LECTURE №4

BASIC LAWS OF KINETICS AND DYNAMICS

ROTARY MOVEMENT. MECHANICAL

PROPERTIES OF BIOTISKS. BIOMECHANICAL

PROCESSES IN THE LOCOMOTOR SYSTEM

HUMAN.

1. Basic laws of kinematics of rotational motion.

Rotational movement of the body around a fixed axis is the simplest type of movement. It is characterized by the fact that any points of the body describe circles, the centers of which are located on one straight line 0 ﺍ 0 ﺍﺍ , which is called the axis of rotation (Fig. 1).

In this case, the position of the body at any moment of time is determined by the angle of rotation φ of the radius vector R of any point A relative to its initial position. Its dependence on time:

(1)

is the equation of rotational motion. The speed of rotation of the body is characterized by the angular velocity ω. The angular velocity of all points of the rotating body is the same. It is a vector quantity. This vector is directed along the axis of rotation and is related to the direction of rotation by the rule of the right screw:

. (2)

With uniform motion of a point along a circle

, (3)

where Δφ=2π is the angle corresponding to one complete rotation of the body, Δt=T is the time of one complete rotation, or the period of rotation. Unit of measurement of angular velocity [ω]=c -1.

With uniform motion, the acceleration of the body is characterized by angular acceleration ε (its vector is located similarly to the angular velocity vector and is directed according to it in accelerated and in the opposite direction - in slow motion):

. (4)

Unit of angular acceleration [ε]=c -2 .

Rotational motion can also be characterized by the linear speed and acceleration of its individual points. The length of the arc dS, described by any point A (Fig. 1) when rotated through an angle dφ, is determined by the formula: dS=Rdφ. (5)

Then the linear velocity of the point :

. (6)

Linear acceleration a:

. (7)

2. Basic laws of rotational motion dynamics.

The rotation of the body around the axis is caused by the force F applied to any point of the body, acting in a plane perpendicular to the axis of rotation and directed (or having a component in this direction) perpendicular to the radius vector of the point of application (Fig. 1).

Moment of force relative to the center of rotation is called a vector quantity numerically equal to the product of the force by the length of the perpendicular d, lowered from the center of rotation to the direction of the force, called the arm of the force. In Fig.1 d=R, therefore

. (8)

Moment the rotating force is a vector quantity. Vector attached to the center of the circle O and directed along the axis of rotation. vector direction is consistent with the direction of the force according to the rule of the right screw. The elementary work dA i , when turning through a small angle dφ, when the body passes a small path dS, is equal to:

A measure of the inertia of a body in translational motion is the mass. When a body rotates, the measure of its inertia is characterized by the moment of inertia of the body about the axis of rotation.

The moment of inertia I i of a material point relative to the axis of rotation is a value equal to the product of the mass of the point and the square of its distance from the axis (Fig. 2):

. (10)

The moment of inertia of the body about the axis is the sum of the moments of inertia of the material points that make up the body:

. (11)

Or in the limit (n→∞):
, (12)

G de integration is performed over the entire volume V. In a similar way, the moments of inertia of homogeneous bodies of regular geometric shape are calculated. The moment of inertia is expressed in kg m 2 .

The moment of inertia of a person relative to the vertical axis of rotation passing through the center of mass (the center of mass of a person is in the sagittal plane slightly ahead of the second cross vertebra), depending on the position of the person, has the following values: 1.2 kg m 2 at attention; 17 kg m 2 - in a horizontal position.

When a body rotates, its kinetic energy is the sum of the kinetic energies of individual points of the body:

Differentiating (14), we obtain an elementary change in the kinetic energy:

. (15)

Equating the elementary work (formula 9) of external forces to the elementary change in kinetic energy (formula 15), we obtain:
, where:
or considering that
we get:
. (16)

This equation is called the basic equation of rotational motion dynamics. This dependence is similar to Newton's II law for translational motion.

The angular momentum L i of a material point relative to the axis is a value equal to the product of the momentum of the point and its distance to the axis of rotation:

. (17)

Angular moment L of a body rotating around a fixed axis:

Angular momentum is a vector quantity oriented along the direction of the angular velocity vector.

