Rotational movement of a body around an axis. Rotational motion of a rigid body

And Savelyeva.

During the forward motion of a body (§ 60 in the textbook by E. M. Nikitin), all its points move along identical trajectories and in each this moment they have equal speeds and equal accelerations.

Therefore, the translational motion of a body is determined by the movement of any one point, usually the movement of the center of gravity.

When considering the movement of a car (problem 147) or a diesel locomotive (problem 141) in any problem, we actually consider the movement of their centers of gravity.

The rotational movement of a body (E.M. Nikitin, § 61) cannot be identified with the movement of any one of its points. The axis of any rotating body (diesel flywheel, electric motor rotor, machine spindle, fan blades, etc.) during movement occupies the same place in space relative to the surrounding stationary bodies.

Movement of a material point or forward movement bodies are characterized depending on time linear quantities s (path, distance), v (speed) and a (acceleration) with its components a t and a n.

Rotational movement bodies depending on time t characterize angular values: φ (angle of rotation in radians), ω (angular velocity in rad/sec) and ε (angular acceleration in rad/sec 2).

The law of rotational motion of a body is expressed by the equation
φ = f(t).

Angular velocity- a quantity characterizing the speed of rotation of a body is defined in the general case as the derivative of the angle of rotation with respect to time
ω = dφ/dt = f" (t).

Angular acceleration- a quantity characterizing the rate of change of angular velocity is defined as the derivative of angular velocity
ε = dω/dt = f"" (t).

When starting to solve problems on the rotational motion of a body, it is necessary to keep in mind that in technical calculations and problems, as a rule, angular displacement is expressed not in radians φ, but in revolutions φ about.

Therefore, it is necessary to be able to move from the number of revolutions to the radian measurement of angular displacement and vice versa.

Since one full revolution corresponds to 2π rad, then
φ = 2πφ about and φ about = φ/(2π).

Angular speed in technical calculations is very often measured in revolutions produced per minute (rpm), so it is necessary to clearly understand that ω rad/sec and n rpm express the same concept - the speed of rotation of a body (angular speed) , but in different units - in rad/sec or in rpm.

The transition from one unit of angular velocity to another is made according to the formulas
ω = πn/30 and n = 30ω/π.

During the rotational motion of a body, all its points move in circles, the centers of which are located on one fixed straight line (the axis of the rotating body). When solving the problems given in this chapter, it is very important to clearly understand the relationship between the angular quantities φ, ω and ε, which characterize the rotational motion of the body, and the linear quantities s, v, a t and an, characterizing the movement of various points of this body (Fig. 205).

If R is the distance from the geometric axis of a rotating body to any point A (in Fig. 205 R = OA), then the relationship between φ - the angle of rotation of the body and s - the distance traveled by a point of the body during the same time is expressed as follows:
s = φR.

The relationship between the angular velocity of a body and the velocity of a point at each given moment is expressed by the equality
v = ωR.

The tangential acceleration of a point depends on angular acceleration and is determined by the formula
a t = εR.

The normal acceleration of a point depends on the angular velocity of the body and is determined by the relationship
a n = ω 2 R.

When solving the problem given in this chapter, it is necessary to clearly understand that rotation is the movement solid, not points. Taken separately material point does not rotate, but moves in a circle - makes a curvilinear movement.

§ 33. Uniform rotational motion

If the angular velocity is ω=const, then the rotational motion is called uniform.

The uniform rotation equation has the form
φ = φ 0 + ωt.

In the particular case when the initial angle of rotation φ 0 =0,
φ = ωt.

Angular velocity of a uniformly rotating body
ω = φ/t
can be expressed like this:
ω = 2π/T,
where T is the period of rotation of the body; φ=2π - angle of rotation for one period.

§ 34. Uniformly alternating rotational motion

Rotational motion with variable angular velocity is called uneven (see below § 35). If the angular acceleration ε=const, then the rotational motion is called equally variable. Thus, uniform rotation of the body is special case uneven rotational movement.

