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Consider a material point of mass m, which is located at a distance r from the fixed axis (Fig. 26). Moment of inertia J material point relative to the axis is called a scalar physical quantity equal to the product of the mass m and the square of the distance r to this axis:

J = mr 2(75)

The moment of inertia of the system N material points will be is equal to the sum moments of inertia of individual points:

Rice. 26.

To the definition of the moment of inertia of a point.

If the mass is distributed continuously in space, then summation is replaced by integration. The body is divided into elementary volumes dv, each of which has a mass dm.

The result is the following expression:

For a body homogeneous in volume, the density ρ is constant, and writing the elementary mass in the form:

dm = ρdv, we transform formula (70) as follows:

The dimension of the moment of inertia - kg * m 2.

The moment of inertia of a body is a measure of the inertia of a body in rotational motion, just as the mass of a body is a measure of its inertia in translational motion.

Moment of inertia - it is a measure of the inert properties of a rigid body during rotational motion, depending on the distribution of mass about the axis of rotation. In other words, the moment of inertia depends on the mass, shape, dimensions of the body and the position of the axis of rotation.

Any body, regardless of whether it rotates or is at rest, has a moment of inertia about any axis, just as a body has mass, regardless of whether it is moving or at rest. Like mass, the moment of inertia is an additive quantity.

In some cases, the theoretical calculation of the moment of inertia is quite simple. Below are the moments of inertia of some solid bodies of regular geometric shape about an axis passing through the center of gravity.

Moment of inertia of an infinitely flat disk of radius R about an axis perpendicular to the disk plane:

Moment of inertia of a ball of radius R:

Moment of inertia of a rod with a length L relative to the axis passing through the middle of the rod perpendicular to it:

Moment of inertia of an infinitely thin hoop of radius R about an axis perpendicular to its plane:

The moment of inertia of a body about an arbitrary axis is calculated using Steiner's theorem:

The moment of inertia of a body about an arbitrary axis is equal to the sum of the moment of inertia about an axis passing through the center of mass parallel to the given one, and the product of the body's mass times the square of the distance between the axes.

Using the Steiner theorem, we calculate the moment of inertia of a rod with a length L about the axis passing through the end perpendicular to it (Fig. 27).

To the calculation of the moment of inertia of the rod

According to the Steiner theorem, the moment of inertia of the rod about the O′O′ axis is equal to the moment of inertia about the OO axis plus md 2. From here we get:

Obviously: the moment of inertia is not the same with respect to different axes, and therefore, when solving problems on the dynamics rotary motion, the moment of inertia of the body relative to the axis of interest to us each time has to be searched separately. So, for example, when designing technical devices containing rotating parts (in railway transport, in aircraft construction, electrical engineering, etc.), knowledge of the values ​​of the moments of inertia of these parts is required. At complex form body, the theoretical calculation of its moment of inertia can be difficult to perform. In these cases, it is preferable to measure the moment of inertia of the non-standard part empirically.

Moment of force F relative to point O

DEFINITION

The measure of the inertia of a rotating body is moment of inertia(J) relative to the axis around which the rotation occurs.

This is a scalar (in the general case, tensor) physical quantity, which is equal to the product of the masses of material points () into which the body under consideration should be partitioned, by the squares of distances () from them to the axis of rotation:

where r is a function of the position of a material point in space; - body density; - the volume of the body element.

For a homogeneous body, expression (2) can be represented as:

Moment of inertia in international system units is measured in:

The value of J is included in the basic laws that describe the rotation of a rigid body.

In general, the magnitude of the moment of inertia depends on the direction of the axis of rotation, and since the vector usually changes its direction relative to the body in the process of motion, the moment of inertia should be considered as a function of time. An exception is the moment of inertia of a body rotating around a fixed axis. In this case, the moment of inertia remains constant.

Steiner's theorem

The Steiner theorem makes it possible to calculate the moment of inertia of a body about an arbitrary axis of rotation, when the moment of inertia of the body under consideration is known with respect to the axis passing through the center of mass of this body and these axes are parallel. In mathematical form, the Steiner theorem is represented as:

where is the moment of inertia of the body about the axis of rotation passing through the center of mass of the body; m is the mass of the considered body; a is the distance between the axles. Be sure to remember that the axes must be parallel. From expression (4) it follows that:

Some expressions for calculating the moments of inertia of a body

When rotating around an axis, a material point has a moment of inertia equal to:

where m is the mass of the point; r is the distance from the point to the axis of rotation.

For a homogeneous thin rod of mass m and length l J relative to the axis passing through its center of mass (the axis is perpendicular to the rod), is equal to:

A thin ring, with a mass rotating about an axis that passes through its center, perpendicular to the plane of the ring, then the moment of inertia is calculated as:

where R is the radius of the ring.

A round homogeneous disk of radius R and mass m has J relative to the axis passing through its center and perpendicular to the disk plane, equal to:

For a uniform ball

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.

If the axes of rotation are the axes of a rectangular Cartesian coordinate system, then for a continuous body the moments of inertia can be calculated as:

where are the coordinates of an infinitely small element of the body.