Now let's return to the main equation (16):

,
.

We bring the constant value I under the sign of the differential and get:
, (19)

where Mdt is called the impulse of the moment of force. If external forces do not act on the body (M=0), then the change in the angular momentum (dL=0) is also equal to zero. This means that the angular momentum remains constant:
. (20)

This conclusion is called the law of conservation of angular momentum about the axis of rotation. It is used, for example, for rotational movements about a free axis in sports, such as acrobatics, etc. Thus, a figure skater on ice, by changing the position of the body during rotation and, accordingly, the moment of inertia relative to the axis of rotation, can regulate his rotation speed.

A rigid body rotating around some axes passing through the center of mass, if it is freed from external influences, maintains rotation indefinitely. (This conclusion is similar to Newton's first law for translational motion).

The occurrence of rotation of a rigid body is always caused by the action of external forces applied to individual points of the body. In this case, the appearance of deformations and the appearance of internal forces are inevitable, which in the case of a solid body ensure the practical preservation of its shape. When the action of external forces ceases, the rotation is preserved: internal forces can neither cause nor destroy the rotation of a rigid body.

The result of the action of an external force on a body with a fixed axis of rotation is an accelerated rotational motion of the body. (This conclusion is similar to Newton's second law for translational motion).

The basic law of the dynamics of rotational motion: in an inertial frame of reference, the angular acceleration acquired by a body rotating about a fixed axis is proportional to the total moment of all external forces acting on the body, and inversely proportional to the moment of inertia of the body about a given axis:

It is possible to give a simpler formulation the basic law of the dynamics of rotational motion(also called Newton's second law for rotational motion): the torque is equal to the product of the moment of inertia and the angular acceleration:

angular momentum(angular momentum, angular momentum) of a body is called the product of its moment of inertia times the angular velocity:

The angular momentum is a vector quantity. Its direction coincides with the direction of the angular velocity vector.

The change in angular momentum is defined as follows:

. (I.112)

A change in the angular momentum (with a constant moment of inertia of the body) can occur only as a result of a change in the angular velocity and is always due to the action of the moment of force.

According to the formula , as well as formulas (I.110) and (I.112), the change in angular momentum can be represented as:

. (I.113)

The product in formula (I.113) is called impulse moment of force or driving moment. It is equal to the change in angular momentum.

Formula (I.113) is valid provided that the moment of force does not change with time. If the moment of force depends on time, i.e. , then

. (I.114)

Formula (I.114) shows that: the change in angular momentum is equal to the time integral of the moment of force. In addition, if this formula is presented in the form: , then the definition will follow from it moment of force: the instantaneous moment of force is the first derivative of the moment of momentum with respect to time,

LABORATORY WORK №107

Verification of the basic equation of dynamics

rotary motion

Objective:Experimental verification of the basic law of the dynamics of rotational motion using the Oberbeck pendulum.

Instruments and accessories: Oberbeck pendulum with millisecond FRM - 15, vernier caliper.

Theoretical Introduction

When considering the rotation of a rigid body from a dynamic point of view, along with the concept of forces, the concept of moments of forces is introduced, and along with the concept of mass, the concept of the moment of inertia.

Let a material point with mass t under the action of an external force, it moves curvilinearly relative to a fixed point O. A moment of force acts on a material point and the point has a moment of momentum. The position of a moving material point is determined by the radius vector drawn to it from the point O (Fig. 1). The moment of force relative to a fixed point O is called a vector quantity equal to the vector product of the radius vector, the force vector

The vector is directed perpendicular to the plane of vectors and and its direction corresponds to the right screw rule. The modulus of the moment of forces is equal to

where a - angle between vectors and , h=rsin a - the shoulder of the force, equal to the shortest distance from the point O to the line of action (along which the force acts) of the force.

The angular momentum relative to the point O is called a vector quantity equal to the vector product of the radius of the vector by the momentum vector, that is

The vector is directed perpendicular to the plane of the vectors and (Fig. 2). The modulus of angular momentum is equal to

where b - the angle between the direction of the vectors and .

