Systems by the squares of their distances to the axis:
- m i- weight i th point,
- r i- distance from i th 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 the shape of a solid 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 relative to this axis. According to Steiner's theorem (Huygens-Steiner theorem), moment of inertia body J relative to an arbitrary axis is equal to the sum moment of inertia this body Jc relative to an 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 of distance d between axes:
where is the total body mass.
For example, the moment of inertia of a rod relative to an axis passing through its end is equal to:
Axial moments of inertia of some bodies
| Body | Description | Axis position a | Moment of inertia J a |
|---|---|---|---|
 |
Material point mass m | On distance r from a point, stationary | |
 |
Hollow thin-walled cylinder or radius ring r and masses m | Cylinder axis | |
 |
Solid cylinder or radius disk r and masses m | Cylinder axis | |
 |
Hollow thick-walled 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 Length Rod l and masses m | The axis is perpendicular to the rod and passes through its center of mass | |
 |
Straight Thin Length Rod l and masses m | The axis is perpendicular to the rod and passes through its end | |
 |
Thin-walled radius sphere r and masses m | The axis passes through the center of the sphere | |
 |
Radius ball r and masses m | The axis passes through the center of the ball | |
| Radius cone r and masses m | Cone axis | ||
| Isosceles triangle with altitude h, 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)
Derivation of the formula
The moment of inertia of a body is equal to the sum of the moments of inertia of its constituent parts. Divide a 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 into the form
Thick-walled cylinder (ring, hoop)
Derivation of the formula
Let there be a homogeneous ring with an outer radius R, inner radius R 1, thick h and density ρ. Let's break it into thin rings thick dr. Mass and moment of inertia of a thin radius ring r will be
Let us find the moment of inertia of the 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)
Derivation of the formula
Considering a cylinder (disk) as a ring with zero internal radius ( R 1 = 0), we obtain the formula for the moment of inertia of the cylinder (disk):
Solid cone
Derivation of the formula
Let's break the cone into thin disks with a thickness dh, perpendicular to the axis of the cone. The radius of such a disk is equal to
Where R– radius of the cone base, H– height of the cone, h– 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
Derivation of the formula
Divide the ball into thin disks of thickness dh, perpendicular to the axis of rotation. The radius of such a disk located at a height h from the center of the sphere, we find it using the formula
The mass and moment of inertia of such a disk will be
We find the moment of inertia of the sphere by integration:
Thin-walled sphere
Derivation of the formula
To derive this, we use the formula for the moment of inertia of a homogeneous 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 infinitesimal amount dR.
Thin rod (axis passes through the center)
Derivation of the formula
Divide the rod into small length fragments dr. The mass and moment of inertia of such a fragment are equal to
Integrating, we get
Thin rod (axis passes through the end)
Derivation of the formula
When the axis of rotation moves from the middle of the rod to its end, the center of gravity of the rod moves relative to the axis by a distance l/2. According to Steiner's theorem, the new moment of inertia will be equal to
Dimensionless moments of inertia of planets and their satellites
Their 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 distribution of mass over depth. One of the methods for measuring it near planets and satellites is to determine the Doppler shift of the radio signal transmitted by an AMS flying near a given planet or satellite. For a thin-walled sphere, the dimensionless moment of inertia is equal to 2/3 (~0.67), for a homogeneous ball - 0.4, and in general, 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), so it is assumed that it is relatively homogeneous, its density changes little with depth. The dimensionless moment of inertia of the Earth is less than that of a homogeneous sphere (equal to 0.335), which is an argument in favor of the existence of a dense core.
Centrifugal moment of inertia
The centrifugal moments of inertia of a body relative 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 main axis of inertia of the body, if 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. Moments of inertia of the body relative to the three main axes of inertia drawn at an arbitrary point O bodies are called main moments of inertia of the body.
The main axes of inertia passing through the center of mass of the body are called main central axes of inertia of the body, and the moments of inertia about these axes are its 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 - 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 related to the movement of the material; it only reflects the degree of rigidity of the section. Used to calculate the radius of gyration, beam deflection, selection of cross-sections of beams, columns, etc.
The SI unit of measurement is m4. In construction calculations, literature and rolled metal assortments, in particular, it is indicated in cm 4.
From it the moment of resistance of the section is expressed:
.| Geometric moments of inertia of some figures | |
|---|---|
| Rectangle height and width: | |
| Rectangular box section with height and width along the external contours and , and along the internal contours and respectively | |
| Circle diameter | |
Central moment of inertia
Central moment of inertia(or moment of inertia relative to point O) is the quantity
The central moment of inertia can be expressed in terms of the main axial or centrifugal moments of inertia: .
