Fluctuations - a process of changing the states of the system around the equilibrium point, repeating to one degree or another in time.

Harmonic oscillation - oscillations in which a physical (or any other) quantity changes over time according to a sinusoidal or cosine law. The kinematic equation of harmonic oscillations has the form

where x is the displacement (deviation) of the oscillating point from the equilibrium position at time t; A - oscillation amplitude, this is the value that determines the maximum deviation of the oscillating point from the equilibrium position; ω - cyclic frequency, a value showing the number of complete oscillations occurring within 2π seconds - the full phase of oscillations, 0 - the initial phase of oscillations.

Amplitude - the maximum value of displacement or change variable from the average value during oscillatory or wave motion.

The amplitude and initial phase of the oscillations are determined by the initial conditions of motion, i.e. position and speed of a material point at the moment t=0.

Generalized harmonic oscillation in differential form

amplitude of sound waves and audio signals usually refers to the amplitude of the air pressure in the wave, but is sometimes described as the amplitude of displacement from equilibrium (air or the speaker's diaphragm)

Frequency - physical quantity, characteristic of the periodic process, equal to the number complete cycles of the process completed per unit of time. The frequency of oscillations in sound waves is determined by the oscillation frequency of the source. High frequency vibrations decay faster than low frequency vibrations.

The reciprocal of the oscillation frequency is called the period T.

The oscillation period is the duration of one complete cycle of oscillations.

In the coordinate system from the point 0 we draw the vector А̅, the projection of which on the OX axis is equal to Аcosϕ. If the vector А̅ rotates uniformly with an angular velocity ω˳ counterclockwise, then ϕ=ω˳t + ϕ˳, where ϕ˳ is the initial value of ϕ (oscillation phase), then the oscillation amplitude is the modulus of the uniformly rotating vector А̅, the oscillation phase (ϕ ) is the angle between the vector А̅ and the ОХ axis, the initial phase (ϕ˳) is the initial value of this angle, the angular frequency of oscillations (ω) is the angular velocity of rotation of the vector А̅..

2. Characteristics of wave processes: wave front, beam, wave speed, wavelength. Longitudinal and transverse waves; examples.

The surface separating at a given moment of time the medium already covered and not yet covered by oscillations is called the wave front. At all points of such a surface, after the departure of the wave front, oscillations are established that are identical in phase.

The beam is perpendicular to the wave front. Acoustic rays, like light rays, are rectilinear in a homogeneous medium. Reflected and refracted at the interface between two media.

Wavelength - the distance between two points closest to each other, oscillating in the same phases, usually the wavelength is indicated by the Greek letter. By analogy with the waves that arise in water from a thrown stone, the wavelength is the distance between two adjacent wave crests. One of the main characteristics of vibrations. Measured in units of distance (meters, centimeters, etc.)

• longitudinal waves (compression waves, P-waves) - the particles of the medium oscillate parallel(along) the direction of wave propagation (as, for example, in the case of sound propagation);
• transverse waves (shear waves, S-waves) - the particles of the medium oscillate perpendicular the direction of wave propagation (electromagnetic waves, waves on media separation surfaces);

The angular frequency of oscillations (ω) is the angular velocity of rotation of the vector А̅(V), the displacement x of the oscillating point is the projection of the vector А̅ onto the OX axis.

V=dx/dt=-Aω˳sin(ω˳t+ϕ˳)=-Vmsin(ω˳t+ϕ˳), where Vm=Аω˳ is the maximum speed (velocity amplitude)

3. Free and forced vibrations. Natural frequency of oscillations of the system. Resonance phenomenon. Examples .

Free (natural) vibrations called those that are committed without external influences due to the initially received heat energy. Typical models of such mechanical vibrations are a material point on a spring (spring pendulum) and a material point on an inextensible thread (mathematical pendulum).

