The rotation of a rigid body around a fixed axis is such a movement in which two points of the body remain motionless during the entire time of movement. In this case, all points of the body located on a straight line passing through its fixed points also remain motionless. This line is called body rotation axis .
Let points A and B be stationary. Let's direct the axis along the axis of rotation. Through the axis of rotation we draw a stationary plane and a movable one, attached to a rotating body (at ).
The position of the plane and the body itself is determined by the dihedral angle between the planes and. Let's denote it . The angle is called body rotation angle .
The position of the body relative to the chosen reference system is uniquely determined at any time if the equation is given, where is any twice differentiable function of time. This equation is called equation of rotation of a rigid body around a fixed axis .
A body rotating around a fixed axis has one degree of freedom, since its position is determined by specifying only one parameter - angle.
An angle is considered positive if it is laid counterclockwise, and negative in the opposite direction. The trajectories of points of a body during its rotation around a fixed axis are circles located in planes perpendicular to the axis of rotation.
To characterize rotational motion solid around a fixed axis we introduce the concepts angular velocity and angular acceleration.
Algebraic angular velocity of a body at any moment in time is called the first derivative with respect to time of the angle of rotation at this moment, that is.
Angular velocity is positive when the body rotates counterclockwise, since the angle of rotation increases with time, and negative when the body rotates clockwise, because the angle of rotation decreases.
The dimension of angular velocity by definition:
In engineering, angular velocity is the rotational speed expressed in revolutions per minute. In one minute the body will rotate through an angle , where n is the number of revolutions per minute. Dividing this angle by the number of seconds in a minute, we get
Algebraic angular acceleration of the body is called the first derivative with respect to time of the angular velocity, that is, the second derivative of the angle of rotation, i.e.
The dimension of angular acceleration by definition:
Let us introduce the concepts of vectors of angular velocity and angular acceleration of a body.
And , where is the unit vector of the rotation axis. Vectors and can be depicted at any point on the rotation axis; they are sliding vectors.
Algebraic angular velocity is the projection of the angular velocity vector onto the axis of rotation. Algebraic angular acceleration is the projection of the angular acceleration vector of velocity onto the axis of rotation.
If at , then the algebraic angular velocity increases with time and, therefore, the body rotates accelerated at the considered moment in time in positive side. The directions of the vectors and coincide, they are both directed in the positive direction of the axis of rotation.
When and the body rotates rapidly in the negative direction. The directions of the vectors and coincide, they are both directed towards the negative side of the rotation axis.
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 the letter Greek alphabet ω (omega). The algebraic value of angular velocity is usually denoted by . The algebraic value of angular velocity is equal to the first derivative with respect to time of the angle of rotation:
. (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 performing 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 forward movement.
The general approach to solving such problems is that motion from one body to another is transmitted through one point - the point of 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 linear and 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 initial and current situation points 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:
At the moment t 1 the angular velocity will be equal
We find the law of change in the angular acceleration of the body by differentiating the law of change in the angular velocity:
At 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 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. By given equation rectilinear translational movement 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 in a clockwise direction.
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 acceleratedAnd Savelyeva.
During the forward motion of a body (§ 60 in the textbook by E. M. Nikitin), all its points move along identical trajectories and at each given 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 motion 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 space relative to surrounding stationary bodies same place.
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 turn 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 in 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 the 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 of a rigid body, not a point. A single material point does not rotate, but moves in a circle - it 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. Uniform 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.
Rotational they call such a movement in which two points associated with the body, therefore, the straight line passing through these points, remain motionless during movement (Fig. 2.16). Fixed straight line A B called axis of rotation.
Rice. 2.1V. Towards the definition of rotational motion of a body
The position of the body during rotational motion determines the angle of rotation φ, rad (see Fig. 2.16). When moving, the angle of rotation changes over time, i.e. the law of rotational motion of a body is defined as the law of change in time of the value of the dihedral angle Ф = Ф(/) between a fixed half-plane TO () , passing through the axis of rotation, and movable n 1 a half-plane connected to the body and also passing through the axis of rotation.
The trajectories of all points of the body during rotational motion are concentric circles located in parallel planes with centers on the axis of rotation.
Kinematic characteristics of the rotational motion of the body. In the same way that kinematic characteristics were introduced for a point, a kinematic concept is introduced that characterizes the rate of change of the function φ(c), which determines the position of the body during rotational motion, i.e. angular velocity co = f = s/f/s//, angular velocity dimension [co] = rad /With.
In technical calculations, the expression of angular velocity with a different dimension is often used - in terms of the number of revolutions per minute: [i] = rpm, and the relationship between n and co can be represented as: co = 27w/60 = 7w/30.