Equation of uniform rotation
(1) φ = φ 0 + ω 0 t + εt 2 /2
and the equation expressing the angular velocity of a body at any time,
(2) ω = ω 0 + εt
represent a set of basic formulas for the rotational uniform motion of a body.

These formulas include only six quantities: three constants for a given problem φ 0, ω 0 and ε and three variables φ, ω and t. Consequently, the condition of each problem for uniform rotation must contain at least four specified quantities.

For the convenience of solving some problems, two more auxiliary formulas can be obtained from equations (1) and (2).

Let us exclude angular acceleration ε from (1) and (2):
(3) φ = φ 0 + (ω + ω 0)t/2.

Let us exclude time t from (1) and (2):
(4) φ = φ 0 + (ω 2 - ω 0 2)/(2ε).

In the particular case of uniformly accelerated rotation starting from a state of rest, φ 0 =0 and ω 0 =0. Therefore, the above basic and auxiliary formulas take the following form:
(5) φ = εt 2 /2;
(6) ω = εt;
(7) φ = ωt/2;
(8) φ = ω 2 /(2ε).

§ 35. Uneven rotational motion

Let's consider an example of solving a problem in which non-uniform rotational motion of a body is specified.

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 P- 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 considered moment in time in positive side(counterclock-wise). 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 always,

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 up. 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 up. 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, to which the moving reference system is attached. The portable speed and 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 transfer 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 is 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.

In nature and technology, we often encounter the manifestation of rotational motion of solid bodies, for example, shafts and gears. How this type of motion is described in physics, what formulas and equations are used for this, these and other issues are covered in this article.

What is rotation?

Each of us intuitively knows what kind of movement we are talking about. Rotation is a process in which a body or material point moves in a circular path around a certain axis. From a geometric point of view, a rigid body is a straight line, the distance to which remains unchanged during movement. This distance is called the radius of rotation. In what follows we will denote it by the letter r. If the axis of rotation passes through the center of mass of the body, then it is called its own axis. An example of rotation around its own axis is the corresponding movement of the planets of the solar system.

For rotation to occur, there must be centripetal acceleration, which arises due to centripetal force. This force is directed from the center of mass of the body to the axis of rotation. The nature of the centripetal force can be very different. So, on a cosmic scale, its role is played by gravity; if the body is secured by a thread, then the tension force of the latter will be centripetal. When a body rotates around its own axis, the role of centripetal force is played by the internal electrochemical interaction between the elements that make up the body (molecules, atoms).

It is necessary to understand that without the presence of a centripetal force, the body will move in a straight line.

Physical quantities describing rotation

Firstly, these are dynamic characteristics. These include:

• angular momentum L;
• moment of inertia I;
• moment of force M.

Secondly, these are kinematic characteristics. Let's list them:

• rotation angle θ;
• angular speed ω;
• angular acceleration α.

Let us briefly describe each of these quantities.

The angular momentum is determined by the formula:

Where p is linear momentum, m is the mass of a material point, v is its linear velocity.

The moment of inertia of a material point is calculated using the expression:

For any body of complex shape, the value of I is calculated as the integral sum of the moments of inertia of material points.

The moment of force M is calculated as follows:

Here F is the external force, d is the distance from the point of its application to the axis of rotation.

The physical meaning of all quantities whose names contain the word “moment” is similar to the meaning of the corresponding linear quantities. For example, the moment of force shows the ability of the applied force to be imparted to a system of rotating bodies.

Kinematic characteristics are mathematically determined by the following formulas:

As can be seen from these expressions, the angular characteristics are similar in meaning to the linear ones (velocity v and acceleration a), only they are applicable for a circular trajectory.

Dynamics of rotation

In physics, the study of the rotational motion of a rigid body is carried out using two branches of mechanics: dynamics and kinematics. Let's start with dynamics.

