In the examples given, the axes pass through the center of inertia of the body. The moment of inertia relative to other axes of rotation is determined using 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 inertiaJcrelative to a parallel axis passing through the center of inertia of the body, and the value of the product of the mass of the body by the square of the distance between them. Wherembody weight, and - distance from the center of inertia of the body to the selected axis of rotation, those.
,Wherem- body weight, and - distance from center
inertia of the body to the selected axis of rotation.
Let us use one example to demonstrate the application of Steiner's theorem. Let us calculate the moment of inertia of a thin rod relative to an axis passing through its edge perpendicular to the rod. Direct calculation is reduced to the same integral (*), but taken within different limits: 
The distance to the axis passing through the center of mass is equal to A = ℓ/2. Using Steiner's theorem we obtain the same result.
.
§22.Basic law of the dynamics of rotational motion.
Statement of the law: The rate of change of angular momentum relative to the pole is equal to the main moment of force relative to the same pole, those.
.
In projections on the coordinate axes: 
.
If the body rotates relative to a fixed axis, then the basic law of the dynamics of rotational motion will take the form: . In this case, the angular momentum can be easily expressed through the angular velocity and the moment of inertia of the body relative to the axis in question: 
. Then the basic law of the dynamics of rotational motion will take the form: 
. If the body does not crumble or deform, then
, as a result of which 
. If to everything 
, That 
and, it is equal to: 
.
The elementary work performed by a moment of force during rotational motion relative to a fixed axis is calculated by the formula: 
(*). Full work 
. If 
, That 
.
Based on formula (*), we obtain an expression for the kinetic energy of the rotational motion of a rigid body relative to a fixed axis. Because 
, That. After integration, we obtain the final result for the kinetic energy of rotational motion relative to a fixed axis 
.
§23. Law of conservation of angular momentum.
As already indicated, the laws of conservation of energy and momentum are associated with the homogeneity of time and space, respectively. But three-dimensional space, unlike one-dimensional time, has another symmetry. The space itself isotropic, there are no dedicated directions. Associated with this symmetry conservation lawmoment of impulse. This connection is manifested in the fact that angular momentum is one of the main quantities describing rotational motion.
By definition, the angular momentum of an individual particle is equal to 
.
Vector direction L is determined by the gimlet (corkscrew) rule, and its value is equal to L = r p sin , Where
the angle between the directions of the radius vector of the particle and its momentum. Magnitude ℓ = r sin equal to the distance from the origin ABOUT to the straight line along which the particle's momentum is directed. This quantity is called impulse shoulder. Vector L depends on the choice of the origin of coordinates, therefore, when talking about it, they usually indicate: “angular momentum relative to the point ABOUT".
Let us consider the time derivative of the angular momentum:
.
The first term is equal to zero, because . In the second term, according to Newton’s second law, the derivative with respect to momentum can be replaced by the force acting on the body. The vector product of the radius vector and the force is called moment of force relative to the point ABOUT:
.
The direction of the moment of force is determined by the same gimlet rule. Its size M = r F sin , Where
angle between radius vector and force. Similarly to how it was done above, we also define shoulder strength
ℓ =
r
sin
- distance from point ABOUT to the line of action of the force. As a result, we obtain the equation of motion for the angular momentum of the particle: 
.
The form of the equation is similar to Newton's second law: instead of the momentum of a particle there is angular momentum, and instead of force there is moment of force. If 
,That 
,
those. the angular momentum is constant in the absence of external torques.
Statement of the law: The angular momentum of a closed system relative to the pole does not change over time.
In the particular case of rotation about a fixed axis, we have: 
, Where
initial moment of inertia and angular velocity of the body relative to the axis under consideration, and
the final moment of inertia and angular velocity of the body relative to the axis under consideration.
Law of conservation of total mechanical energy taking into account rotational motion: the total mechanical energy of a conservative system is constant: .
Example: Find the speed of the system when traveling a distance h.
Given: m, M, h. Find: V - ?
Steiner's theorem - formulation
According to Steiner's theorem, it is established that moment of inertia of a body when calculating a relatively arbitrary axis corresponds to the sum of the moment of inertia of the body relative to an axis that passes through the center of mass and is parallel to this axis, as well as plus the product of the square of the distance between the axes and the mass of the body, according to the following formula (1):
Lesson: Colliding bodies. Absolutely elastic and absolutely inelastic impacts
Introduction
To study the structure of matter, one way or another, various collisions are used. For example, in order to examine an object, it is irradiated with light, or a stream of electrons, and by scattering this light or a stream of electrons, a photograph, or an X-ray, or an image of this object in some physical device is obtained. Thus, the collision of particles is something that surrounds us in everyday life, in science, in technology, and in nature.
