embedded in the body itself and rotating with it. Then the position of the body at any moment of time is uniquely determined by the angle taken with the corresponding sign φ between these planes, which we call body rotation angle. We will consider the angle φ positive if it is delayed from a fixed plane in a counterclockwise direction (for an observer looking from the positive end of the axis Az), and negative if clockwise. Measure angle φ We'll be in radians. To know the position of a body at any moment in time, you need to know the dependence of the angle φ from time to time t, i.e.

.

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

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

9.2.1. Angular velocity and angular acceleration of a body

The quantity characterizing the rate of change in the angle of rotation φ over time is called angular velocity.

If during a period of time
the body rotates through an angle
, then the numerically average angular velocity of the body during this period of time will be
. In the limit at
we get

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.

Sign Rule: When rotation occurs counterclockwise, ω> 0, and when clockwise, then ω< 0.

or, since radian is a dimensionless quantity,
.

In theoretical calculations it is more convenient to use the angular velocity 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 is visible counterclockwise. This vector immediately determines the magnitude of the angular velocity, the axis of rotation, and the direction of rotation around this axis.

The quantity that characterizes the rate of change in angular velocity over time is called the angular acceleration of the body.

If during a period of time
the increment in angular velocity is equal to
, then the relation
, i.e. determines the value of the average acceleration of a rotating body over time
.

When striving
we get the value angular acceleration at the moment t:

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 in time.

The unit of measurement is usually used or, which is also,
.

If the modulus of angular velocity increases with time, the rotation of the body is called accelerated, and if it decreases, - slow When the values ω And ε have the same signs, then the rotation will be accelerated, when they are different, it will be slowed down. By analogy with angular velocity, angular acceleration can also be represented as a vector , directed along the axis of rotation. At the same time

.

If a body rotates in an accelerated direction coincides with , and opposite with slow rotation.

If the angular velocity of a body remains constant during movement ( ω= const), then the rotation of the body is called uniform.

From
we have
. Hence, considering that at the initial moment of time
corner
, and taking the integrals to the left of to , and on the right from 0 to t, we will finally get

.

With uniform rotation, when =0,
And
.

The speed of uniform rotation is often determined by the number of revolutions per minute, denoting this value by n rpm Let's find the relationship between n rpm and ω 1/s. With one revolution the body will rotate by 2π, and with n rpm at 2π n; this turn is done in 1 minute, i.e. t= 1min=60s. It follows from this that

.

If the angular acceleration of a body remains constant throughout its motion (ε = const), then the rotation is called equally variable.

At the initial moment of time t=0 angle
, and the angular velocity
(- initial angular velocity).
;
=ε
. Integrating the left side of to , and the right one from 0 to t, we'll find

Angular velocity ω of this rotation
. If ω and ε have the same signs, the rotation will be uniformly accelerated, and if different - equally slow.

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 such a 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 around the axis of rotation makes one full revolution ([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 determined 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 the 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 called the moment of inertia about axis i 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 axes.

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).

This is a movement in which all points of the body move in circles, the centers of which lie on the axis of rotation.

The position of the body is specified by the dihedral angle (angle of rotation).

 =  (t) - equation of motion.

Kinematic characteristics of the body:

- angular velocity, s -1;

- angular acceleration, s -2.

The quantities  and  can be represented as vectors
, located on the axis of rotation, the direction of the vector such that from its end the rotation of the body is seen to occur counterclockwise. Direction coincides with , If >oh.

P position points of the body: M 0 M 1 = S = h.

Speed points
; at the same time
.

where
;
;
.

Acceleration body points,
- rotational acceleration (in the kinematics of a point - tangent - ):
- point-to-point acceleration (in the kinematics of the point - normal - ).

Modules:
;
;

.

Uniform and uniform rotation

1. Uniform:  = const,
;
;
- equation of motion.

2. Equally variable:  = const,
;
;
;
;
- equation of motion.

2). The mechanical drive consists of pulley 1, belt 2 and stepped wheels 3 and 4. Find the speed of rack 5, as well as the acceleration of point M at time t 1 = 1s. If the angular velocity of the pulley is  1 = 0.2t, s -1; R 1 = 15; R 3 = 40; r 3 = 5; R4 = 20; r 4 = 8 (in centimeters).