Dynamics studies external forces acting on a system of rotating bodies. Let's immediately write down the equation of rotational motion of a rigid body, and then analyze its components. So this equation looks like:

Which acts on a system with a moment of inertia I, causing the appearance of angular acceleration α. The smaller the value of I, the easier it is, with the help of a certain moment M, to spin the system to high speeds in short periods of time. For example, it is easier to rotate a metal rod along its axis than perpendicular to it. However, the same rod is easier to rotate around an axis perpendicular to it and passing through the center of mass than through its end.

Law of conservation of quantity L

This quantity was introduced above; it is called angular momentum. The equation of rotational motion of a rigid body, presented in the previous paragraph, is often written in a different form:

If the moment of external forces M acts on the system during time dt, then it causes a change in the angular momentum of the system by the amount dL. Accordingly, if the moment of force is zero, then L = const. This is the law of conservation of the quantity L. For it, using the relationship between linear and angular velocity, we can write:

L = m*v*r = m*ω*r 2 = I*ω.

Thus, in the absence of torque, the product of angular velocity and moment of inertia is a constant value. This physical law is used by figure skaters in their performances or by artificial satellites that need to be rotated around their own axis in outer space.

Centripetal acceleration

Above, when studying the rotational motion of a rigid body, this quantity was already described. The nature of centripetal forces was also noted. Here we will only supplement this information and provide the corresponding formulas for calculating this acceleration. Let's denote it a c.

Since the centripetal force is directed perpendicular to the axis and passes through it, it does not create a torque. That is, this force has absolutely no effect on the kinematic characteristics of rotation. However, it does create centripetal acceleration. Here are two formulas to determine it:

Thus, the greater the angular velocity and radius, the greater the force that must be applied to keep the body on a circular path. A striking example of this physical process is the skidding of a car during a turn. A skid occurs if the centripetal force, the role of which is played by the friction force, becomes less than the centrifugal force (inertial characteristic).

Three main kinematic characteristics were listed above in the article. solid body is described by the following formulas:

θ = ω*t => ω = const., α = 0;

θ = ω 0 *t + α*t 2 /2 => ω = ω 0 + α*t, α = const.

The first line contains formulas for uniform rotation, which assumes the absence of an external moment of force acting on the system. The second line contains formulas for uniformly accelerated motion in a circle.

Note that rotation can occur not only with positive acceleration, but also with negative one. In this case, in the formulas of the second line, you should put a minus sign before the second term.

Example of problem solution

A moment of force of 1000 N*m acted on the metal shaft for 10 seconds. Knowing that the moment of inertia of the shaft is equal to 50 kg * m 2, it is necessary to determine the angular velocity that the mentioned moment of force imparted to the shaft.

Using the basic equation of rotation, we calculate the acceleration of the shaft:

Since this angular acceleration acted on the shaft for a time t = 10 seconds, to calculate the angular velocity we use the formula for uniformly accelerated motion:

ω = ω 0 + α*t = M/I*t.

Here ω 0 = 0 (the shaft did not rotate until the moment of force M acted).

We substitute the numerical values ​​of the quantities into the equality, and we get:

ω = 1000/50*10 = 200 rad/s.

To convert this number into the usual revolutions per second, you need to divide it by 2*pi. Having performed this action, we find that the shaft will rotate at a frequency of 31.8 rpm.

DEFINITION: Rotational motion of a rigid body we will call such a movement in which all points of the body move in circles, the centers of which lie on the same straight line, called the axis of rotation.

To study the dynamics of the rotational one, we add to the known kinematic quantities two quantities: moment of power(M) and moment of inertia(J).

1. It is known from experience: the acceleration of rotational motion depends not only on the magnitude of the force acting on the body, but also on the distance from the axis of rotation to the line along which the force acts. To characterize this circumstance, a physical quantity called moment of force.

Let's consider the simplest case.