For example, a single collision of lead nuclei in the ALICE detector of the Large Hadron Collider produces tens of thousands of particles, from the movement and distribution of which one can learn about the deepest properties of matter. Considering collision processes using the conservation laws we are talking about allows us to obtain results regardless of what happens at the moment of collision. We don't know what happens when two lead nuclei collide, but we do know what the energy and momentum of the particles that fly apart after these collisions will be.
Today we will look at the interaction of bodies during a collision, in other words, the movement of non-interacting bodies that change their state only upon contact, which we call a collision, or impact.
When bodies collide, in the general case, the kinetic energy of the colliding bodies does not have to be equal to the kinetic energy of the flying bodies. Indeed, during a collision, bodies interact with each other, influencing each other and doing work. This work can lead to a change in the kinetic energy of each body. In addition, the work that the first body does on the second may not be equal to the work that the second body does on the first. This can cause mechanical energy to turn into heat, electromagnetic radiation, or even create new particles.
Collisions in which the kinetic energy of the colliding bodies is not conserved are called inelastic.
Among all possible inelastic collisions, there is one exceptional case when the colliding bodies stick together as a result of the collision and then move as one. This inelastic impact is called absolutely inelastic (Fig. 1).
A) 
b)
Rice. 1. Absolute inelastic collision
Let's consider an example of a completely inelastic impact. Let a bullet of mass fly in a horizontal direction with speed and collide with a stationary box of sand of mass , suspended on a thread. The bullet got stuck in the sand, and then the box with the bullet began to move. During the impact of the bullet and the box, the external forces acting on this system are the force of gravity, directed vertically downward, and the tension force of the thread, directed vertically upward, if the time of impact of the bullet was so short that the thread did not have time to deflect. Thus, we can assume that the momentum of the forces acting on the body during the impact was equal to zero, which means that the law of conservation of momentum is valid:
.
The condition that the bullet is stuck in the box is a sign of a completely inelastic impact. Let's check what happened to the kinetic energy as a result of this impact. Initial kinetic energy of the bullet:
final kinetic energy of bullet and box:
simple algebra shows us that during the impact the kinetic energy changed:
So, the initial kinetic energy of the bullet is less than the final one by some positive value. How did this happen? During the impact, resistance forces acted between the sand and the bullet. The difference in the kinetic energies of the bullet before and after the collision is exactly equal to the work of the resistance forces. In other words, the kinetic energy of the bullet went to heat the bullet and the sand.
If, as a result of the collision of two bodies, kinetic energy is conserved, such a collision is called absolutely elastic.
An example of perfectly elastic impacts is the collision of billiard balls. We will consider the simplest case of such a collision - a central collision.
A collision in which the velocity of one ball passes through the center of mass of the other ball is called a central collision. (Fig. 2.)
Rice. 2. Center ball strike
Let one ball be at rest, and the second fly at it with some speed, which, according to our definition, passes through the center of the second ball. If the collision is central and elastic, then the collision produces elastic forces acting along the line of collision. This leads to a change in the horizontal component of the momentum of the first ball, and to the appearance of a horizontal component of the momentum of the second ball. After the impact, the second ball will receive an impulse directed to the right, and the first ball can move both to the right and to the left - this will depend on the ratio between the masses of the balls. In the general case, consider a situation where the masses of the balls are different.
The law of conservation of momentum is satisfied for any collision of balls:
In the case of an absolutely elastic impact, the law of conservation of energy is also satisfied:
We obtain a system of two equations with two unknown quantities. Having solved it, we will get the answer.
The speed of the first ball after impact is
,
Note that this speed can be either positive or negative, depending on which of the balls has more mass. In addition, we can distinguish the case when the balls are identical. In this case, after hitting the first ball will stop. The speed of the second ball, as we noted earlier, turned out to be positive for any ratio of the masses of the balls:
Finally, let's consider the case of an off-center impact in a simplified form - when the masses of the balls are equal. Then, from the law of conservation of momentum we can write:
And from the fact that kinetic energy is conserved:
An off-central impact will be in which the speed of the oncoming ball will not pass through the center of the stationary ball (Fig. 3). From the law of conservation of momentum, it is clear that the velocities of the balls will form a parallelogram. And from the fact that kinetic energy is conserved, it is clear that it will not be a parallelogram, but a square.
Rice. 3. Off-center impact with equal masses
Thus, with an absolutely elastic off-center impact, when the masses of the balls are equal, they always fly apart at right angles to each other.
The model is a demonstration illustrating the law of conservation of momentum. Elastic and inelastic collisions of balls are considered.