In these examples, oscillations arise either due to the initial energy (deviation of the material point from the equilibrium position and movement without initial velocity), or due to kinetic energy (the body is given speed in the initial position of equilibrium), or due to both of these energies (velocity is communicated to the body deviated from the equilibrium position).

Consider a spring pendulum. In the equilibrium position, the elastic force F1

balances the force of gravity mg. If the spring is pulled a distance x, then a large elastic force will act on the material point. The change in the value of the elastic force (F), according to Hooke's law, is proportional to the change in the length of the spring or the displacement x of the point: F= - rx

Another example. The mathematical pendulum of deviation from the equilibrium position is such a small angle α that it is possible to consider the trajectory of the movement of a material point as a straight line coinciding with the axis OX. In this case, the approximate equality is fulfilled: α ≈sin α≈ tgα ≈x/L

Undamped vibrations. Consider a model in which the drag force is neglected.
The amplitude and initial phase of oscillations are determined by the initial conditions of motion, i.e. position and speed of the material point moment t=0.
Among various kinds oscillation harmonic oscillation is the simplest form.

Thus, a material point suspended on a spring or thread performs harmonic oscillations, if the resistance forces are not taken into account.

The oscillation period can be found from the formula: T=1/v=2P/ω0

damped vibrations. In a real case, resistance (friction) forces act on an oscillating body, the nature of the motion changes, and the oscillation becomes damped.

With regard to one-dimensional motion, we give the last formula the following form: Fс= - r * dx/dt

The rate of decrease in the oscillation amplitude is determined by the damping coefficient: the stronger the retarding effect of the medium, the greater ß and the faster the amplitude decreases. In practice, however, the degree of damping is often characterized by a logarithmic damping decrement, meaning by this value equal to the natural logarithm of the ratio of two successive amplitudes separated by a time interval equal to the oscillation period, therefore, the damping coefficient and the logarithmic damping decrement are related by a fairly simple relationship: λ=ßT

With strong damping, it can be seen from the formula that the oscillation period is an imaginary quantity. The motion in this case will no longer be periodic and is called aperiodic.

Forced vibrations. Forced oscillations are called oscillations that occur in the system with the participation of an external force that changes according to a periodic law.

Let us assume that, in addition to the elastic force and the friction force, an external driving force acts on the material point F=F0 cos ωt

The amplitude of the forced oscillation is directly proportional to the amplitude of the driving force and has a complex dependence on the attenuation coefficient of the medium and the circular frequencies of natural and forced oscillations. If ω0 and ß are given for the system, then the amplitude of forced oscillations has a maximum value at a certain specific frequency of the driving force, called resonant The phenomenon itself - the achievement of the maximum amplitude of forced oscillations for given ω0 and ß - is called resonance.

The resonant circular frequency can be found from the condition of the minimum denominator in: ωres=√ωₒ- 2ß

Mechanical resonance can be both beneficial and detrimental. The harmful effect is mainly related to the destruction it can cause. So, in technology, taking into account different vibrations, it is necessary to provide for the possible occurrence of resonant conditions, otherwise there may be destruction and catastrophes. Bodies usually have several natural vibration frequencies and, accordingly, several resonant frequencies.

Resonant phenomena under the action of external mechanical vibrations occur in the internal organs. This, apparently, is one of the reasons for the negative impact of infrasonic oscillations and vibrations on the human body.

6. Sound research methods in medicine: percussion, auscultation. Phonocardiography.

Sound can be a source of information about the state of the internal organs of a person, therefore, in medicine, such methods of studying the patient's condition as auscultation, percussion and phonocardiography are well spread.

Auscultation

For auscultation, a stethoscope or phonendoscope is used. The phonendoscope consists of a hollow capsule with a sound-transmitting membrane applied to the patient's body, rubber tubes go from it to the doctor's ear. The resonance of the air column occurs in the capsule, as a result of which the sound is amplified and auscultation improves. During auscultation of the lungs, breath sounds, various wheezing, characteristic of diseases, are heard. You can also listen to the heart, intestines and stomach.

