The basic law of the dynamics of a rotating body. Rotational movement of the body

1.8.

The moment of momentum of a body relative to an axis.

The angular momentum of a solid body relative to an 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

Expression of the basic law of the dynamics of rotational motion through a change in the angular momentum of the body.

Let us consider an arbitrary system of bodies. The angular momentum of the system is the quantity L, equal to the vector sum of the angular momentum of its individual parts Li, taken relative to the same point of the selected reference system.

Let's find the rate of change of the angular momentum of the system. Carrying out reasoning similar to the description of the rotational motion of a rigid body, we obtain that

the rate of change of the angular momentum of the system is equal to the vector sum of the moments of external forces M acting on parts of this system.

Moreover, the vectors L and M are specified relative to the same point O in the selected CO. Equation (21) represents the law of change in the angular momentum of the system.

The reason for the change in angular momentum is the resulting torque of external forces acting on the system. The change in angular momentum over a finite period of time can be found using the expression

Law of conservation of angular momentum. Examples.

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 law of conservation of angular momentum is very clear in experiments with a balanced gyroscope - a rapidly rotating body with three degrees of freedom (Fig. 6.9).

It is the law of conservation of angular momentum that is used by ice dancers to change the speed of rotation. Or another well-known example is the Zhukovsky bench (Fig. 6.11).

Work of force.

Work of force -a measure of the effect of force when transforming mechanical motion into another form of motion.

Examples of formulas for the work of forces.

Work of gravity; work of gravity on an inclined surface

Work of elastic force

Work of friction force

Conservative and non-conservative forces.

Conservative are called forces whose work does not depend on the shape of the trajectory, but is determined only by the position of its starting and ending points.

The conservative class includes, for example, gravitational forces, elastic forces, and forces of electrostatic interaction.

There are forces whose work depends on the shape of the path, that is, the work along a closed path is not equal to zero (for example, friction forces). Such forces are called non-conservative .
In this case, the work does not go towards increasing the potential energy (dA dEn), but goes towards heating the bodies, i.e., increasing the kinetic energy of the molecules of the body.

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Derivation of the basic law of the dynamics of rotational motion. To the derivation of the basic equation of the dynamics of rotational motion. Dynamics of rotational motion of a material point. In projection onto the tangential direction, the equation of motion will take the form: Ft = mt.

15. Derivation of the basic law of the dynamics of rotational motion.

Rice. 8.5. To the derivation of the basic equation of the dynamics of rotational motion.

Dynamics of rotational motion of a material point.Consider a particle of mass m rotating around a current O along a circle of radius R , under the action of the resultant force F (see Fig. 8.5). In the inertial reference frame, 2 is valid Ouch Newton's law. Let's write it in relation to an arbitrary moment in time:

F = m·a.

The normal component of the force is not capable of causing rotation of the body, so we will consider only the action of its tangential component. In projection onto the tangential direction, the equation of motion will take the form:

F t = m·a t .

Since a t = e·R, then

F t = m e R (8.6)

Multiplying the left and right sides of the equation scalarly by R, we get:

F t R= m e R 2 (8.7)
M = Ie. (8.8)

Equation (8.8) represents 2 Ouch Newton's law (equation of dynamics) for the rotational motion of a material point. It can be given a vector character, taking into account that the presence of a torque causes the appearance of a parallel angular acceleration vector directed along the axis of rotation (see Fig. 8.5):

M = I·e. (8.9)

The basic law of the dynamics of a material point during rotational motion can be formulated as follows:

the product of the moment of inertia and angular acceleration is equal to the resulting moment of forces acting on a material point.

Moment of inertia about the axis of rotation

Moment of inertia of a material point , (1.8) where is the mass of the point, is its distance from the axis of rotation.

1. Moment of inertia of a discrete rigid body, (1.9) where is the mass element of the rigid body; – the distance of this element from the axis of rotation; – number of body elements.

2. Moment of inertia in the case of continuous mass distribution (solid solid body). (1.10) If the body is homogeneous, i.e. its density is the same throughout the entire volume, then expression (1.11) is used, where is the volume of the body.

3. Steiner's theorem. The moment of inertia of a body of any axis of rotation is equal to the moment of its inertia relative to a parallel axis passing through the center of mass of the body, added to the product of the mass of the body and the square of the distance between them. (1.12)

1. , (1.13) where is the moment of force, is the moment of inertia of the body, is the angular velocity, is the angular momentum.

