Formula for moment of inertia. Moment of force and moment of inertia What is the moment of inertia?

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. Physical meaning of the 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. Forward 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 relative to 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 mass m by the square of the distance d between the axes:

18. Momentum of a rigid body. Angular velocity vector and angular momentum vector. Gyroscopic effect. Angular precession rate

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 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 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 “gimlet rule” (which is also called the “right screw rule”) - if the direction of movement of the gimlet handle (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 rotation axis 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 rocks on the waves, and the gun looks at one point) In navigation, etc.

Observing precession is quite simple. You need to launch 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 lowers and moves in a diverging 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 will stop, and the top will take a stationary position in space. In the example with a top, this will not happen, since in it the force that causes precession - the Earth's gravity - acts constantly.

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

An ideal liquid called imaginary incompressible fluid, which lacks viscosity, internal friction and thermal conductivity. Since there is no internal friction in it, then no shear stress between two adjacent layers of liquid.

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

The equations considered in G. relate. the equilibrium of an incompressible fluid in a field of gravity (relative to the walls of a vessel moving according to a certain known law, for example translational or rotational) makes it possible to solve problems about the shape of the free surface and about the splashing of liquid in moving vessels - in tanks for transporting liquids, fuel tanks of airplanes 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 enclosed in a vessel, in addition to hydrostatic forces. pressure, inertial forces and gravity, it is necessary to take into account the surface tension of the liquid. In the case of rotation of the vessel around the vertical. axes with post. ang. speed, the free surface takes the form of a paraboloid of rotation, and in a vessel moving parallel to the horizontal plane translationally and rectilinearly with a station. acceleration A, the free surface of the liquid is a plane inclined to the horizontal plane at an angle

Let us consider a material point of mass m, which is located at a distance r from the fixed axis (Fig. 26). The moment of inertia J of a material point relative to an axis is a scalar physical quantity equal to the product of mass m by the square of the distance r to this axis:

J = mr 2(75)

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

Rice. 26.

To determine 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:

Dimension of 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 - this is a measure of the inertial properties of a solid body during rotational motion, depending on the distribution of mass relative to the axis of rotation. In other words, the moment of inertia depends on the mass, shape, size of the body and the position of the axis of rotation.

Any body, regardless of whether it is rotating or 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. Similar to 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 relative to an axis perpendicular to the plane of the disk:

Moment of inertia of a ball of radius R:

Moment of inertia of a rod 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 relative to 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 this one and the product of the body mass by the square of the distance between the axes.

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

To calculate the moment of inertia of the rod

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

Obviously: the moment of inertia is not the same relative to different axes, and therefore, when solving problems on the dynamics of rotational motion, the moment of inertia of the body relative to the axis of interest to us must be looked for separately each time. So, for example, when designing technical devices containing rotating parts (in railway transport, aircraft manufacturing, electrical engineering, etc.), knowledge of the values of the moments of inertia of these parts is required. With a complex body shape, theoretical calculation of its moment of inertia may be difficult to perform. In these cases, they prefer to measure the moment of inertia of a non-standard part experimentally.

Moment of force F relative to point O

SI unit: kg m².

Designation: I or J.

For example, the moment of inertia of the disk relative to the O axis in accordance with Steiner’s theorem:

18. Momentum of a rigid body. Angular velocity vector and angular momentum vector. Gyroscopic effect. Angular precession rate

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 .

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

PHYSICAL PENDULUM

Goal of the work: determine the moment of inertia of a physical pendulum in the form of a rod with weights based on the period of its own oscillations.

Equipment: pendulum, stopwatch.

THEORETICAL INTRODUCTION

Moment of inertia of a rigid body is a measure of the inertia of a body during its rotational motion. In this sense, it is an analogue of body mass, which is a measure of the inertia of a body during translational motion. According to the definition, moment of inertia body is equal to the sum of the products of the masses of the particles of the body m i by the squares of their distances to the axis of rotation r i 2:

, or .(1)

The moment of inertia depends not only on the mass, but also on its distribution relative to the axis of rotation. As you can see, the inertia during rotation of a body is greater, the further the particles of the body are located from the axis.

There are various experimental methods for determining the moment of inertia of bodies. The paper proposes a method for determining the moment of inertia from the period of natural oscillations of the body under study as a physical pendulum. Physical pendulum is a body of arbitrary shape, the suspension point of which is located above the center of gravity. If in a gravitational field the pendulum is deflected from the equilibrium position and released, then under the influence of gravity the pendulum tends to the equilibrium position, but, having reached it, by inertia it continues to move and is deflected in the opposite direction. Then the movement process is repeated in the opposite direction. As a result, the pendulum will perform rotational oscillations of its own.

To derive the formula for the moment of inertia of a pendulum through the period of its own oscillations, we use basic law of rotational dynamics: angular acceleration of a body is directly proportional to the moment of force and inversely proportional to the moment of inertia of the body relative to the axis of rotation:

Moment of power by definition equal to the product of the force and the arm of the force. The arm of a force is a perpendicular lowered from the axis of rotation to the line of action of the force. For a pendulum (Fig. 1a), the gravity arm is equal to d = a sin a, Where A– the distance between the axis of rotation and the center of mass of the pendulum. For small oscillations of the pendulum, the angle of deflection a is relatively small, and the sines of small angles are equal to the angles themselves with sufficient accuracy. Then the moment of gravity can be determined by the formula М = −mga∙a. The minus sign is due to the fact that the moment of gravity counteracts the deflection of the pendulum.