Examples of problem solving

EXAMPLE 1

EXAMPLE 2

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 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. physical meaning moment of inertia. The product of the moment of inertia of a body and its angular acceleration is equal to the sum of the moments of all forces applied to the body. Compare. Rotational movement. Progressive movement. 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 axis O "in accordance with the Steiner 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 and the square of the distance d between the axes:

18. Angular moment of a rigid body. Angular velocity vector and angular momentum vector. Gyroscopic effect. Angular speed of precession

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. Considering that , we get .

If the sum of the moments of forces acting on a body rotating around a fixed axis is equal to zero, then the angular momentum is conserved ( law of conservation of angular momentum): . 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 value 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, then the rotation is counterclockwise. Historically 2 , the positive direction of rotation is considered to be “counterclockwise” rotation, although, of course, the choice of this direction is absolutely conditional. To determine the direction of the angular velocity vector, you can also use the "rule of the gimlet" (which is also called the "rule of the right screw") - if the direction of movement of the handle of the gimlet (or corkscrew) is combined with the direction of rotation, then the direction of movement of the entire gimlet will coincide 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 top point gradually descends and moves in a divergent spiral. This is the precession of the axis of the top.

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

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

ideal fluid called imaginary incompressible fluid, in which there are no viscosity, internal friction and thermal conductivity. Since there is no internal friction in it, there is no shear stresses between two adjacent liquid layers.

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

Considered in G. ur-tion refers. equilibrium of an incompressible fluid in the field of gravity (relative to the walls of a vessel moving according to some known law, for example, translational or rotational) make it possible to solve problems about the shape of the free surface and splashing of 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 on space. fly. devices. When determining the shape of the free surface of a liquid enclosed in a vessel, in addition to hydrostatic forces. pressure, inertial forces and gravity must take into account the surface tension of the liquid. In the case of rotation of the vessel around the vertical. axes with d.c. 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 with a post. acceleration a, the free surface of the liquid is a plane inclined to the horizontal plane at an angle

To change the speed of movement of the body in space, you need to make some effort. This fact applies to all types of mechanical motion and is associated with the presence of inertial properties in objects that have mass. This article discusses the rotation of bodies and gives the concept of their moment of inertia.

What is rotation in terms of physics?

Each person can answer this question, because this physical process is no different from its concept in everyday life. The process of rotation is the movement of an object with a finite mass along a circular path around some imaginary axis. The following examples of rotation can be given:

• Movement of the wheel of a car or bicycle.
• The rotation of the blades of a helicopter or fan.
• The movement of our planet around its axis and around the sun.

What physical quantities characterize the process of rotation?

Movement in a circle is described by a set of quantities in physics, the main ones are listed below:

• r - distance to the axis of a material point with mass m.
• ω and α are the angular velocity and acceleration, respectively. The first value shows how many radians (degrees) the body rotates around the axis in one second, the second value describes the rate of change in time of the first.
• L is the angular momentum, which is similar to that of linear motion.
• I is the moment of inertia of the body. This value is discussed in detail below in the article.
• M is the moment of force. It characterizes the degree of change in the value of L if an external force is applied.

The listed quantities are related to each other by the following formulas for rotational motion:

The first formula describes the circular motion of the body in the absence of the action of external moments of forces. In the above form, it reflects the law of conservation of the angular momentum L. The second expression describes the case of acceleration or deceleration of the rotation of the body as a result of the action of the moment of force M. Both expressions are often used in solving problems of dynamics along a circular trajectory.

As can be seen from these formulas, the moment of inertia about the axis (I) is used in them as a certain coefficient. Let's consider this value in more detail.

Where does the value I come from?

In this paragraph, we consider the simplest example of rotation: the circular movement of a material point with mass m, the distance of which from the axis of rotation is r. This situation is shown in the figure.

According to the definition, the angular momentum L is written as the product of the shoulder r and the linear momentum p of the point:

L = r*p = r*m*v since p = m*v

Given that the linear and angular speeds are related to each other through the distance r, this equality can be rewritten as follows:

v = ω*r => L = m*r 2 *ω

The product of the mass of a material point and the square of the distance to the axis of rotation is commonly called the moment of inertia. The formula above would then be rewritten as follows:

That is, we received the expression that was given in the previous paragraph, and introduced the value of I.

General formula for the value I of the body

The expression for the moment of inertia with the mass m of a material point is basic, that is, it allows you to calculate this value for any body that has an arbitrary shape and a non-uniform distribution of mass in it. To do this, it is necessary to divide the object under consideration into small elements of mass m i (an integer i is the element number), then multiply each of them by the square of the distance r i 2 to the axis around which the rotation is considered, and add the results. The described method for finding the value of I can be written mathematically as follows:

I = ∑ i (m i *r i 2)

If the body is broken in such a way that i->∞, then the reduced sum is replaced by the integral over the mass of the body m:

This integral is equivalent to another integral over the volume of the body V, since dV=ρ*dm:

I = ρ*∫ V (r i 2 *dV)

All three formulas are used to calculate the moment of inertia of a body. In this case, in the case of a discrete distribution of masses in the system, it is preferable to use the 1st expression. At continuous distribution masses apply the 3rd expression.