2. In the case of a constant moment of inertia of the body – , (1.14) where is the angular acceleration.

3. In the case of constant moment of force and moment of inertia, the change in the angular momentum of a rotating body is equal to the product of the average moment of force acting on the body during the action of this moment. (1.15)

If the axis of rotation does not pass through the center of mass of the body, then the moment of inertia of the body relative to this axis can be determined by Steiner’s theorem: the moment of inertia of the body relative to an arbitrary axis is equal to the sum of the moments of inertia of this body relative to the axis of rotation O 1 O 2 passing through the center of mass of the body C in parallel axis, and the product of the body mass by the square of the distance between these axes (see Fig. 1), i.e. .

The moment of inertia of the system of individual bodies is equal (for example, the moment of inertia of a physical pendulum is equal to , where the moment of inertia of the rod on which the disk with the moment of inertia is attached).

Table of analogies

Angular momentum (kinetic momentum, angular momentum, orbital momentum, angular momentum) characterizes the amount of rotational motion. A quantity that depends on how much mass is rotating, how it is distributed relative to the axis of rotation, and at what speed the rotation occurs. It should be noted that rotation here is understood in a broad sense, not only as regular rotation around an axis. For example, even when a body moves in a straight line past an arbitrary imaginary point that does not lie on the line of motion, it also has angular momentum. Perhaps the greatest role is played by angular momentum when describing the actual rotational motion; angular momentum relative to a point is a pseudovector, and angular momentum relative to an axis is a pseudoscalar.

The law of conservation of momentum (Law of Conservation of Momentum) states that the vector sum of the momentum of all bodies (or particles) of the system is a constant value if the vector sum of external forces acting on the system is zero.

1) More linear characteristics: path S, speed, tangential and normal acceleration.

2) When a body rotates around a fixed axis, the angular acceleration vector ε is directed along the axis of rotation towards the vector of the elementary increment of angular velocity. When the movement is accelerated, the vector ε is codirectional to the vector ω (Fig. 3), when it is slow, it is opposite to it.

4) The moment of inertia is a scalar quantity that characterizes the distribution of masses in the body. The moment of inertia is a measure of the inertia of a body during rotation (physical meaning).

Acceleration characterizes the rate of change in speed.

5) Moment of force (synonyms: torque, torque, torque, torque) - a vector physical quantity equal to the vector product of the radius vector (drawn from the axis of rotation to the point of application of the force - by definition) and the vector of this force. Characterizes the rotational action of a force on a solid body.

6) If the load is suspended and at rest, then the elastic force \tension\ of the thread is equal in modulus to the force of gravity.

Basic concepts.

Moment of power relative to the axis of rotation - this is the vector product of the radius vector and the force.

The moment of force is a vector , the direction of which is determined by the rule of the gimlet (right screw) depending on the direction of the force acting on the body. The moment of force is directed along the axis of rotation and does not have a specific point of application.

The numerical value of this vector is determined by the formula:

M=r×F× sina(1.15),

where a - the angle between the radius vector and the direction of the force.

If a=0 or p, moment of power M=0, i.e. a force passing through the axis of rotation or coinciding with it does not cause rotation.

The greatest modulus torque is created if the force acts at an angle a=p/2 (M > 0) or a=3p/2 (M< 0).

Using the concept of leverage d- this is a perpendicular lowered from the center of rotation to the line of action of the force), the formula for the moment of force takes the form:

Where (1.16)

Rule of moments of forces(condition of equilibrium of a body having a fixed axis of rotation):

In order for a body with a fixed axis of rotation to be in equilibrium, it is necessary that the algebraic sum of the moments of forces acting on this body be equal to zero.

S M i =0(1.17)

The SI unit for moment of force is [N×m]

During rotational motion, the inertia of a body depends not only on its mass, but also on its distribution in space relative to the axis of rotation.

Inertia during rotation is characterized by the moment of inertia of the body relative to the axis of rotation J.

Moment of inertia material point relative to the axis of rotation is a value equal to the product of the mass of the point by the square of its distance from the axis of rotation:

J i =m i × r i 2(1.18)

The moment of inertia of a body relative to an axis is the sum of the moments of inertia of the material points that make up the body:

J=S m i × r i 2(1.19)

The moment of inertia of a body depends on its mass and shape, as well as on the choice of the axis of rotation. To determine the moment of inertia of a body relative to a certain axis, the Steiner-Huygens theorem is used:

J=J 0 +m× d 2(1.20),

Where J 0– moment of inertia about a parallel axis passing through the center of mass of the body, d– distance between two parallel axes . The moment of inertia in SI is measured in [kg × m 2 ]

The moment of inertia during the rotational movement of the human body is determined experimentally and calculated approximately using the formulas for a cylinder, round rod or ball.