Percussion

In this method, the sound of individual parts of the body is listened to when they are tapped. Imagine a closed cavity inside some body, filled with air. If called in this body sound vibrations, then at a certain frequency of sound, the air in the cavity will begin to resonate, highlighting and amplifying a tone corresponding to the size and position of the cavity. The human body can be represented as a combination of gas-filled (lungs), liquid (internal organs) and solid (bones) volumes. When hitting the surface of the body, oscillations occur, the frequencies of which have a wide range. From this range, some oscillations will die out rather quickly, while others, coinciding with the natural oscillations of the voids, will intensify and, due to resonance, will be audible.

Phonocardiography

It is used to diagnose the state of cardiac activity. The method consists in graphic recording of heart sounds and murmurs and their diagnostic interpretation. The phonocardiograph consists of a microphone, an amplifier, a system of frequency filters and a recording device.

9. Ultrasonic research methods (ultrasound) in medical diagnostics.

1) Methods of diagnostics and research

They include location methods using mainly impulsive radiation. This is echoencephalography - the definition of tumors and swelling of the brain. Ultrasound cardiography - measuring the size of the heart in dynamics; in ophthalmology - ultrasonic location for determining the size of the eye media.

2) Methods of influence

Ultrasonic physiotherapy - mechanical and thermal effects on the tissue.

11. Shock wave. Production and use of shock waves in medicine.
shock wave – discontinuity surface, which moves relative to the gas and at the intersection of which the pressure, density, temperature and velocity experience a jump.
With large disturbances (explosion, supersonic motion of bodies, powerful electric discharge, etc.), the speed of oscillating particles of the medium can become comparable to the speed of sound , a shock wave occurs.

The shock wave can have significant energy, so, in a nuclear explosion, the formation of a shock wave in environment about 50% of the energy of the explosion is expended. Therefore, the shock wave, reaching biological and technical objects, is capable of causing death, injury and destruction.

IN medical technology shock waves are used, which are an extremely short, powerful pressure pulse with high pressure amplitudes and a small stretch component. They are generated outside the patient's body and transmitted deep into the body, producing a therapeutic effect, provided by the specialization of the equipment model: crushing of urinary stones, treatment of pain zones and consequences of injuries of the musculoskeletal system, stimulation of the recovery of the heart muscle after myocardial infarction, smoothing of cellulite formations, etc.

Harmonic Wave Equation

The harmonic oscillation equation establishes the dependence of the body coordinate on time

The cosine graph has a maximum value at the initial moment, and the sine graph has a zero value at the initial moment. If we begin to investigate the oscillation from the equilibrium position, then the oscillation will repeat the sinusoid. If we begin to consider the oscillation from the position of the maximum deviation, then the oscillation will describe the cosine. Or such an oscillation can be described by the sine formula with the initial phase.

Change in speed and acceleration during harmonic oscillation

Not only the coordinate of the body changes with time according to the law of sine or cosine. But such quantities as force, speed and acceleration also change in a similar way. Force and acceleration are maximum when the oscillating body is in extreme positions, where the displacement is maximum, and are equal to zero when the body passes through the equilibrium position. The speed, on the contrary, in the extreme positions is equal to zero, and when the body passes the equilibrium position, it reaches its maximum value.

If the oscillation is described according to the law of cosine

If the oscillation is described according to the sine law

Maximum speed and acceleration values

After analyzing the equations of dependence v(t) and a(t), one can guess that the maximum values ​​of speed and acceleration are taken when the trigonometric factor is equal to 1 or -1. Determined by the formula

An oscillatory movement is any periodically repeating movement. Therefore, the dependences of the coordinate and velocity of the body on time during oscillations are described by periodic functions of time. IN school course physicists consider such oscillations in which the dependences and velocities of the body are trigonometric functions , or a combination of them, where is some number. Such oscillations are called harmonic (functions And often called harmonic functions). To solve problems for vibrations included in the program of a unified state exam in physics, you need to know the definitions of the main characteristics of oscillatory motion: amplitude, period, frequency, circular (or cyclic) frequency and phase of oscillations. Let us give these definitions and connect the enumerated quantities with the parameters of the dependence of the body coordinate on time , which in the case of harmonic oscillations can always be represented as

where , and are some numbers.