When bodies interact, the impulse of one body can be partially or completely transferred to another body. If a system of bodies is not affected by external forces from other bodies, then such a system is called closed.
In a closed system, the vector sum of the impulses of all bodies included in the system remains constant for any interactions of the bodies of this system with each other.
This fundamental law of nature is called the law of conservation of momentum. It is a consequence of Newton's second and third laws .
Let us consider any two interacting bodies that are part of a closed system. We denote the forces of interaction between these bodies by and According to Newton’s third law If these bodies interact over time t, then the impulses of the interaction forces are equal in magnitude and directed in opposite directions:
Let us apply Newton's second law to these bodies:
This equality means that as a result of the interaction of two bodies, their total momentum has not changed. Now considering all possible pair interactions of bodies included in a closed system, we can conclude that the internal forces of a closed system cannot change its total momentum, that is, the vector sum of the momentum of all bodies included in this system.
b) Law of conservation of energy
Conservative forces – forces whose work does not depend on the trajectory, but is determined only by the initial and final coordinates of the point.
In a system in which only conservative forces act, the total energy of the system remains unchanged. Only the transformation of potential energy into kinetic energy and vice versa is possible.
The potential energy of a material point is a function only of its (point’s) coordinates, which means the forces can be defined as follows: . – potential energy of a material point. 
Multiply both sides by and get 
. Let's transform and get an expression proving law of energy conservation
.
c) Loss of mechanical energy
Bernoulli's theorem, together with Euler's theorem, stated in 110, can be used to derive the Borda (1733-1792)-Carnot theorem on the loss of mechanical energy of a fluid flow during its sudden expansion (Fig. 328). This theorem serves as an analogue of the Kar- theorem
The loss of mechanical energy in a forward shock can be characterized by the ratio of the total pressure behind the shock to the total pressure Poi in front of it. The formulas defining this relation have the form
This equation indicates that when a liquid medium moves, its internal energy changes both due to the external influx of heat and due to the dissipation of mechanical energy. The dissipation process, as expression (5-84) shows, is associated with viscosity p and does not take place for an ideal fluid (p = 0). Since this process is irreversible, the dissipated energy Ed can be considered as the amount of loss of mechanical energy.
Since mechanical energy losses are inevitable in any machine, the power expended by the engine to drive the pump (power consumption L) is always greater than the useful power N - These losses are estimated by the overall efficiency of the pump
When deriving equations (136), the viscosity of the liquid and the associated loss of mechanical energy during the movement of a liquid particle were not taken into account.
When fluid moves in a pipe, there is a loss of mechanical energy, therefore, there must be areas in which the influence of viscosity is significant. Due to the adhesion of the liquid to the walls of the pipe, the instantaneous and average velocities of the liquid on the walls are zero. Therefore, there cannot be intensive mixing of the liquid in the immediate vicinity of the pipe walls. This serves as the basis for the conclusion that immediately near the walls, a sharp change in speed should be determined by the viscosity property of the liquid and that a layer with laminar movement should exist near the walls. Experimental data well confirm this conclusion.
The work of viscous forces performed between two sections of the flow and per unit mass, weight or volume of a moving fluid is called mechanical energy losses, or hydraulic losses. If this work is related to a unit of weight, then the hydraulic losses are called pressure losses L.
The model of an inviscid fluid cannot explain the origin of mechanical energy losses when fluid moves through pipelines and the resistance effect in general. To describe these phenomena, a more complex viscous fluid model is used. The simplest and most commonly used model of a viscous fluid is the Newtonian fluid.
The work of pressure forces p is spent on overcoming resistance forces, which causes losses of mechanical energy. These losses are directly proportional to the length of the movement path, therefore they are called specific energy losses along the length. If the losses are expressed in units of pressure, they are called pressure losses along the length and are denoted pi. If energy losses are expressed in linear units EJg), they are called head losses along the length and are denoted /g.
Obtaining regular flows with low losses during braking in diffusers is a much more difficult task than obtaining accelerated flows with low losses in nozzles. In diffusers, ideal reversible movements are violated due to the same reasons and properties of the medium as in nozzles, however, when flows are slowed down, the influence of the above factors manifests itself to a stronger degree. In diffusers, due to the movement against increasing pressure, the conditions for flow separation from the walls are more favorable than in nozzles, in which
A) Friction−− one of the types of interaction between bodies. It occurs when two bodies come into contact. Friction, like all other types of interaction, obeys Newton’s third law: if a friction force acts on one of the bodies, then a force of the same magnitude, but directed in the opposite direction, also acts on the second body. Friction forces, like elastic forces, are of an electromagnetic nature. They arise due to the interaction between atoms and molecules of contacting bodies or the presence of irregularities and roughness.
