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 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 with respect to this axis. According to the Steiner 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 the axis passing through the center of mass of the body parallel to the considered axis, and the product of the body mass m per square distance d between axles:
where is the total mass of the body.
For example, the moment of inertia of a rod about an axis passing through its end is:
Axial moments of inertia of some bodies
| Body | Description | Axis position a | Moment of inertia J a |
|---|---|---|---|
 |
Material point of mass m | On distance r from a point, fixed | |
 |
Hollow thin-walled cylinder or ring of radius r and the masses m | Cylinder axis | |
 |
Solid cylinder or disk radius r and the masses m | Cylinder axis | |
 |
Hollow thick-walled mass cylinder m with outer radius r2 and inner radius r1 | Cylinder axis | |
 |
Solid cylinder length l, radius r and the masses m | ||
 |
Hollow thin-walled cylinder (ring) length l, radius r and the masses m | The axis is perpendicular to the cylinder and passes through its center of mass | |
 |
Straight thin rod length l and the masses m | The axis is perpendicular to the rod and passes through its center of mass | |
 |
Straight thin rod length l and the masses m | The axis is perpendicular to the rod and passes through its end | |
 |
Thin-walled sphere of radius r and the masses m | The axis passes through the center of the sphere | |
 |
ball radius r and the masses m | The axis passes through the center of the ball | |
| Cone radius r and the masses m | cone axis | ||
| Isosceles triangle with height h, base a and weight m | The axis is perpendicular to the plane of the triangle and passes through the vertex | ||
| Right triangle with side a and weight m | The axis is perpendicular to the plane of the triangle and passes through the center of mass | ||
| Square with side a and weight m | The axis is perpendicular to the plane of the square and passes through the center of mass |
Derivation of 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. Dividing a thin-walled cylinder into elements with a 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 converted 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 into thin rings with a thickness 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 disk (solid cylinder)
Formula derivation
Considering the cylinder (disk) as a ring with zero inner radius ( R 1 = 0), we obtain the formula for the moment of inertia of the cylinder (disk):
solid cone
Formula derivation
Divide the cone into thin discs of thickness dh, perpendicular to the axis of the cone. The radius of such a disk is
where R is the radius of the base of the cone, H is the height of the cone, 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 uniform ball
Formula derivation
Divide the ball into thin disks 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 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 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 value dR.
Thin rod (axis passes through the center)
Formula derivation
Divide the rod into small fragments 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 rod to its end, the center of gravity of the rod moves relative to the axis by a distance l/2. According to the Steiner theorem, the new moment of inertia will be equal to
Dimensionless moments of inertia of planets and their satellites
Of great importance for studies of the internal structure of planets and their satellites are their dimensionless moments of inertia. Dimensionless moment of inertia of a body of radius r and the masses m is equal to the ratio of its moment of inertia about 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 in depth. One of the methods for measuring it in planets and satellites is to determine the Doppler shift of the radio signal transmitted by the AMS flying around 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 smaller, the greater the mass of the body is concentrated at its center. For example, the dimensionless moment of inertia of the Moon is 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 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 element of the body with volume dV, density ρ and weight dm.
The OX axis is called main axis of inertia of the body if the centrifugal moments of inertia Jxy and Jxz are simultaneously 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 principal 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 the section of the view
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. It is used to calculate the radius of gyration, beam deflection, section selection of beams, columns, etc.
The SI unit of measurement is m 4 . In construction calculations, literature and assortments of rolled metal, in particular, it is indicated in cm 4.
From it the section modulus is expressed:
.| Geometric moments of inertia of some figures | |
|---|---|
| Rectangle Height and Width: | |
| Rectangular box section with height and width along the outer contours and , and along the inner and respectively | |
| Circle diameter | |
Central moment of inertia
Central moment of inertia(or the moment of inertia about the 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 the 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 inertia tensor. The inertia tensor matrix is symmetrical, has dimensions, and consists of centrifugal moment components:
.
By choosing an appropriate coordinate system, the matrix of the inertia tensor can be reduced to a diagonal form. To do this, you need to solve the eigenvalue problem for the tensor matrix:
,
where -
Often we hear expressions: “it is inert”, “move by inertia”, “moment of inertia”. In a figurative sense, the word "inertia" can be interpreted as a lack of initiative and action. We are interested in direct meaning.
What is inertia
By definition inertia in physics, it is the ability of bodies to maintain a state of rest or motion in the absence of external forces.