The amplitude of oscillation is the maximum deviation of an oscillating body from the equilibrium position. Since the maximum and minimum value of the cosine in (11.1) is equal to ±1, then the amplitude of oscillations of the body that oscillates (11.1) is equal to . The oscillation period is the minimum time after which the movement of the body is repeated. For dependence (11.1), the period can be set from the following considerations. Cosine is a periodic function with period . Therefore, the movement is completely repeated through such a value that . From here we get

Circular (or cyclic) oscillation frequency is the number of oscillations per unit of time. From formula (11.3) we conclude that the circular frequency is the value from formula (11.1).

The oscillation phase is the argument of the trigonometric function that describes the dependence of the coordinate on time. From formula (11.1) we see that the phase of oscillations of the body, the motion of which is described by dependence (11.1), is equal to . The value of the oscillation phase at time = 0 is called the initial phase. For dependence (11.1) the initial phase of oscillations is equal to the value . Obviously, the initial phase of the oscillations depends on the choice of the time reference point (moment = 0), which is always conditional. By changing the origin of the time reference, the initial phase of the oscillations can always be "made" equal to zero, and the sine in the formula (11.1) is "turned" into a cosine or vice versa.

The program of the unified state exam also includes knowledge of the formulas for the oscillation frequency of the spring and mathematical pendulums. It is customary to call a spring pendulum a body that can oscillate on a smooth horizontal surface under the action of a spring, the second end of which is fixed (left figure). A mathematical pendulum is a massive body, the dimensions of which can be neglected, oscillating on a long, weightless and inextensible thread (right figure). The name of this system - "mathematical pendulum" is due to the fact that it is an abstract mathematical real model ( physical) of the pendulum. It is necessary to remember the formulas for the period (or frequency) of oscillations of the spring and mathematical pendulums. For spring pendulum

where is the length of the thread, is the free fall acceleration. Consider the application of these definitions and laws on the example of problem solving.

To find the cyclic frequency of the load in task 11.1.1 let us first find the oscillation period, and then use the formula (11.2). Since 10 m 28 s is 628 s, and during this time the load makes 100 oscillations, the period of oscillation of the load is 6.28 s. Therefore, the cyclic oscillation frequency is 1 s -1 (the answer 2 ). IN task 11.1.2 the load made 60 oscillations in 600 s, so the oscillation frequency is 0.1 s -1 (the answer 1 ).

To understand which way the cargo will go in 2.5 periods ( task 11.1.3), follow its movement. After a period, the load will return back to the point of maximum deflection, making a complete oscillation. Therefore, during this time, the load will cover a distance equal to four amplitudes: to the equilibrium position - one amplitude, from the equilibrium position to the point of maximum deviation in the other direction - the second, back to the equilibrium position - the third, from the equilibrium position to the starting point - the fourth. During the second period, the load will again pass four amplitudes, and for the remaining half of the period - two amplitudes. Therefore, the distance traveled is equal to ten amplitudes (the answer 4 ).

The amount of movement of the body is the distance from the start point to the end point. For 2.5 periods in task 11.1.4 the body will have time to complete two full and half full oscillations, i.e. will be at the maximum deviation, but on the other side of the equilibrium position. Therefore, the amount of displacement is equal to two amplitudes (the answer 3 ).

By definition, the phase of oscillations is an argument of a trigonometric function, which describes the dependence of the coordinate of an oscillating body on time. Therefore the correct answer is task 11.1.5 - 3 .