Dry friction forces are the forces that arise when two solid bodies come into contact in the absence of a liquid or gaseous layer between them. They are always directed tangentially to the contacting surfaces.
Dry friction that occurs when bodies are at relative rest is called static friction. The static friction force is always equal in magnitude to the external force and is directed in the opposite direction.
The static friction force cannot exceed a certain maximum value (Ftr)max(Ftr)max. If the external force is greater than (Ftr)max(Ftr)max, relative slip occurs. The friction force in this case is called sliding friction force. It is always directed in the direction opposite to the direction of motion and, generally speaking, depends on the relative speed of the bodies. However, in many cases, the sliding friction force can be approximately considered independent of the relative velocity of the bodies and equal to the maximum static friction force. This model of dry friction force is used to solve many simple physical problems.
b) Sliding friction force- the force that arises between contacting bodies during their relative motion.
It has been experimentally established that the friction force depends on the force of pressure of bodies on each other (support reaction force), on the materials of the rubbing surfaces, and on the speed of relative movement. Since no body is absolutely smooth, the friction force Not depends on the contact area, and the true contact area is much smaller than the observed one; In addition, by increasing the area, we reduce the specific pressure of bodies on each other.
The quantity characterizing the rubbing surfaces is called friction coefficient, and is most often denoted by a Latin letter (\displaystyle k) or a Greek letter (\displaystyle \mu ). It depends on the nature and quality of processing of the rubbing surfaces. In addition, the coefficient of friction depends on speed. However, most often this dependence is weakly expressed, and if greater measurement accuracy is not required, then (\displaystyle k) can be considered constant. To a first approximation, the magnitude of the sliding friction force can be calculated using the formula:
(\displaystyle F=kN)
(\displaystyle k) - sliding friction coefficient,
(\displaystyle N) - normal ground reaction force.
V) Friction coefficient establishes proportionality between the friction force and the normal pressure force pressing the body to the support. The friction coefficient is a cumulative characteristic of a pair of materials that are in contact and does not depend on the area of contact between the bodies.
Types of friction
Static friction manifests itself when a body that was at rest is set in motion. The coefficient of static friction is designated μ 0 .
Sliding friction manifests itself in the presence of body motion, and it is significantly less than static friction.
The rolling friction force depends on the radius of the rolling object. In typical cases (when calculating the rolling friction of wheels of a train or car), when the radius of the wheel is known and constant, it is taken into account directly in the rolling friction coefficient μ quality.
Static friction coefficient
the body starts to move
(static friction coefficient μ 0
)
A) 5.6. Momentum of a material point and a rigid body
The vector product of the radius vector of a material point and its momentum: called the angular momentum of this point relative to point O (Fig. 5.4)
A vector is sometimes also called the angular momentum of a material point. It is directed along the axis of rotation perpendicular to the plane drawn through the vectors and and forms a right-hand triple of vectors with them (when observed from the vertex of the vector, it is clear that the rotation along the shortest distance from k occurs counterclockwise).
The vector sum of the angular momentum of all material points of the system is called the angular momentum (momentum of motion) of the system relative to point O:
Vectors and are mutually perpendicular and lie in a plane perpendicular to the axis of rotation of the body. That's why . Taking into account the relationship between linear and angular quantities
and is directed along the axis of rotation of the body in the same direction as the vector.
Thus.
Momentum of a body relative to the axis of rotation
| (5.9) |
Consequently, the angular momentum of a body relative to the axis of rotation is equal to the product of the moment of inertia of the body relative to the same axis and the angular velocity of rotation of the body around this axis.
« 5.5. Newton's second law for rotational motion and its analysis
5.7. Basic equation for the dynamics of rotational motion »
Section: Dynamics of rotational motion of a rigid body, Physical foundations of mechanics
B) Equation of dynamics of rotational motion of a rigid body
Moment of force relative to a fixed point O called a pseudovector quantity equal to the vector product of the radius vector , drawn from the point O at the point of application of force, on force
Modulus of moment of force:
- pseudovector, its direction coincides with the direction of the plane of motion of the right propeller as it rotates from to. Direction of the moment of force can also be determined by the rule of the left hand: place four fingers of the left hand in the direction of the first factor, the second factor enters the palm, the thumb bent at a right angle will indicate the direction of the moment of force. The vector of the moment of force is always perpendicular to the plane in which the vectors and lie.
Where is the shortest distance between the line of action of the force and the point ABOUT called the shoulder of force.
