2 the concept of the moment of inertia of a material point. Determination of the moment of inertia

MINISTRY OF EDUCATION AND SCIENCE OF THE RUSSIAN FEDERATION

Federal State Autonomous Educational Institution

higher professional education

Far Eastern Federal University

School of Natural Sciences

Determination of the moments of inertia of bodies of revolution

by torsional vibration method. Verification of the Huygens - Steiner theorem.

Study guide

to laboratory work No. 1.3

Vladivostok

UDC53 (o76.5)

Determination of the moments of inertia of bodies of revolution

by torsional vibration method. Verification of the Huygens - Steiner theorem.

Study guide for laboratory work No. 1.3 on the discipline "physical practice" // comp. V.E. Polischuk, R.F. Polischuk. - Vladivostok: Dalnevost Publishing House. federal University, 2013-p. 12.

The manual, prepared at the Department of General Physics of the School of Natural Sciences of the Far Eastern Federal University, contains methodological instructions for performing laboratory work in mechanics with the aim of experimentally studying the moment of inertia of rigid bodies of revolution and checking the Huygens-Steiner theorem.

For FEFU students of all specialties.

UDC 53 (076.5)

Compiled by V.E. Polishchuk

Polishchuk R.F.

MINISTRY OF EDUCATION AND SCIENCE OF THE RUSSIAN FEDERATION

Federal State Autonomous Educational Institution of Higher Professional Education

Far Eastern Federal University (FEFU)

School of Natural Sciences

Determination of the moments of inertia of bodies of revolution by the method of torsional vibrations.

Verification of the Huygens-Steiner theorem.

Teaching aid to laboratory work No. 1.3

In the discipline "physical workshop"

Vladivostok

Far Eastern Federal University Publishing House

The purpose of this laboratory work is study of the laws of the dynamics of the rotational motion of a rigid body, experimental measurement of the moment of inertia of the simplest bodies of revolution and verification of the Huygens-Steiner theorem.

Basic concepts of the rotational motion of a rigid body.

In addition to the concept of a material point, mechanics uses the model concept absolutely solid - a body, the deformations of which can be neglected under the conditions of this problem. Such a body can be considered as a system of rigidly fixed material points.

Any complex motion of a rigid body can always be decomposed into two main types of motion - translational and rotational. Translational is called a movement of a rigid body in which any straight line drawn through any two points of the body remains parallel to itself during the entire movement (Fig. 1). With such a movement, all points of a rigid body move in exactly the same way, that is, they have the same speed, acceleration, trajectories of movement, make the same movements and travel the same path. Consequently, the translational motion of a rigid body can be regarded as the motion of a material point, the mass of which is equal to the mass of the body m and apply to it Newton's second law of the dynamics of a material point, i.e.

where is the resultant of all external forces acting on the body, is the momentum (momentum) of the body.

Rotational the motion of a rigid body is a movement in which all points of the body describe circles, the centers of which lie on one straight line, called the axis of rotation of the body. During rotational motion, all points of the body move with the same angular velocity and angular acceleration and perform the same angular displacements. However, as experience shows, when a rigid body rotates around a fixed axis, the mass is no longer a measure of its inertia, and the force is insufficient to characterize the external influence. In addition, experiments show that the acceleration during rotational motion depends not only on the mass of the body, but also on its distribution relative to the axis of rotation; depends not only on strength, but also on the point of its application and direction of action. Therefore, to describe the rotational motion of a rigid body, new dynamic characteristics are introduced, such as moment of force, moment of momentum and moment of inertia of a body . It should be borne in mind that there are two different concepts of these quantities: relative to the axis and relative to any point O (pole, origin) taken on this axis.

A moment of power relative to a fixed point ABOUT is called a vector quantity equal to the vector product of the radius vector drawn from point O to the point of application of the resulting force, by the vector of this force:

The vector of the moment of force is always perpendicular to the plane in which the vectors and are located, and its direction relative to this plane is determined by the vector product rule or the gimbal rule. According to the vector product rule, the vector is directed perpendicular to the plane containing vectors and, in such a direction that when viewed from its end, the vector can be aligned with the vector by rotating counterclockwise towards a smaller angle. According to the rule of the right gimlet (Fig. 2), when its handle is rotated in the direction from to in the direction of a smaller angle a, the translational movement of the gimbal will determine the direction of the vector

When applying these rules, it is convenient to start vectors and combine at one point. You can, for example, transfer the vector parallel to itself so that its beginning coincides with the beginning of the vector at point 0 (in Fig. 2, this vector is shown by a dotted line).

Vectors, the direction of which is associated with the direction of rotation (angular velocity, angular acceleration, moment of force, angular momentum, etc.), are called pseudo-vectors or axial in contrast to ordinary vectors (velocity, radius vector, acceleration, etc.), which are called polar or true.