The period is the time of complete oscillation. This means that the return of the body back to the same point from which the body began to move does not mean that the period has passed: the body must return to the same point with the same speed. For example, a body, having started oscillations from an equilibrium position, during the period will have time to deviate by the maximum value in one direction, go back, deviate to the maximum in the other direction and come back again. Therefore, during the period, the body will have time to deviate twice by the maximum value from the equilibrium position and return back. Therefore, the passage from the equilibrium position to the point of maximum deviation ( task 11.1.6) the body spends the fourth part of the period (the answer 3 ).

Such oscillations are called harmonic, in which the dependence of the coordinate of the oscillating body on time is described by a trigonometric (sine or cosine) function of time. IN task 11.1.7 these are the functions and , despite the fact that the parameters included in them are denoted as 2 and 2 . The function is the trigonometric function of the square of time. Therefore, fluctuations of only quantities and are harmonic (the answer 4 ).

With harmonic oscillations, the speed of the body changes according to the law , where is the amplitude of the speed oscillations (the time reference is chosen so that the initial phase of the oscillations would be equal to zero). From here we find the dependence of the kinetic energy of the body on time
(task 11.1.8). Using the well-known trigonometric formula, we get

It follows from this formula that the kinetic energy of the body changes during harmonic oscillations also according to the harmonic law, but with a doubled frequency (the answer is 2 ).

Behind the ratio between the kinetic energy of the load and the potential energy of the spring ( task 11.1.9) can be easily traced from the following considerations. When the body is deviated by the maximum amount from the equilibrium position, the speed of the body is zero, and, therefore, the potential energy of the spring is greater than the kinetic energy of the load. In contrast, when the body passes the equilibrium position, the potential energy of the spring is zero, and therefore the kinetic energy is greater than the potential energy. Therefore, between the passage of the equilibrium position and the maximum deviation, the kinetic and potential energies are compared once. And since during the period the body passes four times from the equilibrium position to the maximum deviation or vice versa, then during the period the kinetic energy of the load and the potential energy of the spring are compared with each other four times (the answer is 2 ).

Amplitude of speed fluctuations ( task 11.1.10) is easiest to find by the law of conservation of energy. At the point of maximum deflection, the energy of the oscillatory system is equal to the potential energy of the spring , where is the spring stiffness coefficient, is the oscillation amplitude. When passing through the equilibrium position, the energy of the body is equal to the kinetic energy , where is the mass of the body, is the speed of the body when passing through the equilibrium position, which is the maximum speed of the body in the process of oscillation and, therefore, represents the amplitude of the speed oscillations. Equating these energies, we find

(answer 4 ).

From formula (11.5) we conclude ( task 11.2.2), which is from the mass mathematical pendulum its period does not depend, and with an increase in length by 4 times, the period of oscillations increases by 2 times (the answer is 1 ).

The clock is an oscillatory process that is used to measure time intervals ( task 11.2.3). The words clock "rush" mean that the period of this process is less than what it should be. Therefore, to clarify the course of these clocks, it is necessary to increase the period of the process. According to formula (11.5), in order to increase the period of oscillation of a mathematical pendulum, it is necessary to increase its length (the answer is 3 ).

To find the amplitude of oscillations in task 11.2.4, it is necessary to represent the dependence of the body coordinate on time in the form of a single trigonometric function. For the function given in the condition, this can be done by introducing an additional angle. Multiplying and dividing this function by and using the addition formula trigonometric functions, we get

where is an angle such that . From this formula it follows that the amplitude of body oscillations is (answer 4 ).

§ 6. MECHANICAL OSCILLATIONSBasic formulas

Harmonic vibration equation

Where X - displacement of the oscillating point from the equilibrium position; t- time; A,ω, φ- respectively amplitude, angular frequency, initial phase of oscillations; - phase of oscillations at the moment t.

Angular oscillation frequency

where ν and T are the frequency and period of oscillations.