In general, angular velocity varies with time. The measure of the rate of change in angular velocity is angular acceleration e = c/co/c//= co = f, the dimension of angular acceleration [e] = rad/s 2 .
The introduced angular kinematic characteristics are completely determined by specifying one function - the angle of rotation versus time.
Kinematic characteristics of body points during rotational motion. Consider the point M body located at a distance p from the axis of rotation. This point moves along a circle of radius p (Fig. 2.17).
Rice. 2.17.
points of the body during its rotation
Arc length M Q M circle of radius p is defined as s= ptp, where f is the angle of rotation, rad. If the law of motion of a body is given as φ = φ(g), then the law of motion of a point M along the trajectory is determined by the formula S= рф(7).
Using the expressions of kinematic characteristics with the natural method of specifying the motion of a point, we obtain kinematic characteristics for points of a rotating body: speed according to formula (2.6)
V= 5 = rf = rso; (2.22)
tangential acceleration according to expression (2.12)
i t = K = sor = er; (2.23)
normal acceleration according to formula (2.13)
a„ = And 2 /р = с 2 р 2 /р = ogr; (2.24)
total acceleration using expression (2.15)
A = -]A + a] = px/e 2 + co 4. (2.25)
The characteristic of the direction of total acceleration is taken to be p - the angle of deviation of the vector of total acceleration from the radius of the circle described by the point (Fig. 2.18).
From Fig. 2.18 we get
tgjLi = aja n=re/pco 2 =g/(o 2. (2.26)
Rice. 2.18.
Note that all kinematic characteristics of the points of a rotating body are proportional to the distances to the axis of rotation. Ve-
Their identities are determined through the derivatives of the same function - the angle of rotation.
Vector expressions for angular and linear kinematic characteristics. For an analytical description of the angular kinematic characteristics of a rotating body, together with the axis of rotation, the concept rotation angle vector(Fig. 2.19): φ = φ(/)A:, where To- eat
rotation axis vector
1; To=sop51 .
The vector f is directed along this axis so that it can be seen from the “end”
rotation occurring counterclockwise.
Rice. 2.19.
characteristics in vector form
If the vector φ(/) is known, then all other angular characteristics of rotational motion can be represented in vector form:
- angular velocity vector co = f = f To. The direction of the angular velocity vector determines the sign of the derivative of the rotation angle;
- angular acceleration vector є = сo = Ф To. The direction of this vector determines the sign of the derivative of the angular velocity.
The introduced vectors с and є allow us to obtain vector expressions for the kinematic characteristics of points (see Fig. 2.19).
Note that the modulus of the point’s velocity vector coincides with the modulus vector product vector of angular velocity and radius vector: |sokh G= sogvіpa = rubbish. Taking into account the directions of the vectors с and r and the rule for the direction of the vector product, we can write an expression for the velocity vector:
V= co xg.
Similarly, it is easy to show that
- ? X
- - egBіpa= єр = a t And
Sosor = co p = i.
(In addition, the vectors of these kinematic characteristics coincide in direction with the corresponding vector products.
Therefore, the tangent vectors and normal acceleration can be represented as vector products:
- (2.28)
- (2.29)
a x = g X G
A= co x V.
This article describes an important section of physics - “Kinematics and dynamics of rotational motion”.
Basic concepts of kinematics of rotational motion
Rotational motion of a material point around a fixed axis is called such motion, the trajectory of which is a circle located in a plane perpendicular to the axis, and its center lies on the axis of rotation.
Rotational motion of a rigid body is a motion in which all points of the body move along concentric (the centers of which lie on the same axis) circles in accordance with the rule for the rotational motion of a material point.
Let an arbitrary rigid body T rotate around the O axis, which is perpendicular to the plane of the drawing. Let us select point M on this body. When rotated, this point will describe a circle with radius around the O axis r.
After some time, the radius will rotate relative to its original position by an angle Δφ.
The direction of the right screw (clockwise) is taken as the positive direction of rotation. The change in the angle of rotation over time is called the equation of rotational motion of a rigid body:
φ = φ(t).
If φ is measured in radians (1 rad is the angle corresponding to an arc of length equal to its radius), then the length of the circular arc ΔS, which the material point M will pass in time Δt, is equal to:
ΔS = Δφr.
Basic elements of the kinematics of uniform rotational motion
A measure of the movement of a material point over a short period of time dt serves as an elementary rotation vector dφ.
The angular velocity of a material point or body is physical quantity, which is determined by the ratio of the vector of an elementary rotation to the duration of this rotation. The direction of the vector can be determined by the rule of the right screw along the O axis. In scalar form:
ω = dφ/dt.
If ω = dφ/dt = const, then such motion is called uniform rotational motion. With it, the angular velocity is determined by the formula
ω = φ/t.
According to the preliminary formula, the dimension of angular velocity
[ω] = 1 rad/s.