The magnitude of the vector of the moment of force (the numerical value of the moment of force) is determined according to the formula of the vector product (2), i.e. , where a is the angle between the directions of vectors and. The value p \u003d r · Sinα is called the arm of the force (Fig. 2). Shoulder of strength p is the shortest distance from point O to the line of action of the force.

The moment of force about the axis is called projection on this axis of the vector of the moment of force found relative to any point belonging to this axis. It is clear that the moment of force relative to the axis is a scalar quantity. In the SI system, the moment of force is measured in Nm. To introduce the concept of angular momentum of a body, we first introduce this concept for a material point belonging to a rotating solid body.

The moment of momentum of a material point Δ m i relative to a fixed point O is the vector product of the radius vector drawn from the point O to the point where the mass Δm i is located by the vector of the momentum of this material point:

where is the momentum of a material point.

The moment of momentum of a rigid body (or mechanical system) relative to a fixed point O is a vector, equal to the geometric sum of the angular momentum relative to the same point O of all material points of a given body, i.e. ...

The moment of momentum of a rigid body about the axis is called the projection onto this axis of the angular momentum vector of the body relative to any point selected on this axis. It is quite obvious that in this case the angular momentum is a scalar quantity. In SI, the angular momentum is measured in.

The measure of inertia of bodies in forward motion is their weight. The inertia of bodies during rotational motion depends not only on the mass of the body, but also on its distribution in space relative to the axis of rotation. A measure of body inertia during rotational motion is the moment of inertia of the body I about the axis of rotation or point. The moment of inertia, like mass, is an additive, scalar quantity.

The moment of inertia of the body about the axis of rotation is called a physical scalar quantity equal to the sum of the products of the masses of material points (into which the whole body can be divided) by the squares of the distances of each of them to the axis of rotation:

where I is the moment of inertia of a material point.

The moment of inertia of the body relative to the point O is called a scalar value equal to the sum of the products of the mass of each material point of a given body by the square of its distance to point O. The calculated formula for the moment of inertia is similar to formula (4). In SI, the moment of inertia is measured in kgm 2.

The moment of inertia of a rigid body depends on the body weight, shape and size of the body.

The basic law of the dynamics of the rotational motion of a rigid body.

Each of the material points of a rotating rigid body will move along a circle in a plane perpendicular to the axis of rotation, and the centers of all these circles will lie on this axis. Moreover, all points of the body at a given time have the same angular velocity and the same angular acceleration.

Consider an i-material point, the mass of which is Δm i, and the radius of the circle along which it moves, r i. It is acted upon by both external forces from other bodies, and internal forces from other material points belonging to the same body. Let us decompose the resulting force acting on a material point of mass Δm i into two mutually perpendicular components of the force, and so that the force vector coincides in the direction with the tangent to the particle's trajectory, and the force is perpendicular to this tangent (Fig. 3). It is quite obvious that the rotation of a given material point is due only to the tangential component of the force, the magnitude of which can be represented as the sum of internal and external forces. In this case, for a material point Δm i, Newton's second law in scalar form will have the form:

(5)

Taking into account the fact that during the rotational motion of a rigid body around an axis, the linear velocities of motion of material points along circular paths are different in magnitude and direction, and the angular velocities w for all these points are the same (both in magnitude and direction), we replace in equation (5) linear velocity per angular (vi \u003d wr i):

. (6)

Let us introduce into equation (6) the moment of the force acting on the particle. To do this, multiply the left and right sides of equation (6) by the radius r i, which is a shoulder in relation to the resulting force:

(7)

Then we get:

where each term on the right-hand side of equation (8) is the moment of the corresponding force about the axis of rotation. If we introduce into this equation the angular acceleration of rotation of a material point of mass Δm i about the axis (\u003d) and its moment of inertia ΔI i about the same axis (\u003d ΔI i), then the equation of the rotational motion of a material point about the axis will take the form:

Similar equations can be written for all other material points included in a given solid. Let us find the sum of these equations, taking into account the fact that the value of the angular acceleration for all material points of a given rotating body will be the same, we get:

The total moment of internal forces is zero, since each internal force, according to Newton's third law, has a force equal in magnitude, but oppositely directed to itself, applied to another material point of the body, with the same shoulder. The total moment is the torque M of all external forces acting on the rotating body. The sum of the moments of inertia \u003d I determines the moment of inertia of the given body about the axis of rotation. After substituting the indicated values \u200b\u200binto equation (10), we finally get:

Equation (11) is called the basic equation of the dynamics of the rotational motion of a rigid body about the axis. Since \u003d, and the moment of inertia of a body relative to a given axis of rotation is a constant value and, therefore, it can be introduced under the sign of the differential, then equation (11) can be written in the form:

The quantity Iw \u003d L (13)

called the angular momentum of the body about the axis. Taking into account (13), equation (12) can be written in the form:

Equations (11-14) are scalar in nature and are used only to describe the rotational motion of bodies about the axis. When describing the rotational motion of bodies relative to a point (or pole, or beginning)belonging to a given axis, these equations are respectively written in vector form:

(11 *); (12 *); (13 *); (14 *).

