Rotation of a rigid body around a fixed axis. Rotational motion of a rigid body around a fixed axis Law of rotational motion around a fixed axis
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 relation 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).
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 a physical quantity that 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 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 the i-th material point are mutually perpendicular,
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 force acting on the 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 mass; 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).
The motion of a rigid body is called rotational if, during motion, all points of the body located on a certain straight line, called the axis of rotation, remain motionless(Fig. 2.15).
The position of the body during rotational movement is usually determined rotation angle body , which is measured as the dihedral angle between the fixed and moving planes passing through the axis of rotation. Moreover, the movable plane is connected to a rotating body.
Let us introduce into consideration moving and fixed coordinate systems, the origin of which will be placed at an arbitrary point O on the rotation axis. The Oz axis, common to the moving and fixed coordinate systems, will be directed along the axis of rotation, the axis Oh of the fixed coordinate system, we direct it perpendicular to the Oz axis so that it lies in the fixed plane, the axis Oh 1 Let's direct the moving coordinate system perpendicular to the Oz axis so that it lies in the moving plane (Fig. 2.15).
If we consider a section of a body by a plane perpendicular to the axis of rotation, then the angle of rotation φ can be defined as the angle between the fixed axis Oh and movable axis Oh 1, invariably associated with a rotating body (Fig. 2.16).
The direction of reference for the angle of rotation of the body is accepted φ counterclockwise is considered positive when viewed from the positive direction of the Oz axis.
Equality φ = φ(t), describing the change in angle φ in time is called the law or equation of rotational motion of a rigid body.
The speed and direction of change in the angle of rotation of a rigid body are characterized by angular speed. The absolute value of angular velocity is usually denoted by a letter of the Greek alphabet ω (omega). The algebraic value of angular velocity is usually denoted by . The algebraic value of the angular velocity is equal to the first time derivative of the rotation angle:
. (2.33)
The units of angular velocity are equal to the units of angle divided by the unit of time, for example, deg/min, rad/h. In the SI system, the unit of measurement for angular velocity is rad/s, but more often the name of this unit of measurement is written as 1/s.
If > 0, then the body rotates counterclockwise when viewed from the end of the coordinate axis aligned with the rotation axis.
If< 0, то тело вращается по ходу часовой стрелки, если смотреть с конца оси координат, совмещенной с осью вращения.
The speed and direction of change in angular velocity are characterized by angular acceleration. The absolute value of angular acceleration is usually denoted by the letter of the Greek alphabet e (epsilon). The algebraic value of angular acceleration is usually denoted by . The algebraic value of angular acceleration is equal to the first derivative with respect to time of the algebraic value of angular velocity or the second derivative of the angle of rotation:
The units of angular acceleration are equal to the units of angle divided by the unit of time squared. For example, deg/s 2, rad/h 2. In the SI system, the unit of measurement for angular acceleration is rad/s 2, but more often the name of this unit of measurement is written as 1/s 2.
If the algebraic values of angular velocity and angular acceleration have the same sign, then the angular velocity increases in magnitude over time, and if it is different, it decreases.
If the angular velocity is constant ( ω = const), then it is customary to say that the rotation of the body is uniform. In this case:
φ = t + φ 0, (2.35)
Where φ 0 - initial rotation angle.
If the angular acceleration is constant (e = const), then it is customary to say that the rotation of the body is uniformly accelerated (uniformly slow). In this case:
Where 0 - initial angular velocity.
In other cases, to determine the dependence φ from And it is necessary to integrate expressions (2.33), (2.34) under given initial conditions.
In drawings, the direction of rotation of a body is sometimes shown with a curved arrow (Fig. 2.17).
Often in mechanics, angular velocity and angular acceleration are considered as vector quantities
And .
Both of these vectors are directed along the axis of rotation of the body. Moreover, the vector
directed in one direction with the unit vector, which determines the direction of the coordinate axis coinciding with the axis of rotation, if >0,
and vice versa if
The direction of the vector is chosen in the same way (Fig. 2.18).
During the rotational motion of a body, each of its points (except for points located on the axis of rotation) moves along a trajectory, which is a circle with a radius equal to the shortest distance from the point to the axis of rotation (Fig. 2.19).
Since the tangent of a circle at any point makes an angle of 90° with the radius, the velocity vector of a point of a body undergoing rotational motion will be directed perpendicular to the radius and lie in the plane of the circle, which is the trajectory of the point’s movement. The tangential component of the acceleration will lie on the same line as the speed, and the normal component will be directed radially towards the center of the circle. Therefore, sometimes the tangential and normal components of acceleration during rotational motion are called respectively rotational and centripetal (axial) components (Fig. 2.19)
The algebraic value of the speed of a point is determined by the expression:
, (2.37)
where R = OM is the shortest distance from the point to the axis of rotation.
The algebraic value of the tangential component of acceleration is determined by the expression:
. (2.38)
The modulus of the normal component of acceleration is determined by the expression:
. (2.39)
The acceleration vector of a point during rotational motion is determined by the parallelogram rule as the geometric sum of the tangent and normal components. Accordingly, the acceleration modulus can be determined using the Pythagorean theorem:
If angular velocity and angular acceleration are defined as vector quantities , , then the vectors of velocity, tangential and normal components of acceleration can be determined by the formulas:
where is the radius vector drawn to point M from an arbitrary point on the axis of rotation (Fig. 2.20).