The basic law of the dynamics of rotational motion

Let the mechanical system consisting of N material points under the action of external forces, the resultant of which, makes a curvilinear motion relative to a fixed point O, that is

where is the radius vector drawn from point O to i th material point, is the vector of the force acting on i-th material point.

You can also find the angular momentum of the system

where is the angular momentum i-th material point.

The angular momentum depends on time t because speed is a function of time. Taking the derivative of the momentum of the system with respect to time t, we get

Formula (7) is a mathematical expression of the basic law of the dynamics of the rotational motion of the system, according to which the rate of change of the angular momentum of the system over time is equal to the resulting moment of external forces acting on the system.

Law (7) is also valid for a rigid body, since a rigid body can be considered as a collection of material points.

Let in a particular case, a rigid body rotates about a fixed axis passing through the center of mass, under the action of an external force. The rigid body is divided into material points. For a material point with mass m i the equation of motion will be written

Angular moment for i- th material point is equal to

Since during rotationb = 90 0 , then the linear velocity is related to the angular velocity by the formula Then (9) can be written as

The value is the moment of inertia of the material point about the Z axis. Then (10) takes the form

Taking into account (11), the basic law of the dynamics of the rotational motion of a rigid body relative to a fixed axis is written

where is the moment of inertia of the rigid body about the Z axis.

At

where is the angular acceleration. According to the main equation dynamics of rotational motion (12), the resulting moment of the external force acting on the body is equal to the product of the moment of inertia J of the body and its angular acceleration.

From equation (12) it follows that at j = const angular acceleration of the body

directly proportional to the moment of external forces relative to the axis of rotation, i.e.

At M = const angular acceleration is inversely proportional to the moment of inertia of the body, i.e.

The purpose of this work is to check relations (13) and (14), and, consequently, the basic equation of the dynamics of rotational motion (12), the consequences of which they are.

Description of operating setup and measurement method

To check relations (13) and (14), an Oberbeck pendulum is used, which is an inertial wheel in the form of a cross. On four mutually perpendicular rods 1 there are four identical cylindrical loads 2, which can be moved along the rods and fixed at a certain distance from the axis. Loads are fixed symmetrically, i.e. so that their center of mass coincides with the axis of rotation. On the horizontal axis of the cross there is a two-stage disk 3, on which the thread is wound. One end of the thread is attached to the disk, and a load 4 is suspended from the second end of the thread, under the action of which the device is driven into rotation. A general view of the Oberbeck FRM-06 pendulum is shown in Fig.3. A braking electromagnet is used to hold the crosshead system together with the weights at rest. In order to read the height of the fall of goods, a millimeter scale 5 is applied on the column. The time of the fall of the load 4 is measured by the FRM-15 millisecond watch, to which photoelectric sensors No. 1 (6) and No. 2 (7) are connected. Photoelectric sensor No. 2 (7) generates an electrical impulse of the end of time measurements and turns on the brake electromagnet.

If you allow the load 4 to move, then this movement will occur with acceleration a.

where t- time of movement of cargo from a height h. In this case, the pulley with the rods and the loads located on them will rotate with an angular acceleratione .

where r- pulley radius.

The torque of the force applied to the cross and reporting the angular acceleration of the rotating part of the device, we find by the formula

where T- the tension force of the cord. According to Newton's second law for load 4 we have

where

where g- acceleration of gravity.

From formulas (12), (15), (16), (17) and (19) we have

The procedure for performing work and processing the measurement results

1. Measure the radius of the large and small pulleys with a caliper r 1 and r 2 .

2. Determine the mass of cargo 4 by weighing on technical scales with accuracy± 0.1 g

3. Check relation (13). For this:

- fix cylindrical movable weights on the rods at the closest distance from the axis of rotation so that the crosspiece is in a position of indifferent equilibrium;

- wind the thread around a large radius pulley r1 and measure the time of movement of the cargo t from high h millisecond watch, why

- connect the power cord of the meter to the power supply;