When comparing the equations of translational (1) and rotational (11-14) body movements, it can be seen that during rotational motion, instead of force in the equations there is its moment, instead of body mass - its moment of inertia, instead of momentum (or momentum) - moment of momentum (or angular momentum).

From equations (14) and (14 *) it follows, respectively, equation of momentsrelative to the axis and relative to the point:

dL \u003d Mdt (15); (fifteen *) .

According to the equation of moments about the axis (15), the change in the angular momentum of the body dL relative to the fixed axis is equal to the angular momentum of the external force Mdt acting on the body relative to the same axis. Regarding the point, the equation of moments (15 *) is formulated: change vector angular momentum relative to the point is equal to the momentum moment vector the force acting on the body relative to the same point.

Equations (15) and (15 *) yield angular momentum conservation law a rigid body both about an axis and a point. From equation (15) it follows: if the total moment of all external forces M relative to the axis is zero (M \u003d 0, hence dL \u003d 0), then the angular momentum of this body relative to the axis of its rotation remains constant (L \u003d Const).

Regarding the point: if the total vector of the moment of all external forces relative to the point of rotation O remains unchanged, then the vector of the angular momentum of this body relative to the same point O remains constant.

In this laboratory work, the moments of inertia are determined for the simplest bodies of revolution. A body of revolution is understood as a volumetric body that arises when a plane figure bounded by an arbitrary curve rotates around an axis lying in the same plane. A body of revolution always has an axis of symmetry. The simplest examples of bodies of revolution are:

ball - formed by a semicircle rotating around the diameter of the cut;

cylinderp - formed by a rectangle rotating around one of its sides;

cone - formed by a right-angled triangle rotating around one of its legs, etc.

In this laboratory work torsional vibration method the moments of inertia for bodies are determined: a sphere, a disk, a rod, a hollow and a solid cylinder. In addition, the Huygens-Steiner theorem is experimentally verified. This theorem allows you to determine the moment of inertia of a body with respect to any axis that does not pass through the center of mass of the body, if the moment of inertia of the given body about the axis passing through the center of mass and parallel to the desired axis is known.

Huygens-Steiner theorem. The moment of inertia of a body about any axis that does not pass through the center of mass of a given body is equal to the moment of inertia of this body about an axis passing through its center of mass and parallel to the first axis, plus the product of the mass of this body by the square of the distance between these axes: I \u003d I o + mɑ 2, where I is the moment of inertia of the body about the desired axis (not passing through the center of the body's mass), Io is the moment of inertia of the body about the axis passing through the center of mass and parallel to the first axis, m is the mass of the body, ɑ is the distance between the axes.

Derivation of a working formula for calculating the moment of inertia of bodies of revolution by the method of torsional vibrations.

The torsion pendulum in this work consists of a spiral spring fixed in a tripod. An axle is rigidly attached to the spring, freely rotating in the tripod. A body is attached to the axis, the moment of inertia of which is determined. If this system is taken out of the equilibrium position by turning the body through a certain angle φ and letting go, then torsional vibrations of the body will arise. During torsional vibrations, a restoring moment of force acts on the body, stopping the deviation of the body from the state of balance, and then imparting a reverse motion to the body. Recovering moment of power Mdue to elastic forces arising in the coil spring.

Experiments show that in the area of \u200b\u200belastic torsional deformations, the angle of rotation of the spiral spring is directly proportional to the projection of the moment of force Mon the axis of rotation z (M z), i.e.

М z \u003d - G φ (16).

The proportionality factor G is called the slope of the coil spring. Equation (11) implies: М z \u003d I z ·, where \u003d is the angular acceleration, I z is the moment of inertia of the body relative to the rotating axis of the installation. Consequently,

М z \u003d I z (17).

From (16) and (17) the equality follows: I z · \u003d - G · φ. Or

Equation (17) is a differential equation of harmonic oscillations, which can be rewritten in the following form

+ ω 2 φ \u003d 0, (19)

where ω 2 \u003d (20)

Equation (18) corresponds to a harmonic oscillator and describes its harmonic oscillations, in this case, oscillations of the angular displacement of the pendulum relative to its equilibrium position. From the solution of the differential equation (18) it follows that the oscillations of the torsional pendulum are harmonic φ \u003d φ о Sin (ω t + α), where φ о is the amplitude of the angular displacement, equal to the initial angular deviation of the pendulum, and ω is the cyclic oscillation frequency, which is related to the oscillation period by the relation

Equations (20) and (21) imply working formula experimental determination of the moment of inertia I z for the proposed bodies of revolution and verification of the Huygens - Steiner theorem:

I z \u003d I \u003d, (22)

Preparation and implementation of laboratory work.