DEFINITION: The moment of a force about a certain point “O” is a vector quantity defined by the expression , where is the radius vector drawn from the point “O” to the point of application of the force.

From the definition it follows that is an axial vector. Its direction is chosen so that the rotation of the vector around the point “O” in the direction of the force and the vector form a right-handed system. The modulus of the moment of force is equal to , where a is the angle between the directions of the vectors and , and l= r sin a is the length of the perpendicular dropped from point “O” to the straight line along which the force acts (called shoulder of strength relative to point “O”) (Fig. 4.2).

2. Experimental data indicate that the magnitude of angular acceleration is influenced not only by the mass of the rotating body, but also by the distribution of mass relative to the axis of rotation. The quantity that takes this circumstance into account is called moment of inertia relative to the axis of rotation.

DEFINITION: Strictly speaking, moment of inertia body relative to a certain axis of rotation is called the value J, equal to the sum of the products of elementary masses by the squares of their distances from a given axis.

The summation is carried out over all elementary masses into which the body was divided. It should be borne in mind that this quantity (J) exists regardless of rotation (although the concept of moment of inertia was introduced when considering the rotation of a rigid body).

Each body, regardless of whether it is at rest or rotating, has a certain moment of inertia relative to any axis, just as a body has mass regardless of whether it is moving or at rest.

Considering that , the moment of inertia can be represented as: . This relationship is approximate and the smaller the elementary volumes and the corresponding mass elements, the more accurate it will be. Consequently, the task of finding moments of inertia comes down to integration: . Here integration is carried out over the entire volume of the body.

Let us write down the moments of inertia of some bodies of regular geometric shape.

1. Uniform long rod.
Rice. 4.3	The moment of inertia about the axis perpendicular to the rod and passing through its middle is equal to
2. Solid cylinder or disk.
Rice. 4.4	The moment of inertia about the axis coinciding with the geometric axis is equal to .
3. Thin-walled cylinder of radius R.
Rice. 4.5
4. Moment of inertia of a ball of radius R relative to an axis passing through its center
Rice. 4.6
5. Moment of inertia of a thin disk (thickness b<
Rice. 4.7
6. Moment of inertia of the block
Rice. 4.8
7. Moment of inertia of the ring
Rice. 4.9

Calculation of the moment of inertia here is quite simple, because The body is assumed to be homogeneous and symmetrical, and the moment of inertia is determined relative to the axis of symmetry.

To determine the moment of inertia of a body relative to any axis, it is necessary to use Steiner’s theorem.

DEFINITION: Moment of inertia J about an arbitrary axis is equal to the sum of the moment of inertia J c relative to an axis parallel to the given one and passing through the center of inertia of the body, and the product of the body mass by the square of the distance between the axes (Fig. 4.10).

Progressive is the movement of a rigid body in which any straight line invariably associated with this body remains parallel to its initial position.

Theorem. During the translational motion of a rigid body, all its points describe identical trajectories and at each given moment have equal velocity and acceleration in magnitude and direction.

Proof. Let's draw through two points and , a linearly moving body segment
and consider the movement of this segment in position
. At the same time, the point describes the trajectory
, and point – trajectory
(Fig. 56).

Considering that the segment
moves parallel to itself, and its length does not change, it can be established that the trajectories of points And will be the same. This means that the first part of the theorem is proven. We will determine the position of the points And vector method relative to a fixed origin . Moreover, these radii - vectors are dependent
. Because. neither the length nor the direction of the segment
does not change when the body moves, then the vector

. Let's move on to determining the speeds using dependence (24):

, we get
.

Let's move on to determining accelerations using dependence (26):

, we get
.

From the proven theorem it follows that the translational motion of a body will be completely determined if the motion of only one point is known. Therefore, the study of the translational motion of a rigid body comes down to the study of the movement of one of its points, i.e. to the point kinematics problem.