If everything is clear with the very concept of inertia on an intuitive level, then moment of inertia- a separate issue. Agree, it is difficult to imagine in the mind what it is. In this article, you will learn how to solve basic problems on the topic "Moment of inertia".
Determining the moment of inertia
It is known from the school curriculum that mass is a measure of the inertia of a body. If we push two carts of different masses, then it will be more difficult to stop the one that is heavier. That is, the greater the mass, the greater the external influence is necessary to change the motion of the body. Considered refers to the translational movement, when the cart from the example moves in a straight line.
By analogy with mass and translational motion, the moment of inertia is a measure of the inertia of a body during rotational motion around an axis.
Moment of inertia- a scalar physical quantity, a measure of the inertia of a body during rotation around an axis. Denoted by letter J and in the system SI measured in kilograms multiplied by a square meter.
How to calculate the moment of inertia? There is a general formula by which the moment of inertia of any body is calculated in physics. If the body is broken into infinitely small pieces of mass dm , then the moment of inertia will be equal to the sum of the products of these elementary masses and the square of the distance to the axis of rotation.
This is the general formula for the moment of inertia in physics. For a material point of mass m , rotating about an axis at a distance r from it, this formula takes the form:
Steiner's theorem
What does the moment of inertia depend on? From the mass, the position of the axis of rotation, the shape and size of the body.
The Huygens-Steiner theorem is a very important theorem that is often used in solving problems.
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The Huygens-Steiner theorem states:
The moment of inertia of a body about an arbitrary axis is equal to the sum of the moment of inertia of the body about an axis passing through the center of mass parallel to an arbitrary axis and the product of the body's mass times the square of the distance between the axes.
For those who do not want to constantly integrate when solving problems of finding the moment of inertia, here is a figure showing the moments of inertia of some homogeneous bodies that are often found in problems:
An example of solving the problem of finding the moment of inertia
Let's consider two examples. The first task is to find the moment of inertia. The second task is to use the Huygens-Steiner theorem.
Problem 1. Find the moment of inertia of a homogeneous disk of mass m and radius R. The axis of rotation passes through the center of the disk.
Decision:
Let us divide the disk into infinitely thin rings, the radius of which varies from 0 before R and consider one such ring. Let its radius be r, and the mass dm. Then the moment of inertia of the ring:
The mass of the ring can be represented as:
Here dz is the height of the ring. Substitute the mass into the formula for the moment of inertia and integrate:
The result was a formula for the moment of inertia of an absolute thin disk or cylinder.
Problem 2. Let there again be a disk of mass m and radius R. Now we need to find the moment of inertia of the disk about the axis passing through the middle of one of its radii.
Decision:
The moment of inertia of the disk about the axis passing through the center of mass is known from the previous problem. We apply the Steiner theorem and find:
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The moment of inertia is a measure of the inertia of a body about an axis during rotational motion (real or imaginary) around this axis3. The moment of inertia is quantitatively equal to the sum of the moments of inertia of the particles of the body - the products of the masses of the particles and the squares of their distances from the axis of rotation: J=Smr 2
When body particles are farther from the axis of rotation, then the angular acceleration of the body under the same moment of force smaller; if particles closer to the axis, then the angular acceleration is greater. This means that if you bring the body (as a whole or its parts) closer to the axis, then it is easier to cause angular acceleration, it is easier to accelerate the body in rotation, and it is easier to stop it. This is used when moving around the axis.
Having found empirically the moment of inertia of the body, it is possible to calculate the radius of gyration, the value of which reflects the distribution of particles in the body relative to a given axis.
The radius of gyration is a comparative measure of the inertia of a given body about its different axes. It is measured by the square root of the ratio of the moment of inertia about a given axis
to body weight:R=ÖJ/m
The quantitative determination of moments of inertia in biomechanics is not always accurate enough. But to understand the physical foundations of human movements, this characteristic must be taken into account.
POWER CHARACTERISTICS
Force
Force is a measure of the mechanical action of one body on another. Numerically, it is determined by the product of the mass of the body and its acceleration caused by the application of this force:F=ma;
Thus, the measurement of force, like the measurement of mass, is based on Newton's 2nd law. Since this law reveals dependencies in translational motion, then the force as a vector is determined only in the case of such a simple type of motion in terms of mass and acceleration,
Sources of strength. It has already been pointed out that acceleration depends on the frame of reference. Therefore, the force determined by acceleration also depends on the frame of reference. In an inertial frame of reference, the source of force for a given body is always another material body. As soon as two material objects interact, then under these conditions Newton's 3rd law3 is manifested.