The moment of inertia of a person relative to the vertical axis of rotation, which passes through the center of mass (the center of mass of the human body is located in the sagittal plane slightly in front of the second sacral vertebra), depending on the position of the person, has the following values: when standing at attention - 1.2 kg × m 2; with the “arabesque” pose – 8 kg × m 2; in horizontal position – 17 kg × m 2.

Work in rotational motion occurs when a body rotates under the influence of external forces.

The elementary work of force in rotational motion is equal to the product of the moment of force and the elementary angle of rotation of the body:

dA i =M i × dj(1.21)

If several forces act on a body, then the elementary work of the resultant of all applied forces is determined by the formula:

dA=M×dj(1.22),

Where M– the total moment of all external forces acting on the body.

Kinetic energy of a rotating bodyW to depends on the moment of inertia of the body and the angular velocity of its rotation:

Angle of impulse (angular momentum) – a quantity numerically equal to the product of the body’s momentum and the radius of rotation.

L=p× r=m× V× r(1.24).

After appropriate transformations, you can write the formula for determining angular momentum in the form:

(1.25).

Angular momentum is a vector whose direction is determined by the right-hand screw rule. The SI unit of angular momentum is [kg×m 2 /s]

Basic laws of the dynamics of rotational motion.

The basic equation for the dynamics of rotational motion:

The angular acceleration of a body undergoing rotational motion is directly proportional to the total moment of all external forces and inversely proportional to the moment of inertia of the body.

(1.26).

This equation plays the same role in describing rotational motion as Newton's second law does for translational motion. From the equation it is clear that under the action of external forces, the greater the angular acceleration, the smaller the moment of inertia of the body.

Newton's second law for the dynamics of rotational motion can be written in another form:

(1.27),

those. the first derivative of the angular momentum of a body with respect to time is equal to the total moment of all external forces acting on a given body.

Law of conservation of angular momentum of a body:

If the total moment of all external forces acting on the body is equal to zero, i.e.

S M i =0, Then dL/dt=0 (1.28).

This implies either (1.29).

This statement constitutes the essence of the law of conservation of angular momentum of a body, which is formulated as follows:

The angular momentum of a body remains constant if the total moment of external forces acting on a rotating body is zero.

This law is valid not only for an absolutely rigid body. An example is a figure skater who performs a rotation around a vertical axis. By pressing his hands, the skater reduces the moment of inertia and increases the angular velocity. To slow down the rotation, he, on the contrary, spreads his arms wide; As a result, the moment of inertia increases and the angular speed of rotation decreases.

In conclusion, we present a comparative table of the main quantities and laws characterizing the dynamics of translational and rotational movements.

Table 1.4.

Forward movement	Rotational movement
Physical quantity	Formula	Physical quantity	Formula
Weight	m	Moment of inertia	J=m×r 2
Force	F	Moment of power	M=F×r, if
Body impulse (amount of movement)	p=m×V	Momentum of a body	L=m×V×r; L=J×w
Kinetic energy		Kinetic energy
Mechanical work	dA=FdS	Mechanical work	dA=Mdj
Basic equation of translational motion dynamics		Basic equation for the dynamics of rotational motion	,
Law of conservation of body momentum	or If	Law of conservation of angular momentum of a body	or SJ i w i =const, If

Centrifugation.

The separation of inhomogeneous systems consisting of particles of different densities can be carried out under the influence of gravity and the Archimedes force (buoyancy force). If there is an aqueous suspension of particles of different densities, then a net force acts on them

F r =F t – F A =r 1 ×V×g - r×V×g, i.e.

F r =(r 1 - r)× V ×g(1.30)

where V is the volume of the particle, r 1 And r– respectively, the density of the substance of the particle and water. If the densities differ slightly from each other, then the resulting force is small and separation (deposition) occurs quite slowly. Therefore, forced separation of particles is used due to rotation of the separated medium.

Centrifugation is the process of separation (separation) of heterogeneous systems, mixtures or suspensions consisting of particles of different masses, occurring under the influence of the centrifugal force of inertia.