Topic 11. Rotational motion of a rigid body

Rotational This is the movement of a rigid body in which two of its points remain motionless throughout the entire movement. In this case, the straight line passing through these two fixed points is called axis of rotation.

During this movement, each point of the body that does not lie on the axis of rotation describes a circle, the plane of which is perpendicular to the axis of rotation, and its center lies on this axis.

We draw through the axis of rotation a fixed plane I and a movable plane II, invariably connected to the body and rotating with it (Fig. 57). The position of plane II, and accordingly the entire body, in relation to plane I in space, is completely determined by the angle . When a body rotates around an axis this angle is a continuous and unambiguous function of time. Therefore, knowing the law of change of this angle over time, we can determine the position of the body in space:

- law of rotational motion of a body. (43)

In this case, we will assume that the angle measured from a fixed plane in the direction opposite to the clockwise movement, when viewed from the positive end of the axis . Since the position of a body rotating around a fixed axis is determined by one parameter, such a body is said to have one degree of freedom.

Angular velocity

The change in the angle of rotation of a body over time is called angular body speed and is designated
(omega):

.(44)

Angular velocity, just like linear velocity, is a vector quantity, and this vector built on the axis of rotation of the body. It is directed along the axis of rotation in that direction so that, looking from its end to its beginning, one can see the rotation of the body counterclockwise (Fig. 58). The modulus of this vector is determined by dependence (44). Application point on the axis can be chosen arbitrarily, since the vector can be transferred along the line of its action. If we denote the orth-vector of the rotation axis by , then we obtain the vector expression for angular velocity:

. (45)

Angular acceleration

The rate of change in the angular velocity of a body over time is called angular acceleration body and is designated (epsilon):

. (46)

Angular acceleration is a vector quantity, and this vector built on the axis of rotation of the body. It is directed along the axis of rotation in that direction so that, looking from its end to its beginning, one can see the direction of rotation of the epsilon counterclockwise (Fig. 58). The modulus of this vector is determined by dependence (46). Application point on the axis can be chosen arbitrarily, since the vector can be transferred along the line of its action.

If we denote the orth-vector of the rotation axis by , then we obtain the vector expression for angular acceleration:

. (47)

If the angular velocity and acceleration are of the same sign, then the body rotates expedited, and if different – slowly. An example of slow rotation is shown in Fig. 58.

Let us consider special cases of rotational motion.

1. Uniform rotation:

,
.

,
,
,

,
. (48)

2. Equal rotation:

.

,
,
,
,
,
,
,
,

,
,
.(49)

Relationship between linear and angular parameters

Consider the movement of an arbitrary point
rotating body. In this case, the trajectory of the point will be a circle with radius
, located in a plane perpendicular to the axis of rotation (Fig. 59, A).

Let us assume that at the moment of time the point is in position
. Let us assume that the body rotates in a positive direction, i.e. in the direction of increasing angle . At a moment in time
the point will take position
. Let's denote the arc
. Therefore, over a period of time
the point has passed the way
. Her average speed , and when
,
. But, from Fig. 59, b, it's clear that
. Then. Finally we get

. (50)

Here - linear speed of the point
. As was obtained earlier, this speed is directed tangentially to the trajectory at a given point, i.e. tangent to the circle.

Thus, the module of the linear (circumferential) velocity of a point of a rotating body is equal to the product of the absolute value of the angular velocity and the distance from this point to the axis of rotation.

Now let's connect the linear components of the acceleration of a point with the angular parameters.

,
. (51)

The modulus of the tangential acceleration of a point of a rigid body rotating around a fixed axis is equal to the product of the angular acceleration of the body and the distance from this point to the axis of rotation.

,
. (52)

The modulus of normal acceleration of a point of a rigid body rotating around a fixed axis is equal to the product of the square of the angular velocity of the body and the distance from this point to the axis of rotation.