Fig.4 General view of the experimental setup and the investigated bodies.

As can be seen from the working formula (22), the main parameters for the experimental determination of the moments of inertia of the above bodies are the period of oscillations of the body T and the angular coefficient of elasticity of the spiral spring G. In this laboratory work, the slope coefficient has already been experimentally determined by the method described on page 12 and has the meaning

Measurement of moments of inertia of bodies

1. Attach a narrow sheet of paper, no more than 3 mm wide, to all test bodies. (fig. 5).

2. Fix the test body on a rotating shaft, fastened with a spring.

3. Set up a tripod with a spring and a fixed rigid body so that the sheet is under the light barrier (Fig. 5).

4. For the photocell, select the measurement mode.

5. Tilt the body to be examined from the equilibrium position by about 90 ° and release it by pressing the “Set” button of the photocell sensor. The light barrier will measure the time span equal to the oscillation period systems.

6. To carry out repeated measurements, reset the light barrier counter by pressing the “Set” button. After a subsequent oscillation cycle, the sensor will again show the value of the system oscillation period.

7. For each investigated body make 5-7 measurements of the oscillation period. According to the formula (22), calculate the moments of inertia of the investigated bodies. For each body, enter the measurement data in a separate table. Determine the mean values \u200b\u200band confidence intervals for each body examined. When calculating the moments of inertia of bodies, use the (previously experimentally found) value of the angular coefficient of elasticity of the spiral spring, equal to: G\u003d 0.0241 ± 0.0009 N M / AHR.

Table No. 1. Determination of the moment of inertia of a homogeneous cylinder.

In the dynamics of the translational motion of a material point, in addition to the kinematic characteristics, the concepts of force and mass were introduced. When studying the dynamics of rotational motion, physical quantities are introduced - moment of forcesand moment of inertia, the physical meaning of which will be revealed below.

Let some body under the action of a force applied at a point AND, comes into rotation around the axis of the OO "(Figure 5.1).

Figure 5.1 - To the conclusion of the concept of moment of force

The force acts in a plane perpendicular to the axis. Perpendicular rdropped from point ABOUT (lying on the axis) on the direction of the force is called shoulder of strength... The product of the force on the shoulder determines the modulus moment of force relative to point ABOUT:

(5.1)

Moment of power is the vector determined by the vector product of the radius vector of the force application point and the force vector:

(5.2)

The unit of moment of force is newton meter (H . m). The direction of the vector of the moment of force is found using right screw rules.

Mass is a measure of inertness of bodies during translational motion. The inertia of bodies during rotational motion depends not only on the mass, but also on its distribution in space relative to the axis of rotation. The measure of inertia during rotational motion is a quantity called moment of inertia of the body about the axis of rotation.

The moment of inertia of a material point relative to the axis of rotation - the product of the mass of this point by the square of the distance from the axis:

Body moment of inertia about the axis of rotation - the sum of the moments of inertia of the material points of which this body consists:

(5.4)

In the general case, if the body is solid and is a collection of points with low masses dm, the moment of inertia is determined by integration:

, (5.5)

where r- distance from the axis of rotation to an element of mass d m.

If the body is homogeneous and its density ρ = m/V, then the moment of inertia of the body

(5.6)

The moment of inertia of a body depends on which axis it rotates and how the body's mass is distributed over its volume.

The moment of inertia of bodies with the correct geometric shape and a uniform distribution of mass over the volume is determined most simply.

Moment of inertia of a homogeneous bar relative to the axis passing through the center of inertia and perpendicular to the bar,

The moment of inertia of a homogeneous cylinder relative to the axis perpendicular to its base and passing through the center of inertia,

(5.8)

Moment of inertia of a thin-walled cylinder or hoop relative to the axis perpendicular to the plane of its base and passing through its center,

Ball moment of inertia relative to diameter

(5.10)

Let us determine the moment of inertia of the disk relative to the axis passing through the center of inertia and perpendicular to the plane of rotation. Let the mass of the disk be m, and its radius is R.

The area of \u200b\u200bthe ring (Figure 5.2) enclosed between r and, is equal to.

Figure 5.2 - To the conclusion of the moment of inertia of the disk

Disc area. With a constant ring thickness,

from where or .