If another body acts on one body, then it changes the motion of the first. But the first body in this interaction also changes the motion of the other. Both forces are applied to different objects, each exhibiting a corresponding effect. They cannot be replaced by one resultant, since they are applied to different objects. That is why they do not balance each other.
In a non-inertial frame of reference, in addition to the interactions of two bodies, special forces of inertia (“fictitious”) are also considered, for which Newton's 3rd law is not applicable.
Force measurement . Applies static force measurement, i.e. measurement with balancing force(when the acceleration is zero), and dynamic - according to the acceleration imparted to the body by its application.
At static action force on the body (M) there are two bodies (A and B); there are three material objects in total (Fig. 23, a). Forces F a and f in, attached to the body M, equal in magnitude and opposite in direction, they are mutually balanced. Their resultant is zero. The acceleration caused by them is also zero. The speed does not change (remains constant - uniform movement or relative immobility).
Strength fa, acting statically can be measured by the balancing force f c.
Consider three cases of manifestation of the static action of force, when all bodies are motionless -
a) a gymnast hanging on the crossbar; support force balances the body's gravity (G);
b) a balanced body moves perpendicular to the balanced force of gravity - the skater slides on the ice; support force balances the body's gravity (G); the latter does not directly affect the sliding speed;
c) a balanced body moves by inertia in the direction of the balanced force; the skier slides at a constant speed down the slope; resistance forces (air and ski friction on snow - Q) balance the rolling component of gravity (G). In all three cases, regardless of the state of rest or the direction of motion of the body, the balanced force does not change the motion; speeds in the direction of its action are constant.
It should be emphasized that in all cases the static action of the force causes deformation body.
At dynamic action force on the body M there is an unbalanced force. In problems in theoretical mechanics, only this one driving force is often considered as a measure of the action of only one driving body.
The driving force is the force that coincides with the direction of movement (passing ) or forms an acute angle with it and at the same time can do positive work (to increase the energy of the body).
However, in real conditions of human movement, there is always a medium (air or water), support and other external bodies (projectiles, equipment, partners, opponents, etc.) operate. All of them can have an inhibitory effect. Moreover, not a single real movement without the participation braking forces it just doesn't happen.
The braking force is directed opposite to the direction of movement (oncoming) or forms an obtuse angle with it. She can do negative work(to reduce the energy of the body).
Part of the driving force, equal in magnitude to the braking force, balances the latter - this is balancing force (Fip).
The excess of the driving force over the braking force is the accelerating force (Fusk)- causes acceleration of the body with mass m according to Newton's 2nd law (Fy=ma).
Consequently, the speed does not remain constant, but changes, i.e., acceleration occurs. This is the dynamic action of force. F.
Strength F usk, acting dynamically, can be measured by the mass of the body and its acceleration.
Classification of forces. The forces that are studied in the analysis of human movements, depending on the general characteristics, are divided into groups. According to the method of interaction of bodies, all forces are divided into d i s t a n t n e, arising at a distance without direct contact of bodies, and contact, which arise only when bodies come into contact.
Distant forces in mechanics include the forces of universal gravitation, of which in biomechanics the forces of terrestrial gravity are studied, manifested in gravity . Contact forces include elastic forces and friction forces .
According to the influence on the movement, forces are distinguished a k t i v n e(or given) and bond reactions. We remind you that connections are restrictions on the movement of an object carried out by other bodies. The force with which the connection opposes the movement is the reaction of the connection. It is not known in advance and depends on the action of other forces on the body and the movement of the body itself.
Coupling reactions in themselves do not cause movement, they only counteract active forces or balance them. If the reactions of connection do not balance the active forces, then the movement begins under the action of the latter.
According to the source of occurrence relative to the system (for example, the human body), forces are distinguished in e s h n i e, caused by the action of bodies external to the system, and internal, caused by interactions within the system. This division is necessary when determining the possibilities of action of certain forces. One and the same force should be considered external or internal, depending on the object in relation to which we consider it.
By way of application forces in mechanics divide by concentrated applied to the body at one point, and distributed. The latter are divided into surface and bulk.