Then the expression for the total acceleration of the point takes the form

. (53)

Vector directions ,,shown in Figure 59, V.

Flat motion of a rigid body is a movement in which all points of the body move parallel to some fixed plane. Examples of such movement:

The motion of any body whose base slides along a given fixed plane;

Rolling of a wheel along a straight section of track (rail).

We obtain the equations of plane motion. To do this, consider a flat figure moving in the plane of the sheet (Fig. 60). Let us relate this movement to a fixed coordinate system
, and with the figure itself we connect the moving coordinate system
, which moves with it.

Obviously, the position of a moving figure on a stationary plane is determined by the position of the moving axes
relative to fixed axes
. This position is determined by the position of the moving origin , i.e. coordinates ,and rotation angle , a moving coordinate system, relatively fixed, which we will count from the axis in the direction opposite to the clockwise movement.

Consequently, the movement of a flat figure in its plane will be completely determined if the values ​​of ,,, i.e. equations of the form:

,
,
. (54)

Equations (54) are equations of plane motion of a rigid body, since if these functions are known, then for each moment of time it is possible to find from these equations, respectively ,,, i.e. determine the position of a moving figure at a given moment in time.

Let's consider special cases:

1.

, then the movement of the body will be translational, since the moving axes move while remaining parallel to their initial position.

2.

,

. With this movement, only the angle of rotation changes , i.e. the body will rotate about an axis passing perpendicular to the drawing plane through the point .

Decomposition of the motion of a flat figure into translational and rotational

Consider two consecutive positions And
occupied by the body at moments of time And
(Fig. 61). Body from position to position
can be transferred as follows. Let's move the body first progressively. In this case, the segment
will move parallel to itself to position
, and then let's turn body around a point (pole) at an angle
until the points coincide And .

Hence, any plane motion can be represented as the sum of translational motion together with the selected pole and rotational motion, relative to this pole.

Let's consider methods that can be used to determine the velocities of points of a body performing plane motion.

1. Pole method. This method is based on the resulting decomposition of plane motion into translational and rotational. The speed of any point of a flat figure can be represented in the form of two components: translational, with a speed equal to the speed of an arbitrarily chosen point -poles , and rotational around this pole.

Let's consider a flat body (Fig. 62). The equations of motion are:
,
,
.

From these equations we determine the speed of the point (as with the coordinate method of specifying)

,
,
.

Thus, the speed of the point - the quantity is known. We take this point as a pole and determine the speed of an arbitrary point
bodies.

Speed
will consist of a translational component , when moving along with the point , and rotational
, when rotating the point
relative to the point . Point speed move to point
parallel to itself, since during translational motion the velocities of all points are equal both in magnitude and direction. Speed
will be determined by dependence (50)
, and this vector is directed perpendicular to the radius
in the direction of rotation
. Vector
will be directed along the diagonal of a parallelogram built on vectors And
, and its module is determined by the dependency:

, .(55)

2. Theorem on the projections of velocities of two points of a body.

The projections of the velocities of two points of a rigid body onto a straight line connecting these points are equal to each other.

Consider two points of the body And (Fig. 63). Taking a point beyond the pole, we determine the direction depending on (55):
. We project this vector equality onto the line
and considering that
perpendicular
, we get

3. Instantaneous velocity center.

Instantaneous velocity center(MCS) is a point whose speed at a given time is zero.

Let us show that if a body does not move translationally, then such a point exists at every moment of time and, moreover, is unique. Let at a moment in time points And bodies lying in section , have speeds And , not parallel to each other (Fig. 64). Then point
, lying at the intersection of perpendiculars to the vectors And , and there will be an MCS, since
.

Indeed, if we assume that
, then according to Theorem (56), the vector
must be perpendicular at the same time
And
, which is impossible. From the same theorem it is clear that no other section point at this moment in time cannot have a speed equal to zero.