Then the moment of inertia of the disk,

For clarity, Figure 5.3 depicts homogeneous solids of various shapes and indicates the moments of inertia of these bodies relative to the axis passing through the center of mass.

Figure 5.3 - Moments of inertia I C some homogeneous solids.

Steiner's theorem

The above formulas for the moments of inertia of bodies are given under the condition that the axis of rotation passes through the center of inertia. To determine the moments of inertia of a body about an arbitrary axis, you should use steiner's theorem : the moment of inertia of a body relative to an arbitrary axis of rotation is equal to the sum of the moment of inertia J 0 about an axis parallel to the given one and passing through the center of inertia of the body and the value md 2:

(5.12)

where m - body mass, d- distance from the center of mass to the selected axis of rotation. Moment of inertia unit - kilogram-meter squared (kg . m 2).

So, the moment of inertia of a homogeneous rod of length l with respect to the axis passing through its end, by Steiner's theorem is

Systems into squares of their distances to the axis:

• m i - weight i-th point,
• r i - distance from ith point to the axis.

Axial moment of inertia body J a is a measure of the inertia of a body in rotational motion around an axis, just as the mass of a body is a measure of its inertia in translational motion.

If the body is homogeneous, that is, its density is the same everywhere, then

Huygens-Steiner theorem

Moment of inertia of a rigid body relative to any axis depends not only on the mass, shape and size of the body, but also on the position of the body in relation to this axis. According to Steiner's theorem (Huygens-Steiner theorem), moment of inertia body J with respect to an arbitrary axis is equal to the sum moment of inertia this body J c relative to the axis passing through the center of mass of the body parallel to the axis under consideration and the product of the body mass m per square distance d between axles:

where is the total body weight.

For example, the moment of inertia of a rod about an axis passing through its end is:

Axial moments of inertia of some bodies

Moments of inertia homogeneous bodies of the simplest shape with respect to some axes of rotation

Body	Description	Axis position a	Moment of inertia J a
	Material point of mass m	On distance r from point, motionless
	Hollow thin-walled cylinder or radius ring r and masses m	Cylinder axis
	Solid cylinder or disc of radius r and masses m	Cylinder axis
	Thick-walled hollow mass cylinder m with outer radius r 2 and inner radius r 1	Cylinder axis
	Solid cylinder length l, radius r and masses m
	Hollow thin-walled cylinder (ring) length l, radius r and masses m	The axis is perpendicular to the cylinder and passes through its center of mass
	Straight thin rod length l and masses m	The axis is perpendicular to the rod and passes through its center of mass
	Straight thin rod length l and masses m	The axis is perpendicular to the bar and passes through its end
	Thin-walled sphere of radius r and masses m	The axis goes through the center of the sphere
	Ball radius r and masses m	The axis goes through the center of the ball
	Radius cone r and masses m	Cone axis
	Isosceles triangle with height h, the basis a and mass m	The axis is perpendicular to the plane of the triangle and passes through the vertex
	Regular triangle with side a and mass m	The axis is perpendicular to the plane of the triangle and passes through the center of mass
	Square with side a and mass m	The axis is perpendicular to the plane of the square and passes through the center of mass

Deriving formulas

Thin-walled cylinder (ring, hoop)

Formula derivation

The moment of inertia of a body is equal to the sum of the moments of inertia of its constituent parts. Divide the thin-walled cylinder into elements with mass dm and moments of inertia dJ i... Then

Since all elements of a thin-walled cylinder are at the same distance from the axis of rotation, formula (1) is transformed to the form

Thick-walled cylinder (ring, hoop)

Formula derivation

Let there be a homogeneous ring with outer radius R, inner radius R 1, thick h and density ρ. Let's break it down into thin rings thick dr... Mass and moment of inertia of a thin ring of radius r will be

We find the moment of inertia of a thick ring as an integral

Since the volume and mass of the ring are equal

we obtain the final formula for the moment of inertia of the ring

Homogeneous disc (solid cylinder)

Formula derivation

Considering the cylinder (disk) as a ring with zero inner radius ( R 1 \u003d 0), we obtain the formula for the moment of inertia of the cylinder (disk):

Solid cone

Formula derivation

Let's break the cone into thin discs thick dhperpendicular to the axis of the cone. The radius of such a disk is

where R - radius of the base of the cone, H - cone height, h Is the distance from the top of the cone to the disk. The mass and moment of inertia of such a disk will be

Integrating, we get

Solid homogeneous ball

Formula derivation

Let's break the ball into thin disks thick dhperpendicular to the axis of rotation. The radius of such a disk located at a height h from the center of the sphere, we find by the formula

The mass and moment of inertia of such a disk will be

We find the moment of inertia of the sphere by integrating:

Thin walled sphere

Formula derivation

For the derivation, we use the formula for the moment of inertia of a uniform ball of radius R:

Let us calculate how much the moment of inertia of the ball will change if, at a constant density ρ, its radius increases by an infinitely small amount dR.