Tensor of inertia and ellipsoid of inertia
The moment of inertia of a body relative to an arbitrary axis passing through the center of mass and having a direction specified by the unit vector can be represented in the form of a quadratic (bilinear) form:
(1),where is the inertia tensor. The inertia tensor matrix is symmetrical, has dimensions and consists of components of centrifugal moments:
.
By choosing the appropriate coordinate system, the inertia tensor matrix can be reduced to diagonal form. To do this, you need to solve the eigenvalue problem for the tensor matrix:
,
Where -
Description
The moment of inertia is a quantity that characterizes the distribution of masses in a body and is, along with mass, a measure of the inertia of a body during non-translational motion.
The moment of inertia of a body relative to the axis of rotation depends on the mass of the body and on the distribution of this mass. The greater the mass of the body and the farther it is from the imaginary axis, the greater the moment of inertia the body has. The moment of inertia of an elementary (point) mass m i, separated from the axis at a distance r i, is equal to:
The moment of inertia of the entire body relative to the axis is equal to:
or, for continuously distributed mass:
The moment of inertia of an entire body of complex configuration is usually determined experimentally.
The moment of inertia of some homogeneous solids is given in Table 1.
Table 1 |
||
Moment of inertia of some symmetrical homogeneous bodies |
||
Solid | Axis of rotation | Moment of inertia I, kg m 2 |
Thin rod length l | Perpendicular to the rod, passing through the center of mass | ml 2/12 |
Thin rod length l | Perpendicular to the rod, passing through the edge | ml 2 /3 |
Solid cylinder of radius R | Coincides with the cylinder axis | mR 2 /2 |
Hollow cylinder of radius R | Coincides with the cylinder axis | mR 2 |
Ball of radius R | Passes through the center of the ball | 2mR 2 /5 |
Hollow ball of radius R | Passes through the center of the ball | 2mR 2 /3 |
Thin disk of radius R | Same as disc diameter | mR 2 /4 |
Thin rectangular plate with sides a and b | Passes through the center of the plate perpendicular to the plate | m (a 2 +b 2 )/12 |
The calculation of moments of inertia can in many cases be simplified using symmetry considerations and Steiner's theorem. According to Steiner's theorem, the moment of inertia of a body relative to any axis I A is equal to the moment of inertia of the body is equal to the inertia of the body relative to a parallel axis passing through the center of mass I C, added to the value ma 2, where a is the distance between the axes:
I A = I C + ma 2 .
The concept of the moment of inertia is widely used in solving many problems in mechanics and technology.
Timing characteristics
Initiation time (log to -20 to 20);
Lifetime (log tc from -20 to 20);
Degradation time (log td from -20 to 20);
Time of optimal development (log tk from -1 to 2).
Diagram:
Technical implementations of the effect
"Soft" super flywheel
Moment of inertia is the main characteristic of rotating mechanisms. So in the flywheel they try to increase the moment of inertia by distributing most of the mass onto the wheel rim to accumulate energy. Flywheels are used to level the course of machines; they are present in any car engine, in tape recorders, in sewing machines, mechanical scissors, presses, gyroscopes (see, for example, 104002), etc.
In Fig. Figure 1 shows a diagram of a “soft” super flywheel designed for smooth acceleration of cars.
"Soft" super flywheel
Rice. 1
1 - outer roll of tape;
2 - intermediate turns of tape;
3 - drum.
An increase or decrease in speed is achieved by changing the inertia of the superflywheel by redistributing the mass of the filler belt.
Applying an effect
A.s. 538 800: A method for regulating impact energy in impact forging machines, which consists in changing the moment of inertia of the flywheel masses, characterized in that in order to improve the quality of the processed products and the durability of the machines, the moment of inertia is changed by supplying or discharging liquid into the internal cavities of the flywheel masses .
A.s. 523 213: A method for balancing the inertial forces of moving elements of machines, which consists in the fact that the balanced element of the machine is connected to an accumulating body and causes them to rotate, characterized in that in order to increase the efficiency of balancing, a flywheel with a variable radius of the center of mass is used as an accumulating body, for example, a centrifugal regulator.
The forces generated during rotational motion can be used to accelerate certain technological processes.