The basis of the centrifuge is a rotor with nests for test tubes, located in a closed housing, which is driven by an electric motor. When the centrifuge rotor rotates at a sufficiently high speed, suspended particles of different masses, under the influence of the centrifugal force of inertia, are distributed in layers at different depths, and the heaviest are deposited at the bottom of the test tube.

It can be shown that the force under the influence of which separation occurs is determined by the formula:

(1.31)

Where w- angular speed of rotation of the centrifuge, r– distance from the axis of rotation. The greater the difference in the densities of the separated particles and liquid, the greater the effect of centrifugation, and also significantly depends on the angular velocity of rotation.

Ultracentrifuges operating at a rotor speed of about 10 5 –10 6 revolutions per minute are capable of separating particles less than 100 nm in size, suspended or dissolved in a liquid. They have found wide application in biomedical research.

Ultracentrifugation can be used to separate cells into organelles and macromolecules. First, larger parts (nuclei, cytoskeleton) settle (sediment). With a further increase in the centrifugation speed, smaller particles sequentially settle out - first mitochondria, lysosomes, then microsomes and, finally, ribosomes and large macromolecules. During centrifugation, different fractions settle at different rates, forming separate bands in the test tube that can be isolated and examined. Fractionated cell extracts (cell-free systems) are widely used to study intracellular processes, for example, to study protein biosynthesis and decipher the genetic code.

To sterilize handpieces in dentistry, an oil sterilizer with a centrifuge is used to remove excess oil.

Centrifugation can be used to sediment particles suspended in urine; separation of formed elements from blood plasma; separation of biopolymers, viruses and subcellular structures; control over the purity of the drug.

Tasks for self-control of knowledge.

Exercise 1 . Questions for self-control.

What is the difference between uniform circular motion and uniform linear motion? Under what condition will a body move uniformly in a circle?

Explain the reason why uniform motion in a circle occurs with acceleration.

Can curvilinear motion occur without acceleration?

Under what condition is the moment of force equal to zero? takes the greatest value?

Indicate the limits of applicability of the law of conservation of momentum and angular momentum.

Indicate the features of separation under the influence of gravity.

Why can the separation of proteins with different molecular weights be carried out using centrifugation, but the method of fractional distillation is unacceptable?

Task 2 . Tests for self-control.

Fill in the missing word:

A change in the sign of the angular velocity indicates a change in_ _ _ _ _ rotational motion.

A change in the sign of angular acceleration indicates a change in_ _ _ rotational motion

Angular velocity is equal to the _ _ _ _ _derivative of the angle of rotation of the radius vector with respect to time.

Angular acceleration is equal to the _ _ _ _ _ _derivative of the angle of rotation of the radius vector with respect to time.

The moment of force is equal to_ _ _ _ _ if the direction of the force acting on the body coincides with the axis of rotation.

Find the correct answer:

The moment of force depends only on the point of application of the force.

The moment of inertia of a body depends only on the mass of the body.

Uniform circular motion occurs without acceleration.

A. Correct. B. Incorrect.

All of the above quantities are scalar, with the exception of

A. moment of force;

B. mechanical work;

C. potential energy;

D. moment of inertia.

The vector quantities are

A. angular velocity;

B. angular acceleration;

C. moment of force;

D. angular momentum.

Answers: 1 – directions; 2 – character; 3 – first; 4 – second; 5 – zero; 6 – B; 7 – B; 8 – B; 9 – A; 10 – A, B, C, D.

Task 3. Get the relationship between units of measurement :

linear speed cm/min and m/s;

angular acceleration rad/min 2 and rad/s 2 ;

moment of force kN×cm and N×m;

body impulse g×cm/s and kg×m/s;

moment of inertia g×cm 2 and kg×m 2.

Task 4. Tasks of medical and biological content.

Task No. 1. Why is it that during the flight phase of a jump an athlete cannot use any movements to change the trajectory of the body’s center of gravity? Do the athlete’s muscles perform work when the position of body parts in space changes?

Answer: By moving in free flight along a parabola, an athlete can only change the location of the body and its individual parts relative to its center of gravity, which in this case is the center of rotation. The athlete performs work to change the kinetic energy of rotation of the body.

Task No. 2. What average power does a person develop when walking if the duration of the step is 0.5 s? Consider that work is spent on accelerating and decelerating the lower extremities. Angular movement of the legs is about Dj=30 o. The moment of inertia of the lower limb is 1.7 kg × m 2. The movement of the legs should be considered as uniformly alternating rotational.