The uniform rotational motion of a body can be described by the period of rotation. The period of rotation T is a physical quantity that determines the time during which a body makes one full revolution around the axis of rotation ([T] = 1 s). If in the formula for angular velocity we take t = T, φ = 2 π (one full revolution of radius r), then
ω = 2π/T,
Therefore, we define the rotation period as follows:
T = 2π/ω.
The number of revolutions that a body makes per unit time is called the rotation frequency ν, which is equal to:
ν = 1/T.
Frequency units: [ν]= 1/s = 1 s -1 = 1 Hz.
Comparing the formulas for angular velocity and rotation frequency, we obtain an expression connecting these quantities:
ω = 2πν.
Basic elements of the kinematics of uneven rotational motion
The uneven rotational motion of a rigid body or material point around a fixed axis is characterized by its angular velocity, which changes with time.
Vector ε , characterizing the rate of change of angular velocity, is called the angular acceleration vector:
ε = dω/dt.
If a body rotates, accelerating, that is dω/dt > 0, the vector has a direction along the axis in the same direction as ω.
If the rotational movement is slow - dω/dt< 0 , then the vectors ε and ω are oppositely directed.
Comment. When uneven rotational motion occurs, the vector ω can change not only in magnitude, but also in direction (when the axis of rotation is rotated).
Relationship between quantities characterizing translational and rotational motion
It is known that the arc length with the angle of rotation of the radius and its value are related by the relation
ΔS = Δφ r.
Then the linear speed of a material point performing rotational motion
υ = ΔS/Δt = Δφr/Δt = ωr.
The normal acceleration of a material point that performs rotational translational motion is defined as follows:
a = υ 2 /r = ω 2 r 2 /r.
So, in scalar form
a = ω 2 r.
Tangential accelerated material point that performs rotational motion
a = ε r.
Momentum of a material point
The vector product of the radius vector of the trajectory of a material point of mass m i and its momentum is called the angular momentum of this point about the axis of rotation. The direction of the vector can be determined using the right screw rule.
Momentum of a material point ( L i) is directed perpendicular to the plane drawn through r i and υ i, and forms a right-hand triple of vectors with them (that is, when moving from the end of the vector r i To υ i the right screw will show the direction of the vector L i).
In scalar form
L = m i υ i r i sin(υ i , r i).
Considering that when moving in a circle, the radius vector and the linear velocity vector for i-th material mutually perpendicular points,
sin(υ i , r i) = 1.
So the angular momentum of a material point for rotational motion will take the form
L = m i υ i r i .
The moment of force that acts on the i-th material point
The vector product of the radius vector, which is drawn to the point of application of the force, and this force is called the moment of the force acting on i-th material point relative to the axis of rotation.
In scalar form
M i = r i F i sin(r i , F i).
Considering that r i sinα = l i ,M i = l i F i .
Magnitude l i, equal to the length of the perpendicular lowered from the point of rotation to the direction of action of the force, is called the arm of the force F i.
Dynamics of rotational motion
The equation for the dynamics of rotational motion is written as follows:
M = dL/dt.
The formulation of the law is as follows: the rate of change of angular momentum of a body that rotates around a fixed axis is equal to the resulting moment relative to this axis of all external forces, attached to the body.
Moment of impulse and moment of inertia
It is known that for the i-th material point the angular momentum in scalar form is given by the formula
L i = m i υ i r i .
If instead of linear speed we substitute its expression through angular speed:
υ i = ωr i ,
then the expression for the angular momentum will take the form
L i = m i r i 2 ω.
Magnitude I i = m i r i 2 called the moment of inertia about axis i material point of an absolutely rigid body passing through its center of mass. Then we write the angular momentum of the material point:
L i = I i ω.
We write the angular momentum of an absolutely rigid body as the sum of the angular momentum material points composing this body:
L = Iω.
Moment of force and moment of inertia
The law of rotational motion states:
M = dL/dt.
It is known that the angular momentum of a body can be represented through the moment of inertia:
L = Iω.
M = Idω/dt.
Considering that the angular acceleration is determined by the expression
ε = dω/dt,
we obtain a formula for the moment of force, represented through the moment of inertia:
M = Iε.
Comment. A moment of force is considered positive if the angular acceleration that causes it is greater than zero, and vice versa.
Steiner's theorem. Law of addition of moments of inertia
If the axis of rotation of a body does not pass through its center of mass, then relative to this axis one can find its moment of inertia using Steiner’s theorem:
I = I 0 + ma 2,
Where I 0- initial moment of inertia of the body; m- body weight; a- distance between axles.
If a system that rotates around a fixed axis consists of n bodies, then the total moment of inertia of this type of system will be equal to the sum of the moments of its components (the law of addition of moments of inertia).