Literature
1. Irodov I.E. Basic laws of mechanics. - M.: Higher School, 1985. - 248 p.
2. Physical encyclopedia.- M.: Great Russian Encyclopedia, 1992.- T.3.- P.206-207.
Keywords
- moment of inertia
- body mass
- axis of rotation
Sections of natural sciences:
| Dynamics |
We come across this concept almost constantly, since it has an extremely great influence on all material objects of our world, including humans. In turn, such a moment of inertia is inextricably linked with the law mentioned above, determining the strength and duration of its effect on solid bodies.
From the point of view of mechanics, any material object can be described as an unchanging and clearly structured (idealized) system of points, the mutual distances between which do not change depending on the nature of their movement. This approach allows you to accurately calculate the moment of inertia of almost all solid bodies using special formulas. Another interesting nuance here is that anything complex, even the most intricate, can be represented as a set of simple movements in space: rotational and translational. This also makes life much easier for physicists when calculating this physical quantity.
The easiest way to understand what a moment of inertia is and what its influence is on the world around us is by the example of a sharp change in the speed of a passenger vehicle (braking). In this case, the legs of a standing passenger will be carried away by friction on the floor. But at the same time, no impact will be exerted on the body and head, as a result of which they will continue to move for some time at the same specified speed. As a result, the passenger will lean forward or fall. In other words, the moment of inertia of the legs, extinguished by the floor, will be significantly less than that of other points of the body. The opposite picture will be observed with a sharp increase in the speed of a bus or tram car.
The moment of inertia can be formulated as a physical quantity equal to the sum of the products of elementary masses (those individual points of a rigid body) by the square of their distance from the axis of rotation. From this definition it follows that this characteristic is an additive quantity. Simply put, the moment of inertia of a material body is equal to the sum of similar indicators of its parts: J = J 1 + J 2 + J 3 + ...
This indicator for bodies of complex geometry is determined experimentally. It is necessary to take into account too many different physical parameters, including the density of the object, which can be non-uniform at different points, which creates the so-called mass difference in different segments of the body. Accordingly, standard formulas are not suitable here. For example, the moment of inertia of a ring with a certain radius and uniform density, having an axis of rotation that passes through its center, can be calculated using the following formula: J = mR 2. But in this way it will not be possible to calculate this value for a hoop, all parts of which are made of different materials.
And the moment of inertia of a ball of a continuous and homogeneous structure can be calculated using the formula: J = 2/5mR 2. When calculating this indicator for bodies relative to two parallel axes of rotation, an additional parameter is introduced into the formula - the distance between the axes, denoted by the letter a. The second axis of rotation is designated by the letter L. For example, the formula may look like this: J = L + ma 2.
Thorough experiments to study the inertial motion of bodies and the nature of their interaction were first carried out by Galileo Galilei at the turn of the sixteenth and seventeenth centuries. They allowed the great scientist, who was ahead of his time, to establish the fundamental law that physical bodies maintain a state of rest or relative to the Earth in the absence of influence on them by other bodies. The law of inertia was the first step in establishing the basic physical principles of mechanics, which at that time were still completely vague, inarticulate and unclear. Subsequently, Newton, formulating the general laws of motion of bodies, included the law of inertia among them.
The moment of inertia of a body (system) relative to a given axis Oz (or axial moment of inertia) is a scalar quantity that is different from the sum of the products of the masses of all points of the body (system) by the squares of their distances from this axis:
From the definition it follows that the moment of inertia of a body (or system) relative to any axis is a positive quantity and not equal to zero.
In the future, it will be shown that the axial moment of inertia plays the same role during rotational motion of a body as mass does during translational motion, i.e., that the axial moment of inertia is a measure of the inertia of a body during rotational motion.
According to formula (2), the moment of inertia of a body is equal to the sum of the moments of inertia of all its parts relative to the same axis. For one material point located at a distance h from the axis, . The unit of measurement of the moment of inertia in SI will be 1 kg (in the MKGSS system -).
To calculate the axial moments of inertia, the distances of points from the axes can be expressed through the coordinates of these points (for example, the square of the distance from the Ox axis will be, etc.).
Then the moments of inertia about the axes will be determined by the formulas:
Often during calculations the concept of radius of gyration is used. The radius of inertia of a body relative to an axis is a linear quantity determined by the equality
where M is body mass. From the definition it follows that the radius of inertia is geometrically equal to the distance from the axis of the point at which the mass of the entire body must be concentrated so that the moment of inertia of this one point is equal to the moment of inertia of the entire body.
Knowing the radius of inertia, you can use formula (4) to find the moment of inertia of the body and vice versa.