Moment of force about a fixed axis Z called a scalar quantity equal to the projection onto this axis of the vector of the moment of force, defined relative to an arbitrary point O of a given axis Z. If the axis Z is perpendicular to the plane in which the vectors and lie, i.e. coincides with the direction of the vector, then the moment of force 
is represented as a vector coinciding with the axis.
An axis whose position in space remains unchanged when rotating around a body in the absence of external forces is called the free axis of the body.
For a body of any shape and with an arbitrary distribution of mass, there are 3 mutually perpendicular axes passing through the center of inertia of the body, which can serve as free axes: they are called the main axes of inertia of the body.
Let's find an expression for rotational work bodies. Let it go to mass m a rigid body is acted upon by an external force. Then the work done by this force in time d t equal to
Let us carry out a cyclic rearrangement of factors in a mixed product of vectors using the rule
The work done when a body rotates is equal to the product of the moment of action of the force and the angle of rotation. When a body rotates, work goes towards increasing its kinetic energy:
Hence,
- equation of dynamics of rotational motion
If the axis of rotation coincides with the main axis of inertia passing through the center of mass, then the vector equality is satisfied
І - main moment of inertia (moment of inertia about the main axis)
Torsional vibrations
TORSIONAL VIBRATIONS- mechanical vibrations, during which elastic elements experience shear deformation. They take place in different machines with rotating shafts: in piston engines, turbines, generators, gearboxes, transmissions of transport vehicles.
K. to. arise as a result of uneven periodicity. moment of both driving forces and resistance forces. The unevenness of the torque causes uneven changes in the angular velocity of the shaft, i.e., either acceleration or deceleration of rotation. Usually the shaft consists of an alternation of sections with low mass and elastic compliance with more rigid sections, which means they are fixed to them. masses. Each section of the shaft will have its own degree of uneven rotation, since in the same period of time the masses pass different angles and, therefore, move at different speeds, which creates variable torsion of the shaft and dynamic. alternating stresses, ch. arr. tangents.
When the natural frequencies coincide. oscillations of the system with a periodic frequency. torque of driving forces and resistance forces, resonant oscillations arise. In this case, the dynamic level increases. alternating voltages; acoustic increases noise emitted by a running machine. Dynamic alternating stresses with incorrectly selected (underestimated) shaft dimensions, insufficient strength of its material and the occurrence of resonance can exceed the endurance limit, which will lead to fatigue of the shaft material and its destruction.
When calculating the torque of machine shafts, a calculation scheme with two disks connected by an elastic rod that acts in torsion is often used. In this case, own. frequency
Where I 1 - moment of inertia of the 1st disk, I 2 - moment of inertia of the 2nd disk, WITH- torsional rigidity of the rod, for a round rod with a diameter d and length l C where G is the shear modulus. More complex calculation schemes contain a larger number of disks connected by rods and forming a series. chains, and sometimes branched and ring chains. Calculation of own frequencies of shapes and forced coherent waves according to these calculation schemes are carried out on a computer.
Dr. An example of a torsion pendulum is a torsion pendulum, which is a disk mounted on one end of a torsion rod and rigidly sealed at the other end. Own the frequency of such a pendulum 
Where I- moment of inertia of the disk. Instruments using a torsion pendulum are used to determine the shear modulus of elasticity, coefficient. internal friction of solid materials during shear, coefficient. fluid viscosity.
K. to. arise in a variety of elastic systems; in some cases, joint oscillations with decomposition are possible. types of deformation of system elements, for example. flexural-torsional vibrations. So, at a certain flight conditions under the influence of aerodynamic. Forces sometimes cause self-excited flexural-torsional vibrations of an aircraft wing (the so-called flutter), which can cause destruction of the wing.
Lit.: Den-Hartog D. P., Mechanical vibrations, trans. from English, M., 1960; Maslov G.S., Calculations of shaft vibrations. Directory, 2nd ed., M., 1980; Vibrations in technology. Handbook, ed. V.V. Bolotina, vol. 1, M., 1978; Power transmissions of transport vehicles, L., 1982. A. V. Sinev
Amplitude of oscillations(lat. amplitude- magnitude) is the greatest deviation of the oscillating body from the equilibrium position.
For a pendulum, this is the maximum distance that the ball moves away from its equilibrium position (figure below). For oscillations with small amplitudes, such a distance can be taken as the length of the arc 01 or 02, and the lengths of these segments.
The amplitude of oscillations is measured in units of length - meters, centimeters, etc. On the oscillation graph, the amplitude is defined as the maximum (modulo) ordinate of the sinusoidal curve (see figure below).
Oscillation period.
Oscillation period- this is the shortest period of time through which a system oscillating returns again to the same state in which it was at the initial moment of time, chosen arbitrarily.