- press the “NETWORK” key and check if all indicators of the meter show zero and if all indicators of both photoelectric sensors are on;

- move the weight to the top position and check if the circuit is at rest;

- press the "START" key and measure the time of movement of the load with a millisecond watch;

- press the "RESET" key and check whether the meter readings have been reset to zero and the lock has been released by the electromagnet;

- move the load to the upper position, press the "START" key and check if the circuit has been blocked again;

- repeat the experiment 5 times. Height h it is not recommended to change during the entire operation;

- using formulas (15), (16), (20) calculate the values a 1 , e 1 , M 1 ;

- without changing the location of the moving loads and thereby leaving the moment of inertia of the system unchanged, repeat the experiment by winding the thread with the load on a small pulley with a radius r2;

- using formulas (15), (16), (20) calculate the values a 2 , e 2 , M 2 ;

- check the validity of the consequence of the basic law of the dynamics of rotational motion:

, at

- enter the data of the results of measurements and calculations in tables 1 and 2.

4. Check ratio (1 four ). For this:

- push the movable weights to the stop at the ends of the rods, but so that the crosspiece is again in a position of indifferent equilibrium;

- for small pulley r2 determine the time of movement of the cargo t/ according to 5 experiments;

- using formulas (15), (20), (21) determine the values a / , e / , J1;

- when checking the ratio when you can use the values ​​of previous experience by setting and ;

- using formula (21) determine the value J 2 ;

- calculate the values ​​of and .

- Record the results of measurements and calculations in Table 3.

Table 1

r1	m	h	t 1	< t 1 >	a 1	e 1	M 1
	kg				m/s 2	from -2	H × m

table 2

r2	t 2	< t 2 >	a 2	e 2	M 2	M 1 /M 2	e 1 / e 2
			m/s 2	from -2	H × m

Table 3

r 2	t /	< t / >	a /	e /	J 1	a //	J 2	e //	e / / e //	J 2 / J 1
			m/s 2	from -2	kg × m 2	m/s 2	kg × m 2	from -2

Questions for admission to work

1. What is the purpose of the work?

2. Formulate the basic law of rotational motion dynamics. Explain the physical meaning of the quantities included in this law, indicate the units of their measurement in "SI".

3. Describe the device of the working installation.

Questions to protect the work

1. Give the definitions of the moment of forces, the moment of momentum of a material point relative to a fixed point O.

2. Formulate the basic law of the dynamics of the rotational motion of a rigid body relative to a fixed point O and a fixed axis Z.

3. Define the moment of inertia of a material point and a rigid body.

4. Derive working formulas.

5. Derive the ratio for and for

6. Are there any criticisms of this work?

Moment of power

The rotating action of a force is determined by its momentum. The moment of force about a point is the cross product

Radius vector drawn from point to point of application of force (Fig. 2.12). The unit of measurement of the moment of force.

Figure 2.12

The magnitude of the moment of force

or you can write

where is the shoulder of the force (the shortest distance from the point to the line of action of the force).

The direction of the vector is determined by the rule of the cross product or by the rule of the “right screw” (we combine the vectors and parallel translation at the point O, the direction of the vector is determined so that from its end the turn from the vector to is visible counterclockwise - in Fig. 2.12 the vector is directed perpendicular to the plane drawing “from us” (similarly, according to the gimlet rule - translational movement corresponds to the direction of the vector, rotational corresponds to a turn from to)).

The moment of a force about a point is zero if the line of action of the force passes through that point.

The projection of a vector on any axis, for example, the z-axis, is called the moment of force about this axis. To determine the moment of force about the axis, first project the force onto a plane perpendicular to the axis (Fig. 2.13), and then find the moment of this projection relative to the point of intersection of the axis with the plane perpendicular to it. If the line of action of the force is parallel to the axis, or crosses it, then the moment of force about this axis is equal to zero.

Figure 2.13

angular momentum

Moment of momentum material point a mass moving at a speed relative to any reference point is called a vector product

The radius vector of a material point (Fig. 2.14) is its momentum.

Figure 2.14

The value of the angular momentum of the material point

where is the shortest distance from the vector line to the point .