Formulas (2) and (3) are valid both for a rigid body and for any system of material points. In the case of a solid body, breaking it into elementary parts, we find that in the limit the sum in equality (2) will turn into an integral. As a result, taking into account that where is the density and V is the volume, we obtain
The integral here extends to the entire volume V of the body, and the density and distance h depend on the coordinates of the points of the body. Similarly, formulas (3) for solid bodies take the form
Formulas (5) and (5) are convenient to use when calculating the moments of inertia of homogeneous bodies of regular shape. In this case, the density will be constant and will fall outside the integral sign.
Let us find the moments of inertia of some homogeneous bodies.
1. A thin homogeneous rod of length l and mass M. Let us calculate its moment of inertia relative to the axis perpendicular to the rod and passing through its end A (Fig. 275). Let us direct the coordinate axis along AB. Then for any elementary segment of length d the value is , and the mass is , where is the mass of a unit length of the rod. As a result, formula (5) gives
Replacing here with its value, we finally find
2. A thin round homogeneous ring of radius R and mass M. Let us find its moment of inertia relative to the axis perpendicular to the plane of the ring and passing through its center C (Fig. 276).
Since all points of the ring are located at a distance from the axis, formula (2) gives
Therefore, for the ring
Obviously, the same result will be obtained for the moment of inertia of a thin cylindrical shell of mass M and radius R relative to its axis.
3. A round homogeneous plate or cylinder of radius R and mass M. Let us calculate the moment of inertia of the round plate relative to the axis perpendicular to the plate and passing through its center (see Fig. 276). To do this, we select an elementary ring with radius and width (Fig. 277, a). The area of this ring is , and the mass is where is the mass per unit area of the plate. Then, according to formula (7) for the selected elementary ring there will be and for the entire plate
DEFINITION
Moment of inertia relative to the axis around which rotation occurs - this is a measure of the inertia of a body performing rotational movements.
The moment of inertia is a scalar (in general, tensor) physical quantity, which is found as the sum of the products of the masses of material points () (into which the body in question should be divided) into the squares of the distances () from them to the axis of rotation:
If the body is considered continuous, then the summation in expression (1) is replaced by integration, the masses of the body elements are denoted as:
where r is a function of the position of a material point in space; - body density; - volume of a body element. If the body is homogeneous:
Moment of inertia of a material point
The role of mass when moving around a circle of a material point is performed by the moment of inertia (J), which is equal to:
where r is the distance from the material point to the axis of rotation. For a material point that moves in a circle, the moment of inertia is a constant value.
The moment of inertia is an additive quantity. This means that if there is not one, but several material points in the system, then the moment of inertia of the system (J) is equal to the sum of the moments of inertia () of individual points:
Examples of moments of inertia of some bodies
The moment of inertia of a thin rod rotating about an axis passing through one end and perpendicular to the rod is equal to:
The moment of inertia of a straight circular cone, mass of height h and radius r, rotating about its axis:
The moment of inertia of a homogeneous solid parallelepiped, with geometric parameters and mass m rotating about its longest diagonal, is calculated by the formula:
The moment of inertia of a thin rectangular plate of mass m, width w and length d, rotating about an axis that passes through the intersection point of the diagonals of this rectangle perpendicular to the plane of the plate:
where m is the mass of the ball; R is the radius of the ball. The ball rotates about an axis that passes through its center.
Examples of formulas for calculating the moments of inertia of other bodies can be found in the section. In the same section you can familiarize yourself with Steiner's theorem.
Examples of solving problems on the topic “Moment of inertia”
EXAMPLE 1
| Exercise | Two small balls of mass m each are connected by a thin weightless rod, the length of which is equal to What will be the moment of inertia of the system relative to the axis that passes perpendicular to the rod through the center of mass of the system? |
| Solution | To solve the problem, we use the formula for the moment of inertia of one material point: where the distance from the point to the axis of rotation is . Consequently, formula (1.1) is transformed to the form:
Since the masses of the first and second material points are equal, the distances from each of them to the axis of rotation are equal, then: The moment of inertia is an additive quantity, which means that we find the moment of inertia of two points as the sum of and:
|
| Answer |
EXAMPLE 2
| Exercise | What is the moment of inertia of the system, which is shown in Fig. 2 and consists of two thin rods with masses m. The angle between the rods is straight. The lengths of the rods are equal to l. The axis of rotation is parallel to one of the rods (Fig. 2). |
| Solution | The moment of inertia of the system can be found as the sum of the moments of inertia of each rod relative to the axis of rotation: The moment of inertia () for a horizontal rod is equal to: |