Using the pole method
- pole, determine the speed of the point (55): because
,
. (57)

A similar result can be obtained for any other point of the body. Therefore, the speed of any point on the body is equal to its rotational speed relative to the MCS:

,
,
, i.e. the velocities of body points are proportional to their distances to the MCS.

From the three considered methods for determining the velocities of points of a flat figure, it is clear that the MCS is preferable, since here the speed is immediately determined both in magnitude and in the direction of one component. However, this method can be used if we know or can determine the position of the MCS for the body.

Determining the position of the MCS

1. If we know for a given position of the body the directions of the velocities of two points of the body, then the MCS will be the point of intersection of the perpendiculars to these velocity vectors.

2. The velocities of two points of the body are antiparallel (Fig. 65, A). In this case, the perpendicular to the velocities will be common, i.e. The MCS is located somewhere on this perpendicular. To determine the position of the MCS, it is necessary to connect the ends of the velocity vectors. The point of intersection of this line with the perpendicular will be the desired MCS. In this case, the MCS is located between these two points.

3. The velocities of two points of the body are parallel, but not equal in magnitude (Fig. 65, b). The procedure for obtaining the MDS is similar to that described in paragraph 2.

d) The velocities of two points are equal in both magnitude and direction (Fig. 65, V). We obtain the case of instantaneous translational motion, in which the velocities of all points of the body are equal. Consequently, the angular velocity of the body in this position is zero:

4. Let us determine the MCS for a wheel rolling without sliding on a stationary surface (Fig. 65, G). Since the movement occurs without sliding, at the point of contact of the wheel with the surface the speed will be the same and equal to zero, since the surface is stationary. Consequently, the point of contact of the wheel with a stationary surface will be the MCS.

Determination of accelerations of points of a plane figure

When determining the accelerations of points on a flat figure, there is an analogy with methods for determining velocities.

1. Pole method. Just as when determining velocities, we take as a pole an arbitrary point of the body whose acceleration we know or we can determine. Then the acceleration of any point of a flat figure is equal to the sum of the accelerations of the pole and the acceleration in rotational motion around this pole:

In this case, the component
determines the acceleration of a point as it rotates around the pole . When rotating, the trajectory of the point will be curvilinear, which means
(Fig. 66).

Then dependence (58) takes the form
. (59)

Taking into account dependencies (51) and (52), we obtain
,
.

2. Instant acceleration center.

Instant acceleration center(MCU) is a point whose acceleration at a given time is zero.

Let us show that at any given moment of time such a point exists. We take a point as a pole , whose acceleration
we know. Finding the angle , lying within
, and satisfying the condition
. If
, That
and vice versa, i.e. corner delayed in direction . Let's postpone from the point at an angle to vector
line segment
(Fig. 67). The point obtained by such constructions
there will be an MCU.

Indeed, the acceleration of the point
equal to the sum of accelerations
poles and acceleration
in rotational motion around a pole :
.

,
. Then
. On the other hand, acceleration
forms with the direction of the segment
corner
, which satisfies the condition
. A minus sign is placed in front of the tangent of the angle , since rotation
relative to the pole counterclockwise, and the angle
is deposited clockwise. Then
.

Hence,
and then
.

Special cases of determining the MCU

1.
. Then
, and therefore the MCU does not exist. In this case, the body moves translationally, i.e. the velocities and accelerations of all points of the body are equal.

2.
. Then
,
. This means that the MCU lies at the intersection of the lines of action of the accelerations of the points of the body (Fig. 68, A).

3.
. Then,
,
. This means that the MCU lies at the intersection of perpendiculars to the accelerations of points of the body (Fig. 68, b).

4.
. Then
,

. This means that the MCU lies at the intersection of rays drawn to the accelerations of points of the body at an angle (Fig. 68, V).

From the considered special cases we can conclude: if we accept the point
beyond the pole, then the acceleration of any point of a flat figure is determined by the acceleration in rotational motion around the MCU:

. (60)

Complex point movement a movement in which a point simultaneously participates in two or more movements is called. With such movement, the position of the point is determined relative to the moving and relatively stationary reference systems.