In other words, the oscillation period ( T) is the time it takes to complete one complete oscillation. For example, in the figure below, this is the time it takes for the pendulum bob to move from the rightmost point through the equilibrium point ABOUT to the far left point and back through the point ABOUT again to the far right.
Over a full period of oscillation, the body thus travels a path equal to four amplitudes. The period of oscillation is measured in units of time - seconds, minutes, etc. The period of oscillation can be determined from a well-known graph of oscillations (see figure below).
The concept of “oscillation period,” strictly speaking, is valid only when the values of the oscillating quantity are exactly repeated after a certain period of time, that is, for harmonic oscillations. However, this concept also applies to cases of approximately repeating quantities, for example, for damped oscillations.
Oscillation frequency.
Oscillation frequency- this is the number of oscillations performed per unit of time, for example, in 1 s.
The SI unit of frequency is named hertz(Hz) in honor of the German physicist G. Hertz (1857-1894). If the oscillation frequency ( v) is equal to 1Hz, this means that every second there is one oscillation. The frequency and period of oscillations are related by the relations:
In the theory of oscillations they also use the concept cyclical, or circular frequency ω . It is related to the normal frequency v and oscillation period T ratios:
.
Cyclic frequency is the number of oscillations performed per 2π seconds
a) Oscillations. Damped and undamped
Repetitive processes define our lives. Winter follows summer, day follows night, inhalation follows exhalation. Time flies, and we also measure it by repeating processes. Repetitive processes are fluctuations.
Oscillations changes in a physical quantity that repeat over time are called.
If these changes are repeated after a certain time interval, then oscillations are called "periodic". Shortest time interval T, through which the values of a physical quantity are repeated A(t), called period her hesitation A(t + T) =A(t). Number of oscillations per unit time v called vibration frequency. The oscillation frequency and period are related by the relation v = 1/T. Oscillations of a system that occur in the absence of external influence are called free. External influence is necessary to excite oscillations. The system is given a supply of energy from the outside, due to which oscillations occur. This external influence takes the system out of the equilibrium position, and subsequently it moves around the equilibrium position, leaving and returning to it, overshooting it by inertia. And this is repeated over and over again. Movement in this context means a change of state. IN mechanical systems this may be a movement in space or a change in pressure, in electrical- change in charge value or field strength. There is an infinite number of different movements and corresponding oscillatory processes.
Any system that undergoes oscillatory motion is called"oscillator" (translated from lat.oscillo- “oscillate”), accordingly, the word “oscillations” is often replaced by the term “oscillations”.
If the amplitude of the oscillations does not change over time, harmonic oscillations are calledundamped .
Differential equation describing harmonic undamped oscillations, has the form:
d 2 A(t) /dt 2+ ω 0 2 A(t) = 0.
Ȧ +ω 0 2 A = 0.
If the amplitude decreases over time, the oscillation is calledfading .
Common example of damped oscillations- oscillations in which the amplitude decreases according to the law
A 0 (t) =a 0 e -βt .
Attenuation coefficient β > 0.
In the SI system, time is measured in s, and frequency, respectively, in reciprocal seconds (s -1). This unit of measurement has a special name"hertz" , 1 Hz = 1 s -1 . German physicist Heinrich Rudolf Gehr
The moment of inertia is defined as, if the distribution of mass is uniform, then it is replaced by – elementary volume, – density of the substance. .
Steiner's theorem: the moment of inertia about an arbitrary axis is equal to the sum of the moment of inertia about an axis parallel to the given one and passing through the center of inertia of the body, and the product of the mass of the body by the square of the distance a between the axes: .
Moment of inertia:
1) a homogeneous thin rod of mass, length relative to the axis passing through the center of mass and perpendicular to the rod:
2) a homogeneous thin rod of mass, length relative to the axis passing through one of the ends of the rod:
3) a thin ring of mass, radius R relative to the axis of symmetry perpendicular to the plane of the ring:
4) a homogeneous disk (cylinder) of mass, radius R, height h relative to the axis of symmetry perpendicular to the base: .
21. Kinetic energy of a rotating rigid body.
When a body rotates with angular velocity, all its elementary masses move at a speed they have kinetic energy, - for a body rotating around a fixed axis. During rotation, both external and internal forces act on the material points of mass that form a rigid body. Over a period of time, it experiences displacement, while forces do work. The work done by all forces will be equal. When adding taking into account Newton's 3rd law, the sum of the work of internal forces = 0. Therefore, . In accordance with the kinetic energy theorem, the increase in kinetic energy = the work of all forces acting on the body.