Solving problems involving the rotational motion of one body usually does not cause any difficulties. Using formulas (2.33)-(2.40), you can easily determine any unknown parameter.
Certain difficulties arise when solving problems associated with the study of mechanisms consisting of several interconnected bodies performing both rotational and translational motion.
The general approach to solving such problems is that motion from one body to another is transmitted through one point - the point of tangency (contact). Moreover, the contacting bodies have equal velocities and tangential acceleration components at the point of contact. The normal components of acceleration for bodies in contact at the point of contact are different; they depend on the trajectory of the points of the bodies.
When solving problems of this type, it is convenient, depending on the specific circumstances, to use both the formulas given in Section 2.3 and the formulas for determining the speed and acceleration of a point when specifying its movement as natural (2.7), (2.14) (2.16) or coordinate (2.3), (2.4), (2.10), (2.11) methods. Moreover, if the movement of the body to which the point belongs is rotational, the trajectory of the point will be a circle. If the motion of the body is rectilinear translational, then the trajectory of the point will be a straight line.
Example 2.4. The body rotates around a fixed axis. The angle of rotation of the body changes according to the law φ = π t 3 glad. For a point located at a distance OM = R = 0.5 m from the axis of rotation, determine the speed, tangent, normal components of acceleration and acceleration at the moment of time t 1= 0.5 s. Show the direction of these vectors in the drawing.
Let us consider a section of a body by a plane passing through point O perpendicular to the axis of rotation (Fig. 2.21). In this figure, point O is the intersection point of the axis of rotation and the cutting plane, point M o And M 1- respectively, the initial and current position of point M. Through points O and M o draw a fixed axis Oh, and through points O and M 1 - movable axis Oh 1. The angle between these axes will be equal to
We find the law of change in the angular velocity of the body by differentiating the law of change in the angle of rotation:
In the moment t 1 the angular velocity will be equal
We will find the law of change in the angular acceleration of the body by differentiating the law of change in angular velocity:
In the moment t 1 the angular acceleration will be equal to:
1/s 2,
We find the algebraic values of the velocity vectors, the tangential component of acceleration, the modulus of the normal component of acceleration and the modulus of acceleration using formulas (2.37), (2.38), (2.39), (2.40):
M/s 2 ;
m/s 2 .
Since the angle φ 1>0, then we will move it from the Ox axis counterclockwise. And since > 0, then the vectors will be directed perpendicular to the radius OM 1 so that we see them rotating counterclockwise. Vector will be directed along the radius OM 1 to the axis of rotation. Vector Let's build according to the parallelogram rule on vectors τ And .
Example 2.5. According to the given equation of rectilinear translational motion of the load 1 x = 0,6t 2 - 0.18 (m) determine the speed, as well as the tangential, normal component of acceleration and the acceleration of point M of the mechanism at the moment of time t 1, when the path traveled by load 1 is s = 0.2 m. When solving the problem, we will assume that there is no slipping at the point of contact of bodies 2 and 3, R 2= 1.0 m, r 2 = 0.6 m, R 3 = 0.5 m (Fig. 2.22).
The law of rectilinear translational motion of load 1 is given in coordinate form. Let's determine the moment in time t 1, for which the path traveled by load 1 will be equal to s
s = x(t l)-x(0),
from where we get:
0,2 = 0,18 + 0,6t 1 2 - 0,18.
Hence,
Having differentiated the equation of motion with respect to time, we find the projections of the velocity and acceleration of load 1 onto the Ox axis:
m/s 2 ;
At moment t = t 1 the projection of the speed of load 1 will be equal to:
that is, it will be greater than zero, as is the projection of the acceleration of load 1. Therefore, load 1 will be at moment t 1 move down uniformly accelerated, respectively, body 2 will rotate uniformly accelerated in a counterclockwise direction, and body 3 will rotate clockwise.
Body 2 is driven into rotation by body 1 through a thread wound on a snare drum. Therefore, the modules of the velocities of the points of body 1, the thread and the surface of the snare drum of body 2 are equal, and the modules of acceleration of the points of body 1, the thread and the tangential component of the acceleration of the points of the surface of the snare drum of body 2 will also be equal. Consequently, the module of the angular velocity of body 2 can be defined as
The modulus of angular acceleration of body 2 will be equal to:
1/s 2 .
Let us determine the modules of velocity and tangential component of acceleration for point K of body 2 - the point of contact of bodies 2 and 3:
m/s, m/s 2
Since bodies 2 and 3 rotate without mutual slipping, the magnitudes of the velocity and the tangential component of the acceleration of the point K - the point of contact for these bodies will be equal.
let's direct it perpendicular to the radius in the direction of rotation of the body, since body 3 rotates uniformly acceleratedProgressive 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 velocities 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 of 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 the 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 
, and the moving coordinate system itself 
relative to a fixed coordinate system 
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:
,
,
.
2. When considering portable motion, we will assume that the coordinates of the point 
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.
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 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 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 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’)
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