The movement of a point relative to a moving reference frame is called relative motion of a point . We agree to denote the parameters of relative motion
.

The movement of that point of the moving reference system with which the moving point relative to the stationary reference system currently coincides is called portable movement of the point .
.

We agree to denote the parameters of portable motion The movement of a point relative to a fixed frame of reference is called absolute (complex) point movement
.

. We agree to denote the parameters of absolute motion

As an example of complex movement, we can consider the movement of a person in a moving vehicle (tram). In this case, the human movement is related to the moving coordinate system - the tram and to the fixed coordinate system - the earth (road). Then, based on the definitions given above, the movement of a person relative to the tram is relative, the movement together with the tram relative to the ground is portable, and the movement of a person relative to the ground is absolute.
We will determine the position of the point
radii - vectors relative to the moving
and motionless coordinate systems (Fig. 69). Let us introduce the following notation:
- radius vector defining the position of the point
,
;relative to the moving coordinate system - radius vector that determines the position of the beginning of the moving coordinate system (point );) (dots
- radius – a vector that determines the position of a point
;
,.

relative to a fixed coordinate system

Let us obtain conditions (constraints) corresponding to relative, portable and absolute motions.
1. When considering relative motion, we will assume that the point
moves relative to the moving coordinate system
- radius – a vector that determines the position of a point
, and the moving coordinate system itself

doesn't move.
Then the coordinates of the point

,

,

.

will change in relative motion, but the orth-vectors of the moving coordinate system will not change in direction:
relative to the moving coordinate system are fixed, and the point moves along with the moving coordinate system
relatively stationary
:

,

,

,.

3. With absolute motion, the point also moves relatively
and together with the coordinate system
relatively stationary
:

Then the expressions for the velocities, taking into account (27), have the form

,
,

Comparing these dependencies, we obtain the expression for absolute speed:
. (61)

We obtained a theorem on the addition of the velocities of a point in complex motion: the absolute speed of a point is equal to the geometric sum of the relative and portable speed components.

Using dependence (31), we obtain expressions for accelerations:

,

Comparing these dependencies, we obtain an expression for absolute acceleration:
.

We found that the absolute acceleration of a point is not equal to the geometric sum of the relative and portable acceleration components. Let us determine the absolute acceleration component in parentheses for special cases.

1. Portable translational movement of the point
. In this case, the axes of the moving coordinate system
move all the time parallel to themselves, then.

,

,

,
,
,
, Then
. Finally we get

. (62)

If the portable motion of a point is translational, then the absolute acceleration of the point is equal to the geometric sum of the relative and portable components of the acceleration.

2. The portable movement of the point is non-translational. This means that in this case the moving coordinate system
rotates around the instantaneous axis of rotation with angular velocity (Fig. 70). Let us denote the point at the end of the vector through . Then, using the vector method of specifying (15), we obtain the velocity vector of this point
.

On the other side,
. Equating the right-hand sides of these vector equalities, we obtain:
. Proceeding similarly for the remaining unit vectors, we obtain:
,
.

In the general case, the absolute acceleration of a point is equal to the geometric sum of the relative and portable acceleration components plus the doubled vector product of the angular velocity vector of the portable motion and the linear velocity vector of the relative motion.

The double vector product of the angular velocity vector of the portable motion and the linear velocity vector of the relative motion is called Coriolis acceleration and is designated

. (64)

Coriolis acceleration characterizes the change in relative speed in translational motion and the change in translational velocity in relative motion.

Headed
according to the vector product rule. The Coriolis acceleration vector is always directed perpendicular to the plane formed by the vectors And , in such a way that, looking from the end of the vector
, see the turn To , through the smallest angle, counterclockwise.

The Coriolis acceleration modulus is equal to.