Thin rod (the axis goes through the center)

Formula derivation

Divide the rod into small pieces of length dr... The mass and moment of inertia of such a fragment is

Integrating, we get

Thin rod (the axis goes through the end)

Formula derivation

When moving the axis of rotation from the middle of the bar to its end, the center of gravity of the bar moves relative to the axis by a distance l/ 2. By Steiner's theorem, the new moment of inertia will be equal to

Dimensionless moments of inertia of planets and their satellites

Dimensionless moments of inertia are of great importance for studies of the internal structure of planets and their satellites. Dimensionless moment of inertia of a body of radius r and masses m is equal to the ratio of its moment of inertia relative to the axis of rotation to the moment of inertia of a material point of the same mass relative to a fixed axis of rotation located at a distance r (equal to mr 2). This value reflects the depth distribution of mass. One of the methods for measuring it in planets and satellites is to determine the Doppler shift of a radio signal transmitted by an AMC flying around a given planet or satellite. For a thin-walled sphere, the dimensionless moment of inertia is 2/3 (~ 0.67), for a homogeneous sphere - 0.4, and generally the less, the greater the mass of the body is concentrated at its center. For example, the Moon has a dimensionless moment of inertia close to 0.4 (equal to 0.391), therefore, it is assumed that it is relatively uniform, its density changes little with depth. The dimensionless moment of inertia of the Earth is less than that of a homogeneous ball (equal to 0.335), which is an argument in favor of the existence of a dense core in it.

Centrifugal moment of inertia

The centrifugal moments of inertia of a body with respect to the axes of a rectangular Cartesian coordinate system are the following quantities:

where x, y and z - coordinates of a small body element with volume dV, density ρ and mass dm.

The OX axis is called the main axis of inertia of the bodyif the centrifugal moments of inertia J xy and J xz are simultaneously equal to zero. Three main axes of inertia can be drawn through each point of the body. These axes are mutually perpendicular to each other. Body moments of inertia with respect to the three main axes of inertia drawn at an arbitrary point O bodies are called the main moments of inertia of the body.

The main axes of inertia passing through the center of mass of the body are called the main central axes of inertia of the body, and the moments of inertia about these axes are the main central moments of inertia... The axis of symmetry of a homogeneous body is always one of its main central axes of inertia.

Geometric moment of inertia

Geometric moment of inertia - a geometric characteristic of a section of the form

where is the distance from the central axis to any elementary area relative to the neutral axis.

The geometric moment of inertia is not associated with the movement of the material, it only reflects the degree of rigidity of the section. It is used to calculate the radius of gyration, beam deflection, section selection of beams, columns, etc.

The SI unit is m 4. In construction calculations, literature and assortments of rolled metal products, in particular, it is indicated in cm 4.

The moment of resistance of the section is expressed from it:

.

Geometric moments of inertia of some figures
Rectangle height and width:
Rectangular box-section with the height and width along the outer contours and, and along the inner contours and, respectively
Circle diameter

Central moment of inertia

Central moment of inertia (or moment of inertia about point O) is the quantity

The central moment of inertia can be expressed in terms of the main axial or centrifugal moments of inertia:.

Inertia tensor and inertia ellipsoid

The moment of inertia of a body about an arbitrary axis passing through the center of mass and having a direction given by a unit vector can be represented as a quadratic (bilinear) form:

(1),

where is the tensor of inertia. The matrix of the tensor of inertia is symmetric, has dimensions and consists of the components of centrifugal moments:

,
.

By choosing the appropriate coordinate system, the matrix of the inertia tensor can be reduced to a diagonal form. To do this, we need to solve the eigenvalue problem for the tensor matrix:
,
where -

Who among us has not followed with surprise and delight how the skaters effectively finish their performances on the ice arena? They begin to rotate, fixing the center of rotation with one skate and pushing off with the other, spreading their arms wide to the sides, reaching a sufficiently high angular rotation rate, and then quickly pressing their hands to the body. After that, their angular velocity of rotation increases sharply.

The moment of inertia of a body relative to a certain axis of rotation is determined by the sum of the moments of inertia of a set of material points.

By changing the moment of inertia of the body by moving her hands, the skater controls the speed of rotation.

What is the matter here? Why, only by pressing his hands to the body and not applying any more efforts, the skater manages to dramatically increase the angular speed of his rotation? Does this not refute the law of conservation of energy? Of course not. The explanation of the described phenomenon is given by one of the branches of Newtonian mechanics - the dynamics of a rigid body. In this case, a rigid body is understood as a system of particles, the mutual distances between which do not change.