By the nature of the force, there are constants and variables. AT An example of a constant force is the force of gravity (at a given point on the Earth). The same force can vary depending on several conditions. In practice, in the movement of a person, constant forces are almost never encountered. All forces are variable. They change depending on time (a muscle changes the traction force over time), distance (at different points on the Earth, even the "constant force" of gravity is different), speed (the resistance of the environment depends on the relative speed of the body and the environment).
Since the interaction of the human body with the external environment, caused by the movements of body parts, is especially important in biomechanics, then the external and internal forces relative to the system (human body) will be considered in detail. The interaction of physical objects is the main reason for the change in movements. Therefore, the measure of interaction - force - is given special attention in biomechanics.
Moment of power
A moment of force is a measure of the mechanical action capable of turning a body (a measure of the rotating action of a force). It is numerically determined by the product of the modulus of force and its shoulder (the distance from the center of the moment1 to the line of action of the force):
The moment of force has a plus sign if the force imparts counterclockwise rotation, and a minus sign if it is in the opposite direction.
The rotational ability of a force is manifested in the creation, change or termination of rotational motion.
Polar moment of force(moment of a force about a point) can be defined for any force about that point (O) (the center of the moment). If the distance from the line of action of the force to the chosen point is zero, then the moment of force is zero. Therefore, a force thus placed has no rotational power about this center. Rectangle area (Fd) numerically equal to the modulus of the moment of force.
When several moments of force are applied to one body, they can be reduced to one moment - the main moment.
To determine the vector of the moment of force1, you need to know: a) moment modulus(the product of the modulus of force on her shoulder); b) plane of rotation(passes through the line of action of the force and the center of the moment) and c) direction of rotation in this planes.
Axial moment of force(moment of force relative to the axis) can be defined for any force, except for coinciding with the axis, parallel to it or crossing it. In other words, the force and the axis must not lie in the same plane.
Apply static measurement a moment of force if it is balanced by a moment of another force lying in the same plane, equal in absolute value and opposite in direction, relative to the same center of the moment (for example, when a lever is in equilibrium). The moments of gravity of the links relative to their proximal joints are called static moments of links.
Apply dynamic measurement moment of force, if the moment of inertia of the body about the axis of rotation and its angular acceleration are known. Like forces, moments of forces about the center can be driving and braking, and therefore, balancing, accelerating and slowing down. The moment of force can be deviating- deflects the plane of rotation in space.
At all accelerations, inertia forces arise: at normal accelerations - centrifugal inertia forces, at tangential accelerations (positive or negative) - tangential inertia forces. The centrifugal force of inertia is directed along the radius of rotation and has no moment relative to the center of rotation. The tangential force of inertia is applied to a solid link in the center of its swings. Thus, there is moment of inertia about the axis of rotation.
Action of force
MOMENT OF INERTIA I of the body relative to a point, axis or plane is the sum of the products of the mass of the points of the body m i , by the squares of their distances r i to the point, axis or plane:
The moment of inertia of a body about an axis is a measure of the inertia of a body in rotation around that axis.
The moment of inertia of a body can also be expressed in terms of the mass M of the body and its radius of gyration r:
MOMENTS OF INERTIA REGARDING THE AXES, PLANES AND THE ORIGIN OF THE CARTESIS COORDINATES.
Moment of inertia about the origin (polar moment of inertia):
RELATIONSHIP BETWEEN AXIAL, PLANE AND POLAR MOMENTS OF INERTIA:
The values of the axial moments of inertia of some geometric bodies are given in Table. one.
Table 1. Moment of inertia of some bodies
figure or body |
|
|
With c→0, a rectangular plate is obtained |
|
CHANGING THE MOMENTS OF INERTIA WHEN THE AXIS IS CHANGED
Moment of inertia I u 1 about the axis u 1 parallel to the given axis u (Fig. 1):
where I u is the moment of inertia of the body about the u axis; l (l 1) - distance from the u axis (from the u axis 1) to the u axis parallel to them, passing through the center of mass of the body; a is the distance between the axes u and u 1 .
Picture 1.
If the u axis is central (l=0), then
i.e., for any group of parallel axes, the moment of inertia about the central axis is the smallest.
Moment of inertia I u relative to the u axis, making angles α, β, γ with the axes of Cartesian coordinates x, y, z (Fig. 2):
Figure 2.