Rack speed

;

;
;
.

Where
;
;
, s -1 .

From (1) and (2) we obtain, see.

Acceleration of point M.

, s -2 at t 1 = 1 s; a = 34.84 cm/s 2 .

3.3 Plane-parallel (plane) motion of a rigid body

E that movement in which all points of the body move in planes parallel to some fixed plane.

All points of the body on any straight line perpendicular to a fixed plane move equally. Therefore, the analysis of the plane motion of a body is reduced to the study of the motion of a plane figure (section S) in its plane (xy).

This movement can be represented as a set of translational movement together with some arbitrarily selected point a, called pole, and rotational motion around the pole.

Equations of motion flat figure

x a = x a (t); y a = y a; j = j(t)

Kinematic characteristics ki of a flat figure:

- speed and acceleration of the pole; w, e - angular velocity and angular acceleration (do not depend on the choice of pole).

U alignment of movement of any point plane figure (B) can be obtained by projecting the vector equality
on the x and y axes

x 1 B , y 1 B - coordinates of the point in the coordinate system associated with the figure.

Determining point velocities

1). Analytical method.

Knowing the equations of motion x n = x n (t); y n = y n (t), we find
;
;
.

2). Velocity distribution theorem.

D differentiating equality
, we get
,

- the speed of point B when rotating a flat figure around pole A;
;

Formula for the distribution of velocities of points of a plane figure
.

WITH speed of point M of a wheel rolling without slipping

;
.

3). Velocity projection theorem.

The projections of the velocities of two points of the body onto the axis passing through these points are equal. Designing equality
on the x-axis, we have

P example

Determine the speed of water flow v N onto the rudder of the ship, if known (vessel center of gravity speed), b and b K (drift angles).

Solution: .

4). Instantaneous velocity center (IVC).

The velocities of points during plane motion of a body can be determined from the formulas of rotational motion, using the concept of MCS.

MCS is a point associated with a flat figure, the speed of which at a given time is zero (v p = 0).

In general, the MCS is the point of intersection of perpendiculars to the velocity directions of two points of the figure.

Taking point P as a pole, we have for an arbitrary point

, Then

Where
- angular velocity of the figure and
,those. the velocities of the points of a flat figure are proportional to their distances to the MCS.

Possible cases of finding the MCS
			Rolling without slipping
		MCS - at infinity

Case b corresponds to an instantaneous translational velocity distribution.

1). For a given position of the mechanism, findv B, v C, v D, w 1, w 2, w 3, if at the moment v A = 20 cm/s; BC = CD = 40 cm; OC = 25 cm; R = 20 cm.

Solution of the MCS of roller 1 - point P 1:

s -1 ;
cm/s.

MCS of link 2 - point P 2 of intersection of perpendiculars to the speed directions of points B and C:

s -1 ;
cm/s;
cm/s;
s -1 .

2). The load Q is lifted using a stepped drum 1, the angular velocity of which is w 1 = 1 s -1 ; R 1 = 3r 1 = 15 cm; AE || B.D. Find the speed v C of the axis of moving block 2.

Find the speeds of points A and B:

v A = v E = w 1* R 1 = 15 cm/s; v B = v D = w 1* r 1 = 5 cm/s.

MCS of block 2 - point P. Then
, where
;
;
cm/s.

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 both planes coincide at the initial moment of time. 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’)

Related information.

Rotational motion of a rigid body. Rotational motion is the motion of a rigid body in which all its points lying on a certain straight line, called the axis of rotation, remain motionless.

During rotational motion, all other points of the body move in planes perpendicular to the axis of rotation and describe circles whose centers lie on this axis.

To determine the position of a rotating body, we draw two half-planes through the z-axis: half-plane I - stationary and half-plane II - connected to the rigid body and rotating with it (Fig. 2.4). Then the position of the body at any moment of time will be uniquely determined by the angle j between these half-planes, taken with the corresponding sign, which is called the angle of rotation of the body.