The speed of a point making harmonic oscillations,

Harmonic acceleration

Amplitude A the resulting oscillation obtained by adding two oscillations with the same frequencies occurring along one straight line is determined by the formula

Where a 1 And A 2 - amplitudes of oscillation components; φ 1 and φ 2 - their initial phases.

The initial phase φ of the resulting oscillation can be found from the formula

The frequency of beats arising from the addition of two oscillations occurring along the same straight line with different, but close in value, frequencies ν 1 and ν 2,

The equation of the trajectory of a point participating in two mutually perpendicular oscillations with amplitudes A 1 and A 2 and initial phases φ 1 and φ 2,

If the initial phases φ 1 and φ 2 of the oscillation components are the same, then the trajectory equation takes the form

i.e., the point moves in a straight line.

In the event that the phase difference , the equation takes the form

i.e., the point moves along an ellipse.

Differential equation of harmonic vibrations of a material point

, or , where m is the mass of the point; k- coefficient of quasi-elastic force ( k=Tω 2).

The total energy of a material point making harmonic oscillations,

The period of oscillation of a body suspended on a spring (spring pendulum),

Where m- body mass; k- spring stiffness. The formula is valid for elastic vibrations within the limits in which Hooke's law is fulfilled (with a small mass of the spring in comparison with the mass of the body).

The period of oscillation of a mathematical pendulum

Where l- pendulum length; g- acceleration of gravity. Oscillation period of a physical pendulum

Where J- the moment of inertia of the oscillating body about the axis

fluctuations; A- distance of the center of mass of the pendulum from the axis of oscillation;

Reduced length of a physical pendulum.

The above formulas are exact for the case of infinitely small amplitudes. For finite amplitudes, these formulas give only approximate results. At amplitudes no greater than the error in the value of the period does not exceed 1%.

The period of torsional vibrations of a body suspended on an elastic thread,

Where J- the moment of inertia of the body about the axis coinciding with the elastic thread; k- the stiffness of the elastic thread, equal to the ratio the elastic moment that occurs when the thread is twisted to the angle by which the thread is twisted.

Differential equation of damped oscillations , or ,

Where r- coefficient of resistance; δ - damping coefficient: ;ω 0 - natural angular frequency of vibrations *

Damped oscillation equation

Where A(t)- amplitude of damped oscillations at the moment t;ω is their angular frequency.

Angular frequency of damped oscillations

О Dependence of the amplitude of damped oscillations on time

I

Where A 0 - amplitude of oscillations at the moment t=0.

Logarithmic oscillation decrement

Where A(t) And A(t+T)- the amplitudes of two successive oscillations separated in time from each other by a period.

Differential equation of forced vibrations

where is an external periodic force acting on an oscillating material point and causing forced oscillations; F 0 - its amplitude value;

Amplitude of forced vibrations

Resonant frequency and resonant amplitude And

Examples of problem solving

Example 1 The point oscillates according to the law x(t)=, Where A=2 see Determine initial phase φ if

x(0)=cm and X , (0)<0. Построить векторную диаграмму для мо-­ мента t=0.

Solution. We use the equation of motion and express the displacement at the moment t=0 through initial phase:

From here we find the initial phase:

* In the previously given formulas for harmonic oscillations, the same value was simply denoted by ω (without the index 0).

Substitute the given values ​​into this expression x(0) and A:φ= = . The value of the argument is satisfied by two angle values:

In order to decide which of these values ​​of the angle φ also satisfies the condition , we first find:

Substituting into this expression the value t=0 and alternately the values ​​of the initial phases and, we find

T ok as always A>0 and ω>0, then only the first value of the initial phase satisfies the condition. Thus, the desired initial phase