It turns out that despite the complexity of the problem of the rotational motion of a rigid body, it can be reduced to solving equations similar in form to Newton's equations for translational motion. The role of acceleration, force and mass in this case is played by angular acceleration, moment of force and moment of inertia. You can get acquainted with these important concepts on a simple example of the motion of one material point A with mass m, which is held on a circle of radius r by means of a weightless rod. Let the constant force $ \\ overrightarrow (F) act on the point $ A $. $ If at the moment it makes an angle $ α $ with the radius vector of the material point $ A, $ then its component $ F_r \u003d F⋅ \\ cos α $ is simply compresses the rod, and the component $ F_t \u003d F⋅ \\ sin α $ leads to the appearance of tangential acceleration $ a_t, $ changing the particle velocity. (This acceleration is directed tangentially to the trajectory of the particle. It should be distinguished from centripetal acceleration, which is always directed towards the center of rotation and only changes the direction of the particle's velocity vector.)

According to newton's second law, for tangential acceleration, you can write:

$ m⋅a_t \u003d F_t \u003d F⋅ \\ sin α. $

By analogy with the angular velocity, we introduce the angular acceleration $ ε \u003d \\ frac (a_t) (r). $ It characterizes the rate of change in the angular velocity ω with time. Then equality (1) will have the form:

$ F⋅ \\ sin α \u003d m⋅r⋅ \\ frac (a_t) (r) \u003d m⋅r⋅ε. $

Multiplying both sides of this equation by the radius, we get:

$ F⋅r⋅ \\ sin α \u003d m⋅r ^ 2⋅ε, $

or $ M \u003d J⋅ε. $

The value $ M \u003d F⋅r⋅ \\ sin α, $ numerically equal to the product of the force $ F $ and the length of the perpendicular $ d \u003d r⋅ \\ sin α, $ lowered to the direction of the force from the center of rotation (force shoulder), is called moment of power relative to the point $ O. $ The value $ J \u003d m⋅r ^ 2, $ equal to the product of the mass of the material point $ A $ by the square of its distance to the center of rotation, is called moment of inertia material point relative to point $ O. $

In the case of an arbitrary rigid body, the moment of inertia is characterized by the distribution of mass in this body and is determined by the sum of the moments of inertia of a set of material points into which a rigid body can be broken:

$ J \u003d \\ sum \\ limits_ (i \u003d 1) ^ (N) (\\ Delta ((m) _ (i)) r_ (i) ^ (2)), $

where $ Δm_i $ is the mass of the $ i $ -th point, $ r_i $ is its distance to the axis of rotation.

The moment of inertia serves as a measure of the inertia of a body during rotation and, thus, plays the same role as mass in the case of translational motion. However, unlike body mass, which under normal conditions remains unchanged, the moment of inertia can be easily changed. Indeed, even in the simplest case of a material point on a rod considered above, the moment of inertia depended not only on the magnitude of the mass, but also on how far it was located from the axis of rotation. Therefore, moving the material point along the rod from the center of rotation, it is possible to increase the inertia of rotation of such a system.

Solid bodies of the same mass can have different moments of inertia, depending on the shape and the chosen axis of rotation. So, the moment of inertia of a hollow cylinder of radius $ r $ about its axis of symmetry is equal to $ mr ^ 2; $ of a homogeneous ball rotating about an axis passing through its center is $ \\ frac (2) (5) mr ^ 2; $ of a homogeneous cylinder rotating about its axis of symmetry - $ \\ frac (1) (2) mr ^ 2. $

And the moment of force $ \\ overrightarrow (M), $ and angular velocity $ \\ overrightarrow (ω), $ and angular acceleration $ \\ overrightarrow (ε) $, as well as the corresponding values \u200b\u200bof force, velocity and acceleration when describing translational motion, are vectors. These vectors are directed along the axis of rotation ( axial vectors), and their direction is determined by gimlet rule, i.e., coincides with the direction of the translational movement of the gimbal, the handle of which rotates in the same direction as the body.

One more important vector can be introduced: $ L \u003d J⋅ \\ overrightarrow (ω), $ called moment of momentum... Being an analogue of an impulse for rotational motion, it has a remarkable property: the angular momentum of a closed system remains constant in magnitude and direction. It changes only under the influence of the uncompensated moments of external forces applied to the system under consideration.