The direction of the angular momentum is determined similarly to the direction of the moment of force.

If the expression for L 0 is multiplied and divided by l, we get:

Where - the moment of inertia of a material point - an analogue of mass in rotational motion.

Angular velocity.

Moment of inertia of a rigid body

It can be seen that the resulting formulas are very similar to the expressions for the momentum and for Newton's second law, respectively, only instead of linear velocity and acceleration, angular velocity and acceleration are used, and instead of mass, the quantity I=mR 2, called moment of inertia of a material point .

If the body cannot be considered a material point, but can be considered absolutely rigid, then its moment of inertia can be considered the sum of the moments of inertia of its infinitely small parts, since the angular speeds of rotation of these parts are the same (Fig. 2.16). The sum of infinitesimals is the integral:

For any body, there are axes passing through its center of inertia, which have the following property: when the body rotates around such axes in the absence of external influences, the axes of rotation do not change their position. Such axes are called free axes of the body . It can be proved that for a body of any shape and with any density distribution, there are three mutually perpendicular free axes, called main axes of inertia body. The moments of inertia of a body about the principal axes are called main (intrinsic) moments of inertia body.

The main moments of inertia of some bodies are given in the table:

Huygens-Steiner theorem.

This expression is called Huygens-Steiner theorems : moment of inertia of the body about an arbitrary axis is equal to the sum the moment of inertia of the body 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 by the square of the distance between the axes.

The basic equation of the dynamics of rotational motion

The basic law of the dynamics of rotational motion can be obtained from Newton's second law for the translational motion of a rigid body

Where F is the force applied to the body by the mass m; a is the linear acceleration of the body.

If to a rigid body of mass m at point A (Fig. 2.15) apply force F, then as a result of a rigid connection between all material points of the body, all of them will receive an angular acceleration ε and the corresponding linear accelerations, as if a force F 1 …F n acts on each point. For each material point, you can write:

Where therefore

Where m i- weight i- th point; ε is the angular acceleration; r i is its distance to the axis of rotation.

Multiplying the left and right sides of the equation by r i, we get

Where - the moment of force - is the product of the force on her shoulder.

Rice. 2.15. A rigid body rotating under the action of a force F about the “ОО” axis

- moment of inertia i th material point (analogous to mass in rotational motion).

The expression can be written like this:

Let's sum the left and right parts over all points of the body:

The equation is the basic law of the dynamics of the rotational motion of a rigid body. Value - the geometric sum of all the moments of forces, that is, the moment of force F, giving acceleration ε to all points of the body. is the algebraic sum of the moments of inertia of all points of the body. The law is formulated as follows: "The moment of force acting on a rotating body is equal to the product of the moment of inertia of the body and the angular acceleration."

On the other hand

In turn - a change in the angular momentum of the body.

Then the basic law of the dynamics of rotational motion can be rewritten as:

Or - the impulse of the moment of force, acting on a rotating body, is equal to the change in its angular momentum.

Law of conservation of angular momentum

Similar to ZSI.

According to the basic equation of the dynamics of rotational motion, the moment of force about the Z axis: . Hence, in a closed system and, therefore, the total angular momentum about the Z axis of all bodies included in a closed system is a constant value. It expresses law of conservation of angular momentum . This law is valid only in inertial frames of reference.

Let us draw an analogy between the characteristics of translational motion and rotational motion.

Bases and foundations are calculated according to 2 limit states

	By bearing capacity:	N- the specified design load on the base in the most unfavorable combination; - bearing capacity (ultimate load) of the foundation for a given direction of load N; - coefficient of working conditions of the foundation (<1); - коэффициент надежности (>1).
	According to limit deformations:	- estimated absolute settlement of the foundation; - calculated relative difference of foundation settlements; , - limit values, respectively, of the absolute and relative difference of foundation settlements (SNiP 2.02.01-83 *)

Rotational dynamics

Foreword

I draw students' attention to the fact that THIS material was not considered ABSOLUTELY at school (except for the concept of moment of force).