Let us calculate the kinetic energy of a rigid body performing arbitrary plane motion. all points move in parallel planes. Rotation occurs around an axis, perpendicular to the planes, and moves together with a certain point O. Let us represent the speed of a material point of mass in the form . The body moves translationally, therefore, , is an expression of the kinetic energy of a body performing arbitrary plane motion. If we choose the center of mass as point O, then .
Gyroscopes.
Gyroscope(or top) is a massive solid body, symmetrical to a certain axis, rotating around it at a high angular velocity. Due to the symmetry of the gyroscope, . When trying to rotate a rotating gyroscope around a certain axis, gyroscopic effect– under the influence of forces that, it would seem, should cause a rotation of the axis of the gyroscope OO around the straight line O'O', the axis of the gyroscope rotates around the straight line O''O'' (the axis OO and the straight line O'O' are assumed to lie in the plane of the drawing, and the straight line O''O'' and the forces f1 and f2 are perpendicular to this plane). The explanation of the effect is based on the use of the moment equation. The angular momentum rotates around the OX axis due to the relationship. Together with the OX, the gyroscope also rotates. Due to the gyroscopic effect, the bearing on which the gyroscope rotates begins to act gyroscopic forces. Under the influence of gyroscopic forces, the gyroscope axis tends to take a position parallel to the angular velocity of the Earth's rotation.
The described behavior of the gyroscope is the basis gyroscopic compass. Advantages of the gyroscope: indicates the exact direction to the geographic north pole, its operation is not affected by metal objects.
Gyroscope precession– a special type of gyroscope motion occurs if the moment of external forces acting on the gyroscope, while remaining constant in magnitude, rotates simultaneously with the gyroscope axis, forming a right angle with it all the time. Let us consider the movement of a gyroscope with one fixed point on the axis under the influence of gravity, is the distance from the fixed point to the center of inertia of the gyroscope, and is the angle between the gyroscope and the vertical. the moment is directed perpendicular to the vertical plane passing through the axis of the gyroscope. Equation of motion: momentum increment = Consequently, changes its position in space in such a way that its end describes a circle in the horizontal plane. Over a period of time, the gyroscope rotates through an angle 
The gyroscope axis describes a cone around a vertical axis with an angular velocity - the angular velocity of precession.
Dear visitors to the site, I bring to your attention a work on mathematics on the topic , where materials of a theoretical and practical nature are presented, recommendations for solving problems using the specified theorem.
Steiner's theorem, or, as it is called in other sources, the Huygens-Steiner theorem, received its name in honor of its author, Jakob Steiner (Swiss mathematician), and also thanks to additions by Christian Huygens (Dutch physicist, astronomer and mathematician). Let us briefly consider their contributions to other sciences.
Steiner's theorem - about the authors of the theorem
Jacob Steiner
(1796—1863)
Jacob Steiner (1796-1863) is one of the scientists who is considered the founder of both the synthetic geometry of curved lines and surfaces of the second and higher orders.
As for Christiaan Huygens, his contribution to various sciences is also not small. He significantly improved (up to 92-fold magnification of the image), discovered the rings of Saturn and its satellite, Titan, and in 1673, in his rather informative work “Pendulum Clocks,” he presented work on the kinematics of accelerated .
Steiner's theorem - formulation
According to Steiner's theorem, it is established that moment of inertia of a body when calculating a relatively arbitrary axis corresponds to the sum of the moment of inertia of the body relative to an axis that passes through the center of mass and is parallel to this axis, as well as plus the product of the square of the distance between the axes and the mass of the body, according to the following formula (1):
J=J0+md 2 (1)
Where in the formula we take the following values: d
– distance between the axes OO 1 ║О’O 1 ’;
J 0
– moment of inertia of the body, calculated relative to the axis that passes through the center of mass and will be determined by relation (2):
J 0 =J d =mR 2 /2(2)
Since d = R, then the moment of inertia about the axis that passes through the point A indicated in the figure will be determined by formula (3):
J=mR 2 +mR 2 /2 = 3 / 2mR 2(3)
More detailed information about the theorem is presented in the abstract and presentation, which can be downloaded from the links before the article.
Steiner's theorem. Moment of inertia - content of work
Introduction
Part 1. Dynamics of rotation of a rigid body
1.1. Moments of inertia of the ball and disk
1.2. Huygens-Steiner theorem
1.3. Dynamics of rotational motion of a rigid body - theoretical foundations
Momentum
Moment of power
Moment of inertia about the axis of rotation
The main law of the dynamics of the rotational motion of a rigid body relative to a fixed axis
When describing rotational motion mathematically, it is important to know the moment of inertia of the system relative to the axis. In the general case, the procedure for finding this quantity involves the implementation of the integration process. The so-called Steiner theorem allows us to simplify calculations. Let's look at it in more detail in the article.