When a body rotates, the angle of rotation j changes depending on time, i.e., it is a function of time t:

This equation is called equation rotational motion of a rigid body.

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

If during time D t= t1 + t the body makes a turn by Dj = j1 –j, then the average angular velocity of the body during this period of time will be equal to

(1.16)

To determine the value of the angular velocity of a body at a given time t let's find the limit of the ratio of the increment of the rotation angle Dj to the time interval D t as the latter tends to zero:

(2.17)

Thus, the angular velocity of the body at a given time is numerically equal to the first derivative of the angle of rotation with respect to time. The sign of the angular velocity w coincides with the sign of the angle of rotation of the body j: w > 0 at j > 0, and vice versa, if j < 0. then w < 0. The dimension of angular velocity is usually 1/s, so radians are dimensionless.

Angular velocity can be represented as a vector w , the numerical value of which is equal to dj/dt which is directed along the axis of rotation of the body in the direction from which the rotation can be seen occurring counterclockwise.

The change in the angular velocity of a body over time is characterized by angular acceleration e. By analogy with finding the average value of angular velocity, we will find an expression for determining the value of the average acceleration:

(2.18)

Then the acceleration of the rigid body at a given time is determined from the expression

(2.19)

i.e., the angular acceleration of the 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/s 2.

The angular acceleration of a rigid body, like the angular velocity, can be represented as a vector. The angular acceleration vector coincides in direction with the angular velocity vector during accelerated motion of a solid top and is directed in the opposite direction during slow motion.

Having established the characteristics of the motion of a rigid body as a whole, let us move on to studying the motion of its individual points. Let's consider some point M solid body located at a distance h from the axis of rotation r (Fig. 2.3).

When the body rotates, point M will describe a circular point of radius h centered on the axis of rotation and lying in a plane perpendicular to this axis. If during time dt an elementary body whipping occurs at an angle dj , then point M at the same time makes an elementary movement along its trajectory dS = h*dj ,. Then the speed of point M was determined from the expression

(2.20)

The speed is called the linear or circumferential speed of point M.

Thus, the linear velocity of a point on a rotating rigid body is numerically equal to the product of the angular velocity of the body and the distance from this point to the axis of rotation. Since for all points of the body the angular velocity w; has the same value, then from the formula for linear speed it follows that the linear speeds of the points of a rotating body are proportional to their distances from the axis of rotation. The linear velocity of a point of a rigid body is a vector n directed tangentially to the circle described by the point M.

Beli the distance from the axis of rotation of a solid pel to a certain point M considered as the radius vector h of point M, then the linear velocity vector of point v can be represented as the vector product of the angular velocity vector w radius vector h:

V = w * h (2/21)

Indeed, the result of the vector product (2.21) is a vector equal in modulus to the product w*h and directed (Fig. 2.5) perpendicular to the plane in which the two factors lie, in the direction from which the closest combination of the first factor with the second is observed to occur counterclockwise , i.e., tangent to the trajectory of point M.

Thus, the vector resulting from the vector product (2.21) corresponds in magnitude and direction to the linear velocity vector of point M.

Rice. 2.5

To find an expression for acceleration A point M, we differentiate with respect to time the expression (2.21) for the speed of the point

(2.22)

Taking into account that dj/dt=e, and dh/dt = v, we write expression (2.22) in the form

where аг and аn are, respectively, the tangent and normal components of the total acceleration of a point of a body during rotational motion, determined from the expressions

The tangential component of the total acceleration of a body point (tangential acceleration) at characterizes the change in the velocity vector in magnitude and is directed tangentially to the trajectory of the body point in the direction of the velocity vector during accelerated motion or in the opposite direction during slow motion. The magnitude of the tangential acceleration vector of a point of a body during rotational motion of a rigid body is determined by the expression

(2,25)

Normal component of total acceleration (normal acceleration) A" arises due to a change in the direction of the velocity vector of a point when painting a solid body. As follows from expression (2.24) for normal acceleration, this acceleration is directed along the radius h to the center of the circle along which the point moves. The modulus of the normal acceleration vector of a point during rotational motion of a rigid body is determined taking into account (2.20) by the expression