1. The law of dynamics of rotational motion

a. Law of rotational motion dynamics

b. Moment of power

c. Moment of a pair of forces

d. Moment of inertia

2. Moments of inertia of some bodies:

a. Ring (thin-walled cylinder)

b. Thick wall cylinder

c. solid cylinder

e. thin rod

3. Steiner's theorem

4. Angular moment of the body. Change in the angular momentum of the body. momentum impulse. Law of conservation of angular momentum

5. Rotary operation

6. Kinetic energy of rotation

7. Comparison of quantities and laws for translational and rotational motion

1a. Consider a rigid body that can rotate around a fixed axis OO (Fig. 3.1). Let's break this solid body into separate elementary masses Δ m i . The resultant of all forces applied to Δ m i , denoted by . It suffices to consider the case when the force lies in a plane perpendicular to the axis of rotation: the force components parallel to the axis cannot affect the rotation of the body, since the axis is fixed. Then the equation of Newton's second law for the tangential components of force and acceleration will be written as:

The normal component of the force provides centripetal acceleration and does not affect the angular acceleration. From (1.27): , where is the radius of rotation i- that point. Then

Let's multiply both sides of (3.2) by:

notice, that

where α is the angle between the force vector and the radius vector of the point (Fig. 3.1), is the perpendicular dropped to the line of action of the force from the center of rotation (shoulder of the force). Let's introduce the concept of moment of force.

1b. Moment of force relative to the axis is called a vector directed along the axis of rotation and associated with the direction of the force by the gimlet rule, the module of which is equal to the product of the force and its arm: . Shoulder of Strength l relative to the axis of rotation is the shortest distance from the line of action of the force to the axis of rotation. Dimension of the moment of force:

In vector form, the moment of force about a point:

The vector of the moment of force is perpendicular to both the force and the radius vector of the point of its application:

If the force vector is perpendicular to the axis, then the vector of the moment of force is directed along the axis according to the rule of the right screw, and the magnitude of the moment of force relative to this axis (projection onto the axis) is determined by formula (3.4):

The moment of force depends both on the magnitude of the force and on the arm of the force. If the force is parallel to the axis, then .

1c. Power couple - these are two equal in magnitude and opposite in direction forces, the lines of action of which do not coincide (Fig. 3.2). The arm of a pair of forces is the distance between the lines of action of the forces. Let's find the total moment of the pair of forces and () in the projection onto the axis passing through the point O:

That is, the moment of a pair of forces is equal to the product of the magnitude of the force and the plccho of the pair:

Let us return to (3.3). Taking into account (3.4) and (3.6):

1d. Definition: a scalar value equal to the product of the mass of a material point and the square of its distance to the axis is called moment of inertia of a material point relative to the OO axis:

Dimension of moment of inertia

The vectors and coincide in direction with the axis of rotation, are related to the direction of rotation according to the gimlet rule, so equality (3.9) can be rewritten in vector form:

Let us sum (3.10) over all elementary masses into which the body is divided:

Here it is taken into account that the angular acceleration of all points of a rigid body is the same, and it can be taken out of the sum sign. On the left side of the equation is the sum of the moments of all forces (both external and internal) applied to each point of the body. But according to Newton's third law, the forces with which the points of the body interact with each other (internal forces) are equal in magnitude and opposite in direction and lie on the same straight line, so their moments cancel each other out. Thus, in the left part of (3.11) the total moment of only external forces remains: .

The sum of the products of elementary masses and the square of their distances from the axis of rotation is called moment of inertia of a rigid body about this axis:

In this way, ; - this is the basic law of the dynamics of the rotational motion of a rigid body (analogous to Newton's second law): angular acceleration of a body is directly proportional to the total moment of external forces and inversely proportional to the moment of inertia of the body :

Moment of inertia Isolid is a measure of the inert properties of a solid body during rotational motion and is similar to the mass of a body in Newton's second law. It essentially depends not only on the mass of the body, but also on its distribution relative to the axis of rotation (in the direction perpendicular to the axis).