Let's go back to the beginning of this article, where it was talked about the rotating skater. Neglecting the small moments of the resistance forces acting on it, we can assume that it is a closed system. Therefore, the angular momentum moment $ J_1⋅ \\ overrightarrow (ω_1) $ achieved by him during the initial acceleration must be preserved ($ ω_1 $ is his initial angular velocity, $ J_1 $ is the moment of inertia in the position with arms out). By pressing his hands to the body, the skater obviously reduces his moment of inertia to a certain value $ J_2 $ and thereby increases his angular velocity: $ ω_2 \u003d \\ frac (J_1) (J_2). $ However, at this moment he has to "work", since the initial kinetic energy of its rotation was $ \\ frac (J_1⋅ω_1 ^ 2) (2), $ and the final one becomes $ \\ frac (J_2⋅ω_2 ^ 2) (2). $ The difference of these energies is the amount of the skater's work.

Moment of inertia - a scalar (in the general case - tensor) physical quantity, a measure of inertia in rotational motion around an axis, just as the mass of a body is a measure of its inertia in translational motion. It is characterized by the distribution of masses in the body: the moment of inertia is equal to the sum of the products of elementary masses by the square of their distances to the base set (point, line or plane).

SI unit: kg · m².

Designation: I or J.

2. The physical meaning of the moment of inertia. The product of the moment of inertia of a body by its angular acceleration is equal to the sum of the moments of all forces applied to the body. Compare. Rotational motion. Translational motion. The moment of inertia is a measure of the inertia of a body in rotational motion

For example, the moment of inertia of the disk about the O axis in accordance with Steiner's theorem:

Steiner's theorem: The moment of inertia I about an arbitrary axis is equal to the sum of the moment of inertia I0 about an axis parallel to the given one and passing through the center of mass of the body, and the product of the body's mass m by the square of the distance d between the axes:

18. Moment of impulse of a rigid body. The angular velocity vector and the angular momentum vector. Gyroscopic effect. Angular rate of precession

Moment of momentum of a rigid body relative to the axis is the sum of the angular momentum of the individual particles that make up the body relative to the axis. Given that, we get.

If the sum of the moments of forces acting on a body rotating around a fixed axis is zero, then the angular momentum is conserved ( angular momentum conservation law):. The derivative of the angular momentum of a rigid body with respect to time is equal to the sum of the moments of all forces acting on the body:

angular velocity as a vector, the magnitude of which is numerically equal to the angular velocity, and directed along the axis of rotation, and, if viewed from the end of this vector, the rotation is directed counterclockwise. Historically, 2 that the positive direction of rotation is considered to be counterclockwise rotation, although, of course, the choice of this direction is absolutely arbitrary. To determine the direction of the angular velocity vector, you can also use the "gimbal rule" (also called the "right screw rule") - if the direction of movement of the gimbal handle (or corkscrew) is combined with the direction of rotation, then the direction of movement of the entire gimbal coincides with the direction of the angular velocity vector.

A rotating body (motorcycle wheel) strives to keep the position of the axis of rotation in space unchanged. (Gyroscopic effect) Therefore, movement on 2 wheels is possible, but standing on two wheels is not possible. This effect is used in ship and tank gun guidance systems. (the ship sways on the waves, and the gun looks at one point) In navigation, etc.

Precession is easy to observe. You need to start the top and wait until it starts to slow down. Initially, the axis of rotation of the top is vertical. Then its upper point gradually descends and moves in an expanding spiral. This is the precession of the top's axis.

The main property of precession is inertialessness: as soon as the force causing the precession of the top disappears, the precession stops, and the top takes a fixed position in space. In the spinning top example, this will not happen, since the force causing the precession - the gravity of the Earth - acts constantly in it.

19. Ideal and viscous liquid. Incompressible fluid hydrostatics. Stationary motion of an ideal fluid. Birnoulli's equation.

Ideal liquid called imaginary incompressible fluidwhich lacks viscosity, internal friction and thermal conductivity... Since there is no internal friction in it, then no shear stresses between two adjacent layers of liquid.

viscous liquid characterized by the presence of frictional forces that arise during its movement. viscous called. liquid, in which, during motion, in addition to normal stresses, tangential stresses are also observed

Considered in G. ur-niya refers. equilibrium of an incompressible fluid in a gravity field (relative to the walls of a vessel moving according to a certain well-known law, for example, translational or rotational) make it possible to solve problems about the shape of a free surface and about splashing liquid in moving vessels - in tanks for transporting liquids, fuel tanks of aircraft and rockets, etc., as well as in conditions of partial or complete weightlessness in space. fly. devices. When determining the shape of the free surface of a liquid contained in a vessel, in addition to hydrostatic forces. pressure, inertial forces and gravity, the surface tension of the fluid must be taken into account. In the case of rotation of the vessel around the vertices. axles with DC ang. speed, the free surface takes the form of a paraboloid of revolution, and in a vessel moving parallel to the horizontal plane translationally and rectilinearly from the post. acceleration and, the free surface of the liquid is a plane inclined to the horizontal plane at an angle