Solution:

1) Let’s write down a brief condition of the problem: Dt= 0.5s; DJ=30 0 =p/ 6; I=1.7kg × m 2

2) Define the work in one step (right and left leg): A= 2×Iw 2 / 2=Iw 2 .

Using the average angular velocity formula w av =Dj/Dt, we get: w= 2w av = 2×Dj/Dt; N=A/Dt= 4×I×(Dj) 2 /(Dt) 3

3) Substitute the numerical values: N=4× 1,7× (3,14) 2 /(0,5 3 × 36)=14.9(W)

Answer: 14.9 W.

Task No. 3. What is the role of arm movement when walking?

Answer: The movement of the legs, moving in two parallel planes located at some distance from each other, creates a moment of force that tends to rotate the human body around a vertical axis. A person swings his arms “towards” the movement of his legs, thereby creating a moment of force of the opposite sign.

Task No. 4. One of the areas for improving drills used in dentistry is to increase the rotation speed of the bur. The rotation speed of the boron tip in foot drills is 1500 rpm, in stationary electric drills - 4000 rpm, in turbine drills - already reaches 300,000 rpm. Why are new modifications of drills with a large number of revolutions per unit of time being developed?

Answer: Dentin is several thousand times more susceptible to pain than skin: there are 1-2 pain points per 1 mm of skin, and up to 30,000 pain points per 1 mm of incisor dentin. Increasing the number of revolutions, according to physiologists, reduces pain when treating a carious cavity.

Z task 5 . Fill out the tables:

Table No. 1. Draw an analogy between the linear and angular characteristics of rotational motion and indicate the relationship between them.

Table No. 2.

Task 6. Fill out the indicative action card:

Chapter 2. Fundamentals of biomechanics.

Questions.

Levers and joints in the human musculoskeletal system. The concept of degrees of freedom.

Types of muscle contraction. Basic physical quantities describing muscle contractions.

Principles of motor regulation in humans.

Methods and instruments for measuring biomechanical characteristics.

2.1. Levers and joints in the human musculoskeletal system.

The anatomy and physiology of the human musculoskeletal system have the following features that must be taken into account in biomechanical calculations: body movements are determined not only by muscle forces, but also by external reaction forces, gravity, inertial forces, as well as elastic forces and friction; the structure of the locomotor system allows exclusively rotational movements. Using the analysis of kinematic chains, translational movements can be reduced to rotational movements in the joints; the movements are controlled by a very complex cybernetic mechanism, so that there is a constant change in acceleration.

The human musculoskeletal system consists of skeletal bones articulated with each other, to which muscles are attached at certain points. The bones of the skeleton act as levers that have a fulcrum at the joints and are driven by the traction force generated by muscle contraction. Distinguish three types of lever:

1) Lever to which the acting force F and resistance force R applied on opposite sides of the fulcrum. An example of such a lever is the skull viewed in the sagittal plane.

2) A lever that has an active force F and resistance force R applied on one side of the fulcrum, and the force F applied to the end of the lever, and the force R- closer to the fulcrum. This lever gives a gain in strength and a loss in distance, i.e. is lever of power. An example is the action of the arch of the foot when lifting onto the half-toes, the levers of the maxillofacial region (Fig. 2.1). The movements of the masticatory apparatus are very complex. When closing the mouth, the raising of the lower jaw from the position of maximum lowering to the position of complete closure of its teeth with the teeth of the upper jaw is carried out by the movement of the muscles that lift the lower jaw. These muscles act on the lower jaw as a lever of the second kind with a fulcrum in the joint (giving a gain in chewing strength).

3) A lever in which the acting force is applied closer to the fulcrum than the resistance force. This lever is speed lever, because gives a loss in strength, but a gain in movement. An example is the bones of the forearm.

Rice. 2.1. Levers of the maxillofacial region and arch of the foot.

Most of the bones of the skeleton are under the action of several muscles, developing forces in different directions. Their resultant is found by geometric addition according to the parallelogram rule.

The bones of the musculoskeletal system are connected to each other at joints or joints. The ends of the bones that form the joint are held together by the joint capsule that tightly encloses them, as well as ligaments attached to the bones. To reduce friction, the contacting surfaces of the bones are covered with smooth cartilage and there is a thin layer of sticky liquid between them.