In the case of a continuous distribution of mass, the sum in (3.12) reduces to an integral over the entire volume of the body:

2a. The moment of inertia of a thin ring about an axis passing through its center perpendicular to the plane of the ring.

since for any element of the ring its distance to the axis is the same and equal to the radius of the ring: .

2b. Thick-walled cylinder (disk) with inner radius and outer radius .

Let us calculate the moment of inertia of a homogeneous disk with density ρ , height h, inner radius and outer radius (Fig.3.3) relative to the axis passing through the center of mass perpendicular to the plane of the disk. Let us divide the disk into thin rings of thickness and height so that the inner radius of the ring is , and the outer one is . The volume of such a ring is , where is the area of ​​the base of the thin ring. Its mass:

We substitute into (3.14) and integrate over r():

Mass of the disk, then finally:

2c. Solid cylinder (disk).

In the special case of a solid disk or cylinder with a radius R let's substitute into (3.17) R 1 =0, R 2 =R and get:

Moment of inertia of a ball of radius R and mass relative to the axis passing through its center (Fig. 3.4), is (without proof):

2e. The moment of inertia of a thin rod with mass and length relative to the axis passing through its end perpendicular to the rod (Fig. 3.5).

We divide the rod into infinitely small segments of length . The mass of such an area. Substitute in (3.14) and integrate from 0 to :

If the axis passes through the center of the rod perpendicular to it, you can calculate the moment of inertia of half of the rod using (3.20) and then double:

3. If the rotation axis does not pass through the center of mass of the body (Fig.3.6), calculations using formula (3.14) can be quite complicated. In this case, the calculation of the moment of inertia is facilitated by using Steiner's theorems : the moment of inertia of the body about an arbitrary axis is equal to the sum of the moment of inertia I c body about an axis passing through the center of mass of the body parallel to this axis, and the product of body mass by the square of the distance between axles:

Let's see how Steiner's theorem works if we apply it to a rod:

It is easy to see that the identity is obtained, since in this case the distance between the axes is equal to half the length of the rod.

4. Angular moment of the body. Change in the angular momentum of the body. momentum impulse. Law of conservation of angular momentum.

From the law of dynamics of rotational motion and the definition of angular acceleration, it follows:

If , then . Let us introduce the angular momentum of the rigid body as

Relation (3.24) is the basic law of rigid body dynamics for rotational motion. It can be rewritten like this:

and then it will be an analogue of Newton's second law for translational motion in impulsive form (2.5)

Expression (3.24) can be integrated:

and formulate the law of change of angular momentum: the change in the moment of momentum of the body is equal to the momentum of the total moment of external forces . The quantity is called the impulse of the moment of force and is similar to the impulse of the force in the formulation of Newton's second law for translational motion (2.2); angular momentum is analogous to momentum.

Dimension of angular momentum

The angular momentum of a rigid body about its axis of rotation is a vector directed along the axis of rotation according to the gimlet rule.

The angular momentum of a material point relative to the point O (Fig. 3.6) is:

where is the radius vector of a material point, is its momentum. The angular momentum vector is directed according to the gimlet rule perpendicular to the plane in which the vectors and lie: in Fig. 3.7 - to us because of the figure. The magnitude of the angular momentum

We divide a rigid body rotating about an axis into elementary masses and sum the angular momentum of each mass over the entire body (the same can be written as an integral; this is not fundamental):

Since the angular velocity of all points is the same and is directed along the axis of rotation, it can be written in vector form:

Thus, the equivalence of definitions (3.23) and (3.26) is proved.

If the total moment of external forces is zero, then the angular momentum of the system does not change(see 3.25):

. This is the law of conservation of momentum . This is possible when:

a) the system is closed (or);

b) external forces have no tangential components (the force vector passes through the axis/center of rotation);

c) external forces are parallel to the fixed axis of rotation.

Examples of the use / operation of the law of conservation of angular momentum:

1. gyroscope;

2. Zhukovsky's bench;

3. ice skater.

5. Work with rotary motion.

Let the body turn through an angle under the action of a force and the angle between the displacement and the force is ; - the radius vector of the point of application of the force (Fig. 3.8), then the work of the force is equal to: