Centrifugal moment of inertia of the section about the y-axis. Geometric characteristics of plane sections. Geometric moments of inertia

The centrifugal moment of inertia about two coordinate axes is the sum of the products of the mass of each of the points of the body and the coordinates along the corresponding axes.

If the body has an axis of symmetry, then the centrifugal moment of inertia of the body is zero and the y, x axes are the main

17. The Huygens-Steiner theorem on the calculation of moments about parallel axes.

The moment of inertia of a rigid body about an axis not passing through the center of mass is equal to the sum of the moments of inertia about the central axis passing through the center of mass and parallel to the given one and the product of the body mass times the square of the distance between the axes.

JC - known moment of inertia about the axis passing through the center of mass of the body,

J - desired moment of inertia about a parallel axis,

m - body weight,

d is the distance between the indicated axes.

18. Calculation of the moments of inertia of homogeneous bodies: thin plate, thin rod, ring, cylinder, cone.

Thin rod: Slim Cylinder:

Thin plate: Cone:

Thin ring: Ball:

Calculation of moments of inertia about arbitrary axes.

Allows you to find the moment of inertia about any axis passing through the coordinate axes and the components of the coal

With these axes, through the magnitude of the axial and centrifugal moments of inertia of these axes.

Ellipsoid of inertia. Central axis of inertia. Extreme properties of moments of inertia.

The center of the ellipsoid is at the origin.

The 3 axes of symmetry of the ellipsoid are called the main axes of inertia, the moments of inertia about the main axes are called the main moments of inertia.

If we take the main axes of inertia as the coordinate axes, then the centrifugal moments of inertia about these axes will be equal to zero.

ELLIPSOID OF INERTIA is a surface that characterizes the distribution of moments of inertia of a body relative to a beam of axes passing through a fixed point O. E. and. like geom. the place of the ends of the segments OK= 1/ laid along Ol from the point O, where Ol is any axis passing through the point O; Il is the moment of inertia of the body about this axis (Fig.). Center E. and. coincides with point O, and its equation in arbitrarily drawn coordinate axes Oxyz has the form

where Ix, Iy, Iz - axial, and Ixу, Iyz, Lzx - centrifugal moments of inertia of the body relative to the indicated coordinate axes. In turn, knowing E. and. for a point O, you can find the moment of inertia about any axis Ol passing through this point, from the equality Il \u003d 1 / R2, by measuring the distance R \u003d OK in the appropriate units.

product of inertia, one of the quantities characterizing the distribution of masses in a body (mechanical system). C. m. and. are calculated as sums of products of masses m to points of the body (system) into two of the coordinates x k , y k, z k these points:

Values ​​C. m. depend on the directions of the coordinate axes. Moreover, for each point of the body there are at least three such mutually perpendicular axes, called the principal axes of inertia, for which the C. m. are equal to zero.

The concept of C. m. and. plays an important role in the study of the rotational motion of bodies. From the values ​​of C. m. depend on the magnitude of the pressure forces on the bearings, in which the axis of the rotating body is fixed. These pressures will be the smallest (equal to static) if the axis of rotation is the main axis of inertia passing through the center of mass of the body.

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• - geometric characteristic of the cross-section of an open thin-walled rod, equal to the sum of the products of elementary sections of sections by squares of sectorial areas - sectoral inertial moment -...

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• - geometric characteristic of the cross section of the rod, equal to the sum of the products of the elementary areas of the section by the squares of their distances to the considered axis - moment of inertia - moment setrvačnosti - Trägheitsmoment -...

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• - a value that characterizes the distribution of masses in the body and, along with the mass, is a measure of the inertia of the body when it does not arrive. movement. Distinguish axial and centrifugal M. and. Axial M. and. is equal to the sum of the products...
• - the main, three mutually perpendicular axes, which can be drawn through any point of the TV. bodies, characterized in that if the body fixed at this point is brought into rotation around one of them, then in the absence of ...

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• - axis in the plane of the cross section of a solid body, relative to which the moment of inertia of the section is determined

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• - a quantity that characterizes the distribution of masses in the body and, along with mass, is a measure of the inertia of the body during non-translational motion. In mechanics distinguish M. and. axial and centrifugal...
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• - a value that characterizes the distribution of masses in the body and, along with the mass, is a measure of the inertia of the body when it does not arrive. movement. Distinguish between axial and centrifugal moments of inertia...
• - main - three mutually perpendicular axes that can be drawn through any point of a rigid body, characterized in that if the body fixed at this point is brought into rotation around one of them, then when ...

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"Centrifugal moment of inertia" in books

Against inertia

author Petrov Rem Viktorovich

Against inertia

From the book Sphinxes of the XX century author Petrov Rem Viktorovich

Contrary to inertia "In the past two decades, the immunological nature of tissue graft rejection has become generally accepted and all aspects of rejection processes are under tight experimental control." Leslie Brent Fingerprints So, to the question "What

By inertia

From the book How much does a person cost. The story of the experience in 12 notebooks and 6 volumes. author

By inertia

From the book How much does a person cost. Book ten: Under the "wing" of the mine author Kersnovskaya Evfrosiniya Antonovna

By inertia To appreciate the landscape, you need to look at the picture from a distance. In order to correctly evaluate this or that event, a known distance is also needed. The law of inertia worked. While the spirit of change reached Norilsk, for a long time it seemed that everything was slipping along the

24. Force of Inertia

From the book Ethereal Mechanics author Danina Tatiana

24. Force of Inertia Ether emitted by the rear hemisphere of an inertially moving particle, this is the Force of Inertia. This Force of Inertia is the repulsion of the Ether filling the particle with the Ether emitted by itself. The magnitude of the Inertial Force is proportional to the speed of emission

3.3.1. Submersible centrifugal pump

From the book Himself a plumber. Plumbing country communications author Kashkarov Andrey Petrovich

3.3.1. Submersible centrifugal pump In this section, we will consider the option with a submersible centrifugal pump NPTs-750. I use water from the spring from April to October. I pump it with a submersible centrifugal pump NPTs-750 / 5nk (the first digit indicates the power consumption in watts,

If we draw coordinate axes through the point O, then with respect to these axes, the centrifugal moments of inertia (or products of inertia) are the quantities defined by the equalities:

where are the masses of points; - their coordinates; while it is obvious that, etc.

For solid bodies, formulas (10), by analogy with (5), take the form

In contrast to axial centrifugal moments of inertia, they can be both positive and negative values ​​and, in particular, can vanish for certain axes.

Principal axes of inertia. Consider a homogeneous body with an axis of symmetry. Let's draw the coordinate axes Oxyz so that the axis is directed along the axis of symmetry (Fig. 279). Then, due to symmetry, each point of the body with mass mk and coordinates will correspond to a point with a different index, but with the same mass and with coordinates equal to . As a result, we obtain that since in these sums all terms are pairwise identical in absolute value and opposite in sign; hence, taking into account equalities (10), we find:

Thus, the symmetry in the distribution of masses about the z-axis is characterized by the vanishing of two centrifugal moments of inertia . The axis Oz, for which the centrifugal moments of inertia containing the name of this axis in their indices, are equal to zero, is called the main axis of inertia of the body for the point O.

It follows from the foregoing that if the body has an axis of symmetry, then this axis is the main axis of inertia of the body for any of its points.

The main axis of inertia is not necessarily the axis of symmetry. Consider a homogeneous body with a plane of symmetry (in Fig. 279, the plane of symmetry of the body is the plane). Let's draw in this plane some axes and an axis perpendicular to them. Then, due to symmetry, each point with mass and coordinates will correspond to a point with the same mass and coordinates equal to . As a result, as in the previous case, we find that or from where it follows that the axis is the main axis of inertia for the point O. Thus, if the body has a plane of symmetry, then any axis perpendicular to this plane will be the main axis of inertia of the body for the point O, in which the axis intersects the plane.

Equalities (11) express the conditions that the axis is the main axis of inertia of the body for the point O (the origin of coordinates).

Similarly, if then the Oy axis will be the main axis of inertia for the point O. Therefore, if all centrifugal moments of inertia are equal to zero, i.e.

then each of the coordinate axes is the main axis of inertia of the body for the point O (the origin of coordinates).

For example, in fig. 279 all three axes are for the point O the main axes of inertia (the axis as the axis of symmetry, and the axes Ox and Oy as perpendicular to the planes of symmetry).

The moments of inertia of the body about the main axes of inertia are called the main moments of inertia of the body.

The main axes of inertia constructed for the center of mass of the body are called the main central axes of inertia of the body. From the above it follows that if the body has an axis of symmetry, then this axis is one of the main central axes of inertia of the body, since the center of mass lies on this axis. If the body has a plane of symmetry, then the axis perpendicular to this plane and passing through the center of mass of the body will also be one of the main central axes of inertia of the body.

In the examples given, symmetrical bodies were considered, which is enough to solve the problems that we will face. However, it can be proved that at least three such mutually perpendicular axes can be drawn through any point of any body, for which equalities (11) will be satisfied, i.e., which will be the main axes of inertia of the body for this point.

The concept of the principal axes of inertia plays an important role in the dynamics of a rigid body. If the coordinate axes Oxyz are directed along them, then all centrifugal moments of inertia turn to zero and the corresponding equations or formulas are significantly simplified (see § 105, 132). Also connected with this concept is the solution of problems on the dynamic equation of rotating bodies (see § 136), on the center of impact (see § 157), etc.