Properties of the quantity I and its physical meaning

The described procedure for obtaining a general expression for I allows us to draw some conclusions about the properties of this physical quantity:

• it is additive, that is, the total moment of inertia of the system can be represented as the sum of the moments of its individual parts;
• it depends on the distribution of mass within the system, as well as on the distance to the axis of rotation, the larger the latter, the larger I;
• it does not depend on the moments of forces acting on the system M and on the rotation speed ω.

The physical meaning of I is how much the system prevents any change in its rotation speed, that is, the moment of inertia characterizes the degree of "smoothness" of the resulting accelerations. For example, a bicycle wheel can be easily spun up to high angular speeds and also easy to stop, but to change the rotation of the flywheel on the crankshaft of a car, it will take considerable effort and some time. In the first case, there is a system with a small moment of inertia, in the second - with a large one.

The I value of some bodies for an axis of rotation passing through the center of mass

If we apply volume integration for any bodies with an arbitrary mass distribution, then we can obtain the value I for them. In the case of homogeneous objects that have an ideal geometric shape, this problem has already been solved. Below are the formulas for the moment of inertia for a rod, a disk and a ball of mass m, in which the substance that makes them is distributed uniformly:

• Kernel. The axis of rotation runs perpendicular to it. I \u003d m * L 2 / 12, where L is the length of the rod.
• Disc of arbitrary thickness. The moment of inertia with the axis of rotation passing perpendicular to its plane through the center of mass is calculated as follows: I = m*R 2 /2, where R is the disk radius.
• Ball. In view of the high symmetry of this figure, for any position of the axis passing through its center, I \u003d 2/5 * m * R 2, here R is the radius of the ball.

The problem of calculating the value of I for a system with a discrete mass distribution

Imagine a rod 0.5 meters long, which is made of a hard and light material. This rod is fixed on the axis in such a way that it runs perpendicular to it exactly in the middle. 3 weights are suspended on this rod as follows: on one side of the axle there are two weights with masses of 2 kg and 3 kg, located at distances of 10 cm and 20 cm from its end, respectively; on the other hand, one weight of 1.5 kg is suspended from the end of the rod. For this system, it is necessary to calculate the moment of inertia I and determine with what speed ω the rod will rotate if a force of 50 N is applied to one of its ends for 10 seconds.

Since the mass of the rod can be neglected, then it is necessary to calculate the moment I for each load and add the results obtained to get the total moment of the system. According to the condition of the problem, a load of 2 kg is at a distance of 0.15 m (0.25-0.1) from the axis, a load of 3 kg is 0.05 m (0.25-0.20), a load of 1.5 kg is 0.25 m. Using the formula for the moment I of a material point, we obtain:

I \u003d I 1 + I 2 + I 3 \u003d m 1 * r 1 2 + m 2 * r 2 2 + m 3 * r 3 2 \u003d 2 * (0.15) 2 + 3 * (0.05) 2 + 1.5 * (0.25) 2 \u003d 0.14 625 kg * m 2.

Please note that when performing calculations, all units of measurement were converted to the SI system.

To determine the angular velocity of rotation of the rod after the action of a force, one should apply the formula with the moment of force, which was given in the second paragraph of the article:

Since α = Δω/Δt and M = r*F, where r is the arm length, we get:

r*F = I*Δω/Δt => Δω = r*F*Δt/I

Given that r = 0.25 m, we substitute the numbers into the formula, we get:

Δω \u003d r * F * Δt / I \u003d 0.25 * 50 * 10 / 0.14625 \u003d 854.7 rad / s

The resulting value is quite large. To get the usual rotational speed, you should divide Δω by 2 * pi radians:

f \u003d Δω / (2 * pi) \u003d 854.7 / (2 * 3.1416) \u003d 136 s -1

Thus, the applied force F to the end of the rod with weights in 10 seconds will spin it up to a frequency of 136 revolutions per second.

Calculation of the I value for a bar when the axis passes through its end

Let there be a homogeneous rod with mass m and length L. It is necessary to determine the moment of inertia if the axis of rotation is located at the end of the rod perpendicular to it.

Let's use general expression for i:

I = ρ*∫ V (r i 2 *dV)

Dividing the object under consideration into elementary volumes, we note that dV can be written as dr*S, where S is the sectional area of ​​the rod, and dr is the thickness of the partition element. Substituting this expression into the formula, we have:

I = ρ*S*∫ L (r 2 *dr)

This integral is quite easy to calculate, we get:

I \u003d ρ * S * (r 3 / 3) ∣ 0 L => I \u003d ρ * S * L 3 / 3

Since the volume of the rod is S*L, and the mass is ρ*S*L, we get the final formula:

It is curious to note that the moment of inertia for the same rod, when the axis passes through its center of mass, is 4 times less than the obtained value (m*L 2 /3/(m*L 2 /12)=4).