Based on the found value of φ, we will construct a vector diagram (Fig. 6.1). Example 2 Material point with mass T\u003d 5 g performs harmonic oscillations with a frequency ν =0.5 Hz. Oscillation amplitude A=3 cm. Determine: 1) speed υ points at the time when the offset x== 1.5 cm; 2) the maximum force F max acting on the point; 3) Fig. 6.1 total energy E oscillating point.

and we obtain the velocity formula by taking the first time derivative of the displacement:

To express the speed in terms of displacement, time must be excluded from formulas (1) and (2). To do this, we square both equations, divide the first by A 2 , the second on A 2 ω 2 and add:

, or

Solving the last equation for υ , find

Having performed calculations according to this formula, we obtain

The plus sign corresponds to the case when the direction of the velocity coincides with the positive direction of the axis X, minus sign - when the direction of speed coincides with the negative direction of the axis X.

Displacement during harmonic oscillation, in addition to equation (1), can also be determined by the equation

Repeating the same solution with this equation, we get the same answer.

2. The force acting on a point, we find according to Newton's second law:

Where A - acceleration of a point, which we get by taking the time derivative of the speed:

Substituting the acceleration expression into formula (3), we obtain

Hence the maximum value of the force

Substituting into this equation the values ​​of π, ν, T And A, find

3. The total energy of an oscillating point is the sum of the kinetic and potential energies calculated for any moment of time.

The easiest way to calculate the total energy is at the moment when the kinetic energy reaches its maximum value. At this point, the potential energy is zero. So the total energy E oscillating point is equal to the maximum kinetic energy

We determine the maximum speed from formula (2), setting: . Substituting the speed expression into formula (4), we find

Substituting the values ​​of the quantities into this formula and performing calculations, we obtain

or mcJ.

Example 3 At the ends of a thin rod l= 1 m and weight m 3 =400 g small balls are reinforced with masses m 1=200 g And m 2 =300g. The rod oscillates about the horizontal axis, perpendicular to

dicular rod and passing through its middle (point O in Fig. 6.2). Define period T vibrations made by the rod.

Solution. The oscillation period of a physical pendulum, which is a rod with balls, is determined by the relation

Where J- T - its weight; l WITH - distance from the center of mass of the pendulum to the axis.

The moment of inertia of this pendulum is equal to the sum of the moments of inertia of the balls J 1 and J 2 and rod J 3:

Taking balls for material points, we express the moments of their inertia:

Since the axis passes through the middle of the rod, then its moment of inertia about this axis J 3 = =. Substituting the resulting expressions J 1 , J 2 And J 3 into formula (2), we find the total moment of inertia of the physical pendulum:

Performing calculations using this formula, we find

Rice. 6.2 The mass of the pendulum consists of the masses of the balls and the mass of the rod:

Distance l WITH we find the center of mass of the pendulum from the axis of oscillation, based on the following considerations. If the axis X direct along the rod and align the origin with the point ABOUT, then the desired distance l is equal to the coordinate of the center of mass of the pendulum, i.e.

Substituting the values ​​of quantities m 1 , m 2 , m, l and performing calculations, we find

Having made calculations according to formula (1), we obtain the oscillation period of a physical pendulum:

Example 4 The physical pendulum is a rod with a length l= 1 m and weight 3 T 1 With attached to one of its ends by a hoop with a diameter and mass T 1 . Horizontal axis Oz

pendulum passes through the middle of the rod perpendicular to it (Fig. 6.3). Define period T oscillations of such a pendulum.

Solution. The oscillation period of a physical pendulum is determined by the formula

(1)

Where J- the moment of inertia of the pendulum about the axis of oscillation; T - its weight; l C - the distance from the center of mass of the pendulum to the axis of oscillation.

The moment of inertia of the pendulum is equal to the sum of the moments of inertia of the rod J 1 and hoop J 2:

(2).