The first stage of biomechanical analysis of motor processes is the determination of their kinematics. Based on such an analysis, abstract kinematic chains are constructed, the mobility or stability of which can be checked based on geometric considerations. There are closed and open kinematic chains formed by joints and rigid links located between them.

The state of a free material point in three-dimensional space is given by three independent coordinates - x, y, z. Independent variables that characterize the state of a mechanical system are called degrees of freedom. For more complex systems, the number of degrees of freedom may be higher. In general, the number of degrees of freedom determines not only the number of independent variables (which characterizes the state of a mechanical system), but also the number of independent movements of the system.

Number of degrees freedom is the main mechanical characteristic of the joint, i.e. defines number of axles, around which mutual rotation of the articulated bones is possible. It is caused mainly by the geometric shape of the surface of the bones in contact at the joint.

The maximum number of degrees of freedom in the joints is 3.

Examples of uniaxial (flat) joints in the human body are the humeroulnar, supracalcaneal and phalangeal joints. They only allow flexion and extension with one degree of freedom. Thus, the ulna, with the help of a semicircular notch, covers a cylindrical protrusion on the humerus, which serves as the axis of the joint. Movements in the joint are flexion and extension in a plane perpendicular to the axis of the joint.

The wrist joint, in which flexion and extension, as well as adduction and abduction occurs, can be classified as joints with two degrees of freedom.

Joints with three degrees of freedom (spatial articulation) include the hip and scapulohumeral joint. For example, at the scapulohumeral joint, the ball-shaped head of the humerus fits into the spherical cavity of the protrusion of the scapula. Movements in the joint are flexion and extension (in the sagittal plane), adduction and abduction (in the frontal plane) and rotation of the limb around the longitudinal axis.

Closed flat kinematic chains have a number of degrees of freedom f F, which is calculated by the number of links n in the following way:

The situation for kinematic chains in space is more complex. Here the relation holds

(2.2)

Where f i - number of degrees of freedom restrictions i- th link.

In any body, you can select axes whose direction during rotation will be maintained without any special devices. They have a name free rotation axes

LECTURE No. 4

BASIC LAWS OF KINETICS AND DYNAMICS

ROTATIONAL MOTION. MECHANICAL

PROPERTIES OF BIO-TISSUES. BIOMECHANICAL

PROCESSES IN THE MUSTOCULAR SYSTEM

PERSON.

1. Basic laws of kinematics of rotational motion.

Rotational movements of the body around a fixed axis are 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 the same 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 time is determined by the angle of rotation φ of the radius of the 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 a body is characterized by 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)

When a point moves uniformly around a circle

, (3)

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

In uniform motion, the acceleration of a body is characterized by angular acceleration ε (its vector is located similar to the angular velocity vector and is directed in accordance with it during accelerated motion and in the opposite direction during slow motion):

. (4)

The unit of measurement for angular acceleration is [ε]=c -2.

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

Then the linear speed of the point :

. (6)

Linear acceleration A:

. (7)

2. Basic laws of the dynamics of rotational motion.

Rotation of a body around an axis is caused by a 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).

A moment of power relative to the center of rotation is a vector quantity numerically equal to the product of 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 rotational force is a vector quantity. Vector applied to the center of the circle O and directed along the axis of rotation. Vector direction consistent with the direction of force according to the right-hand screw rule. Elementary work dA i , when turning through a small angle dφ, when the body passes a small path dS, is equal to:

The measure of the inertia of a body during translational motion is mass. When a body rotates, the measure of its inertia is characterized by the moment of inertia of the body relative to 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 by the square of its distance from the axis (Fig. 2):

. (10)

The moment of inertia of a body relative to an 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 carried out over the entire volume V. The moments of inertia of homogeneous bodies of regular geometric shape are calculated in a similar way. 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 located in the sagittal plane slightly in front of the second cruciate 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 consists of the kinetic energies of individual points of the body:

Differentiating (14), we obtain an elementary change in 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, given 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 point’s momentum and its distance to the axis of rotation:

. (17)

Momentum of impulse L of a body rotating around a fixed axis:

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

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

,
.

Let's bring the constant value I under the differential sign and get:
, (19)

where Mdt is called the moment impulse. If the body is not acted upon by external forces (M=0), then the change in angular momentum (dL=0) is also zero. This means that the angular momentum remains constant:
. (20)

This conclusion is called the law of conservation of angular momentum relative to the axis of rotation. It is used, for example, during rotational movements relative to a free axis in sports, for example in 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.