Since angular acceleration is the second derivative of the angle of rotation with respect to time, the basic law of the dynamics of rotational motion (1) takes the form

. (3)

This is a second order differential equation. Its solution must be a function that, upon substitution, turns the equation into an identity. As can be seen from equation (3), for this, the solution function and its second derivative must have the same form. In mathematics, such a function can be the cosine, sine function

a = a 0 sin( w t + j), (4)

provided that the cyclic frequency is equal to . Cyclic frequency is related to period of oscillation, that is, the time of one oscillation, the ratio T= 2p/w. From here

Oscillation period T and the distance from the axis of rotation to the center of gravity of the pendulum A can be measured. Then from (5) the moment of inertia of the pendulum relative to the axis of rotation WITH can be determined experimentally using the formula

. (6)

The pendulum, the moment of inertia of which is determined in the work, is a rod with two disks placed on it. Theoretically, the moment of inertia of a pendulum can be defined as the sum of the moments of inertia of the individual parts. The moment of inertia of the disks can be calculated using the formula for the moment of inertia of a material point, since they are small compared to the distance to the axis of rotation: , . Moment of inertia of the rod relative to an axis located at a distance b from the middle of the rod, can be determined by Steiner’s theorem . As a result, the total moment of inertia of the pendulum can be theoretically calculated using the formula

. (7)

Here m 1 , m 2 and m 0 – masses of the first, second disks and rod, l 1 , l 2 – distances from the middle of the disks to the axis of rotation, l 0 – length of the rod.

Distance from the suspension point to the center of gravity of the pendulum A, necessary for the experimental determination of the moment of inertia in formula (6), can be determined using the concept of the center of gravity. Center of gravity body is the point to which the resultant force of gravity is applied. Therefore, if the pendulum is placed horizontally on a support located under the center of gravity, then the pendulum will be in equilibrium. Then just measure the distance from the axis WITH to the support.

But you can determine the distance A by calculation. From the equilibrium condition of the pendulum on the support (Fig. 1b) it follows that the moment of the resulting force of gravity relative to the axis WITH (m 1 +m 2 +m 0)ga equal to the sum of the moments of gravity of the loads and the rod m 1 gl 1 +m 2 gl 2 +m 0 gb. Where do we get it from?

. (8)

COMPLETING OF THE WORK

1. By weighing on a scale, determine the masses of the disks and the rod. Place the discs on the rod and secure them. Measure the distances from the axis of rotation to the middle of the disks l 1 , l 2 and to the middle of the rod b, rod length l 0 according to centimeter divisions on the rod. Record the measurement results in the table. 1.

Table 1

2.Connect the electronic unit to a 220 V network.

Measure the period of oscillation. To do this, move the pendulum from the equilibrium position to a small angle and release it. Press the button Start stopwatch. To measure time t, for example, ten oscillations, after the ninth oscillation, press the button Stop. The period is
T = t/ 10. Record the result in the table. 2, press the button Reset. Repeat the experiment at least three times at other angles of deflection of the pendulum.

Turn off the installation.

4. Perform calculations in the SI system. Determine the average value<T> oscillation period. Determine distance A from the axis to the center of gravity of the pendulum according to formula (8), or place the pendulum on a support so that it is in equilibrium, and measure the distance using the divisions on the rod A.

A, m	T 1 , With	T 2 , s	T 3, s	<T>,s	, kg∙m 2	J theor, kg∙m 2

table 2

5. Determine the average experimental value of the moment of inertia of the pendulum<J ex> according to formula (6) according to the average value of the oscillation period<T>.

6. Determine the theoretical value of the moment of inertia of the pendulum J theor according to formula (7).

7. Draw a conclusion by comparing the theoretical and experimental values of the moment of inertia of the pendulum. Estimate measurement error D J= – J theor.

8. Write the result in the form J exp =< J > ±D J.

CONTROL QUESTIONS

1. Give the definition of a physical pendulum, explain why natural oscillations of a pendulum are possible.

2. Write down the basic law of the dynamics of rotational motion for a physical pendulum.

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

Let some body under the influence of a force applied at a point A, comes into rotation around the OO axis" (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 R, dropped from the point ABOUT(lying on the axis) to the direction of the force is called shoulder of strength. The product of force by the arm determines the modulus moment of force relative to the point ABOUT:

(5.1)

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

(5.2)

Unit of moment of force - newton meter(N . m). The direction of the force moment vector can be found using right propeller rules.

The measure of inertia of bodies during translational motion is mass. The inertia of bodies during rotational motion depends not only on 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 relative to the axis of rotation.

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:

Moment of inertia of the body relative to the axis of rotation - the sum of the moments of inertia of the material points that make up this body:

(5.4)

In the general case, if the body is solid and represents a collection of points with small 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 about and how the mass of the body is distributed throughout the volume.

The moment of inertia of bodies that have a regular geometric shape and a uniform distribution of mass over the volume is most easily determined.

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

Moment of inertia of a homogeneous cylinder relative to an 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 an axis perpendicular to the plane of its base and passing through its center,

Moment of inertia of the ball 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 the 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

Disk area. With constant ring thickness,

from where or .

Then the moment of inertia of the disk,

For clarity, Figure 5.3 shows homogeneous solid bodies 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 of 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 relative to an arbitrary axis, you should use Steiner's theorem : the moment of inertia of the body relative to an arbitrary axis of rotation is equal to the sum of the moment of inertia J 0 relative to the 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. Unit of moment of inertia - kilogram meter squared (kg . m 2).

Thus, the moment of inertia of a homogeneous rod of length l relative to the axis passing through its end, according to Steiner’s theorem is equal to