The moment of inertia of the rod relative to the axis perpendicular to the rod and passing through its center of mass is determined by the formula . In this case t= 3T 1 and

We find the moment of inertia of the hoop using the Steiner theorem ,Where J- moment of inertia about an arbitrary axis; J 0 - moment of inertia about the axis passing through the center of mass parallel to the given axis; A - the distance between the specified axes. Applying this formula to the hoop, we get

Substituting expressions J 1 and J 2 into formula (2), we find the moment of inertia of the pendulum about the axis of rotation:

Distance l WITH from the axis of the pendulum to its center of mass is

Substituting into formula (1) the expressions J, l c and the mass of the pendulum , we find the period of its oscillation:

After calculating by this formula, we get T\u003d 2.17 s.

Example 5 Two oscillations of the same direction are added, expressed by the equations ; X 2 = =, where A 1 = 1 cm, A 2 \u003d 2 cm, s, s, ω \u003d \u003d. 1. Determine the initial phases φ 1 and φ 2 of the components of the oscillation

bani. 2. Find the amplitude A and the initial phase φ of the resulting oscillation. Write the equation for the resulting oscillation.

Solution. 1. The equation of harmonic oscillation has the form

Let's transform the equations given in the condition of the problem to the same form:

From the comparison of expressions (2) with equality (1), we find the initial phases of the first and second oscillations:

Glad and glad.

2. To determine the amplitude A of the resulting fluctuation, it is convenient to use the vector diagram presented in rice. 6.4. According to the cosine theorem, we get

where is the phase difference of the oscillation components. Since , then, substituting the found values ​​φ 2 and φ 1 we get rad.

Substitute the values A 1 , A 2 and into formula (3) and perform the calculations:

A= 2.65 cm.

The tangent of the initial phase φ of the resulting oscillation can be determined directly from Figs. 6.4: , whence the initial phase

fluctuations called movements or processes that are characterized by a certain repetition in time. Oscillatory processes are widespread in nature and technology, for example, the swing of a clock pendulum, variable electricity etc. When oscillatory motion pendulum, the coordinate of its center of mass changes; in the case of alternating current, the voltage and current in the circuit fluctuate. physical nature oscillations can be different, therefore, mechanical, electromagnetic, etc. oscillations are distinguished. However, various oscillatory processes are described by the same characteristics and the same equations. From this comes the feasibility unified approach to the study of vibrations different physical nature.

The fluctuations are called free, if they are made only under the influence of internal forces acting between the elements of the system, after the system is removed from the equilibrium position external forces and left to itself. Free vibrations always damped oscillations because energy losses are inevitable in real systems. In the idealized case of a system without energy loss, free oscillations (continuing as long as desired) are called own.

The simplest type of free undamped oscillations are harmonic oscillations - fluctuations in which the fluctuating value changes with time according to the sine (cosine) law. Oscillations encountered in nature and technology often have a character close to harmonic.

Harmonic vibrations are described by an equation called the equation of harmonic vibrations:

Where A- amplitude of fluctuations, the maximum value of the fluctuating value X; - circular (cyclic) frequency of natural oscillations; - the initial phase of the oscillation at a moment of time t= 0; - the phase of the oscillation at the moment of time t. The phase of the oscillation determines the value of the oscillating quantity at a given time. Since the cosine varies from +1 to -1, then X can take values ​​from + A before - A.

Time T, for which the system completes one complete oscillation, is called period of oscillation. During T oscillation phase is incremented by 2 π , i.e.

Where . (14.2)

The reciprocal of the oscillation period

i.e., the number of complete oscillations per unit time is called the oscillation frequency. Comparing (14.2) and (14.3) we obtain

The unit of frequency is hertz (Hz): 1 Hz is the frequency at which one complete oscillation takes place in 1 s.

Systems in which free vibrations can occur are called oscillators . What properties must a system have in order for free oscillations to occur in it? The mechanical system must have position of stable equilibrium, upon exiting which appears restoring force towards equilibrium. This position corresponds, as is known, to the minimum of the potential energy of the system. Let us consider several oscillatory systems that satisfy the listed properties.