What is moment of inertia?
Before presenting the formulation of Steiner's theorem, it is necessary to understand the very concept of moment of inertia. Let's say there is a body of a certain mass and arbitrary shape. This body can be either a material point or any two-dimensional or three-dimensional object (rod, cylinder, ball, etc.). If the object in question is in circular motion around some axis with constant angular acceleration α, then the following equation can be written:
Here the value M represents the total torque that imparts acceleration α to the entire system. The proportionality coefficient between them is I, called the moment of inertia. This physical quantity is calculated using the following general formula:
Here r is the distance between an element with mass dm and the axis of rotation. This expression means that it is necessary to find the sum of the products of the squares of the distances r 2 by the elementary mass dm. That is, the moment of inertia is not a pure characteristic of the body, which distinguishes it from linear inertia. It depends on the distribution of mass throughout the object that is rotating, as well as on the distance to the axis and on the orientation of the body relative to it. For example, a rod will have a different I if it is rotated relative to the center of mass and relative to the end.
Moment of inertia and Steiner's theorem
The famous Swiss mathematician, Jakob Steiner, proved the theorem about parallel axes and moment of inertia, which now bears his name. This theorem postulates that the moment of inertia for absolutely any rigid body of arbitrary geometry relative to some axis of rotation is equal to the sum of the moment of inertia about the axis that intersects the center of mass of the body and is parallel to the first, and the product of the mass of the body by the square of the distance between these axes. Mathematically, this formulation is written as follows:
I Z and I O are the moments of inertia relative to the Z axis and the O axis parallel to it, which passes through the center of mass of the body, l is the distance between straight lines Z and O.
The theorem allows, knowing the value of I O, to calculate any other moment I Z relative to the axis that is parallel to O.
Proof of the theorem
The formula of Steiner's theorem can be easily obtained on your own. To do this, consider an arbitrary body on the xy plane. Let the origin of coordinates pass through the center of mass of this body. Let's calculate the moment of inertia I O which passes through the origin perpendicular to the xy plane. Since the distance to any point on the body is expressed by the formula r = √ (x 2 + y 2), then we obtain the integral:
I O = ∫ m (r 2 *dm) = ∫ m ((x 2 +y 2) *dm)
Now we move the axis parallel to the x axis by a distance l, for example, in the positive direction, then the calculation for the new axis of the moment of inertia will look like this:
I Z = ∫ m (((x+l) 2 +y 2)*dm)
Let's open the complete square in brackets and divide the integrands, we get:
I Z = ∫ m ((x 2 +l 2 +2*x*l+y 2)*dm) = ∫ m ((x 2 +y 2)*dm) + 2*l*∫ m (x*dm) + l 2 *∫ m dm
The first of these terms is the value of I O, the third term, after integration, gives the term l 2 *m, but the second term is equal to zero. The zeroing of this integral is due to the fact that it is taken from the product of x and mass elements dm, which on average gives zero, since the center of mass is located at the origin of coordinates. As a result, the formula of Steiner's theorem is obtained.
The considered case on a plane can be generalized to a volumetric body.
Checking Steiner's formula using the example of a rod
Let us give a simple example to demonstrate how to use the theorem considered.
It is known that for a rod of length L and mass m, the moment of inertia I O (the axis passes through the center of mass) is equal to m*L 2 /12, and the moment I Z (the axis passes through the end of the rod) is equal to m*L 2 /3. Let's check these data using Steiner's theorem. Since the distance between the two axes is L/2, then we get the moment I Z:
I Z = I O + m*(L/2) 2 = m*L 2 /12 + m*L 2 /4 = 4*m*L 2 /12 = m*L 2 /3
That is, we checked the Steiner formula and obtained the same value for I Z as in the source.
Similar calculations can be carried out for other bodies (cylinder, ball, disk), while obtaining the necessary moments of inertia, and without performing integration.
Moment of inertia and perpendicular axes
The theorem discussed concerns parallel axes. To complete the information, it is also useful to present the theorem for perpendicular axes. It is formulated as follows: for a flat object of arbitrary shape, the moment of inertia about the axis perpendicular to it will be equal to the sum of two moments of inertia about two mutually perpendicular axes lying in the plane of the object, while all three axes must pass through one point. Mathematically it is written like this:
Here z, x, y are three mutually perpendicular axes of rotation.
The significant difference between this theorem and Steiner's theorem is that it only applies to flat (two-dimensional) solid objects. Nevertheless, in practice it is used quite widely, mentally cutting the body into separate layers, and then adding up the resulting moments of inertia.