The x, y, z axes are the main ones, if
The moment of inertia about the u axis, which makes angles α, β, γ with the main axes of inertia x, y, z:
CHANGE IN CENTRIFUGAL MOMENTS OF INERTIA DURING PARALLEL AXIS TRANSFER:
where is the centrifugal moment of inertia about the central axes x c, y c, parallel to the axes x, y; M - body weight; x s, y s - coordinates of the center of mass in the system of axes x, y.
CHANGE OF THE CENTRIFUGAL MOMENT OF INERTIA DURING THE ROTATION OF THE AXIS x, y AROUND THE AXIS z BY THE ANGLE α TO THE POSITION x 1 y 1(Fig. 3):
Figure 3
DETERMINATION OF THE POSITION OF THE MAIN AXES OF INERTIA. The axis of material symmetry of the body is the main axis of inertia of the body.
If the xOz plane is the plane of material symmetry of the body, then any of the y axes is the main axis of inertia of the body.
If the position of one of the main axes z ch is known, then the position of the other two axes x ch and y ch is determined by the rotation of the x and y axes around the z axis ch by an angle φ (Fig. 3):
ELLIPSOID AND PARALLELEPIPED OF INERTIA. An ellipsoid of inertia is an ellipsoid whose symmetry axes coincide with the main central axes of the body x main, y main, z main, and the semiaxes a x, a y, and z are equal, respectively:
where r уО z , r x Oz , r xOy are the radii of inertia of the body relative to the main planes of inertia.
A parallelepiped of inertia is a parallelepiped described around an ellipsoid of inertia and having common axes of symmetry with it (Fig. 4).
Figure 4
REDUCTION (REPLACEMENT TO SIMPLIFY THE CALCULATION) OF A RIGID BODY WITH POINTED MASSES. When calculating the axial, planar, centrifugal and polar moments of inertia, a body of mass M can be reduced by eight concentrated masses M/8 located at the vertices of the parallelepiped of inertia. The moments of inertia about any axes, planes, poles are calculated by the coordinates of the vertices of the parallelepiped of inertia x i , y i , z i (i=1, 2, ..., 8) according to the formulas:
EXPERIMENTAL DETERMINATION OF MOMENTS OF INERTIA
1. Determination of the moments of inertia of bodies of revolution using the differential equation of rotation - see the formulas ("Rotational motion of a rigid body").
The body under study is fixed on the horizontal axis x, coinciding with its axis of symmetry, and is rotated around it with the help of a load P attached to a flexible thread wrapped around the body under study (Fig. 5), while the time t of lowering the load to a height h is measured. . To eliminate the influence of friction at the points of body fixation on the x axis, the experiment is performed several times at different values of the weight of the load P.
Figure 5
In two experiments with loads P 1 and P 2
2. Experimental determination of the moments of inertia of bodies by studying the oscillations of a physical pendulum (see 2.8.3) .
The body under study is fixed on the horizontal x-axis (non-central) and the period of small oscillations about this axis T is measured. The moment of inertia about the x-axis is determined by the formula
where P - body weight; l 0 - distance from the axis of rotation to the center of mass C of the body.
The moment of inertia of a body (system) about a given axis Oz (or the axial moment of inertia) is a scalar value that is different from the sum of the products of the masses of all points of the body (system) and the squares of their distances from this axis:
It follows from the definition that the moment of inertia of a body (or system) about any axis is a positive quantity and not equal to zero.
Later it will be shown that the axial moment of inertia plays the same role during the rotational motion of the body as the mass during translational motion, i.e., that the axial moment of inertia is a measure of the inertia of the 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 about 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 in terms of 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 in the course of calculations, the concept of the radius of gyration is used. The radius of gyration of a body relative to an axis is a linear quantity determined by the equality
where M is the mass of the body. It follows from the definition 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, it is possible to find the moment of inertia of the body using formula (4) 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, dividing it into elementary parts, we find that in the limit the sum in equality (2) turns into an integral. As a result, given that where is the density and V is the volume, we get
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 will 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 go out from under 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 about 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 its value, we finally find
2. A thin round homogeneous ring with radius R and mass M. Let's find its moment of inertia about the axis perpendicular to the plane of the ring and passing through its center C (Fig. 276).
Since all points of the ring are 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 with mass M and radius R about its axis.
3. Round homogeneous plate or cylinder with radius R and mass M. Let us calculate the moment of inertia of the round plate about the axis perpendicular to the plate and passing through its center (see Fig. 276). To do this, we select an elementary ring with a 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, and for the entire plate
