Harmonic oscillation is a phenomenon of periodic change of some quantity, in which the dependence on the argument has the character of a sine or cosine function. For example, a quantity that varies in time as follows harmonically fluctuates:
where x is the value of the changing quantity, t is time, the remaining parameters are constant: A is the amplitude of the oscillations, ω is the cyclic frequency of the oscillations, is the full phase of the oscillations, is the initial phase of the oscillations.
Generalized harmonic oscillation in differential form
(Any non-trivial solution of this differential equation is a harmonic oscillation with a cyclic frequency)
Types of vibrations
Free oscillations are performed under the action of the internal forces of the system after the system has been taken out of equilibrium. For free oscillations to be harmonic, it is necessary that the oscillatory system be linear (described by linear equations of motion), and there should be no energy dissipation in it (the latter would cause damping).
Forced oscillations are performed under the influence of an external periodic force. For them to be harmonic, it is sufficient that the oscillatory system be linear (described by linear equations of motion), and the external force itself changes over time as a harmonic oscillation (that is, that the time dependence of this force is sinusoidal).
Harmonic vibration equation
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Equation (1)
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gives the dependence of the fluctuating value S on time t; this is the equation of free harmonic oscillations in explicit form. However, the equation of oscillations is usually understood as a different record of this equation, in differential form. For definiteness, we take equation (1) in the form
Differentiate it twice with respect to time:
It can be seen that the following relation holds:
which is called the equation of free harmonic oscillations (in differential form). Equation (1) is a solution to differential equation (2). Since equation (2) is a second-order differential equation, two initial conditions are necessary to obtain a complete solution (that is, to determine the constants A and included in equation (1); for example, the position and speed of an oscillatory system at t = 0.
A mathematical pendulum is an oscillator, which is a mechanical system consisting of a material point located on a weightless inextensible thread or on a weightless rod in a uniform field of gravitational forces. The period of small eigenoscillations of a mathematical pendulum of length l, motionlessly suspended in a uniform gravitational field with free fall acceleration g, is equal to
and does not depend on the amplitude and mass of the pendulum.
A physical pendulum is an oscillator, which is a rigid body that oscillates in the field of any forces about a point that is not the center of mass of this body, or a fixed axis perpendicular to the direction of the forces and not passing through the center of mass of this body.
Harmonic oscillations - oscillations performed according to the laws of sine and cosine. The following figure shows a graph of the change in the coordinate of a point over time according to the law of cosine.
picture
Oscillation amplitude
The amplitude of a harmonic oscillation is the largest value of the displacement of the body from the equilibrium position. The amplitude can take on different values. It will depend on how much we displace the body at the initial moment of time from the equilibrium position.
The amplitude is determined by the initial conditions, that is, the energy imparted to the body at the initial moment of time. Since the sine and cosine can take values in the range from -1 to 1, then the equation must contain the factor Xm, which expresses the amplitude of the oscillations. Equation of motion for harmonic vibrations:
x = Xm*cos(ω0*t).
Oscillation period
The period of oscillation is the time it takes for one complete oscillation. The period of oscillation is denoted by the letter T. The units of the period correspond to the units of time. That is, in SI it is seconds.
Oscillation frequency - the number of oscillations per unit time. The oscillation frequency is denoted by the letter ν. The oscillation frequency can be expressed in terms of the oscillation period.
v = 1/T.
Frequency units in SI 1/sec. This unit of measurement is called Hertz. The number of oscillations in a time of 2 * pi seconds will be equal to:
ω0 = 2*pi* ν = 2*pi/T.
Oscillation frequency
This value is called the cyclic oscillation frequency. In some literature, the name circular frequency is found. The natural frequency of an oscillatory system is the frequency of free oscillations.
The frequency of natural oscillations is calculated by the formula:
The frequency of natural oscillations depends on the properties of the material and the mass of the load. The greater the stiffness of the spring, the greater the frequency of natural oscillations. The greater the mass of the load, the lower the frequency of natural oscillations.
These two conclusions are obvious. The stiffer the spring, the greater the acceleration it will impart to the body when the system is unbalanced. The greater the mass of the body, the slower this speed of this body will change.
Period of free oscillations:
T = 2*pi/ ω0 = 2*pi*√(m/k)
It is noteworthy that at small deflection angles, the period of oscillation of the body on the spring and the period of oscillation of the pendulum will not depend on the amplitude of the oscillations.
Let's write down the formulas for the period and frequency of free oscillations for a mathematical pendulum.
then the period will be
T = 2*pi*√(l/g).
This formula will be valid only for small deflection angles. From the formula we see that the period of oscillation increases with the length of the pendulum thread. The longer the length, the slower the body will oscillate.
The period of oscillation is completely independent of the mass of the load. But it depends on the free fall acceleration. As g decreases, the oscillation period will increase. This property is widely used in practice. For example, to measure the exact value of free acceleration.
>> Harmonic vibrations
§ 22 HARMONIC OSCILLATIONS
Knowing how the acceleration and the coordinate of an oscillating body are related, it is possible, on the basis of mathematical analysis, to find the dependence of the coordinate on time.
Acceleration is the second derivative of the coordinate with respect to time. The instantaneous speed of a point, as you know from the course of mathematics, is the derivative of the coordinate of the point with respect to time. The acceleration of a point is the derivative of its velocity with respect to time, or the second derivative of the coordinate with respect to time. Therefore, equation (3.4) can be written as follows:
where x " is the second derivative of the coordinate with respect to time. According to equation (3.11), during free oscillations, the x coordinate changes with time so that the second derivative of the coordinate with respect to time is directly proportional to the coordinate itself and opposite in sign to it.
It is known from the course of mathematics that the second derivatives of the sine and cosine with respect to their argument are proportional to the functions themselves, taken with the opposite sign. In mathematical analysis, it is proved that no other functions have this property. All this allows us to assert with good reason that the coordinate of a body that performs free oscillations changes over time according to the law of sine or pasine. Figure 3.6 shows the change in the coordinate of a point over time according to the cosine law.
Periodic changes in a physical quantity depending on time, occurring according to the law of sine or cosine, are called harmonic oscillations.
Oscillation amplitude. The amplitude of harmonic oscillations is the module of the greatest displacement of the body from the equilibrium position.
The amplitude can have different values depending on how much we displace the body from the equilibrium position at the initial moment of time, or on what speed is reported to the body. The amplitude is determined by the initial conditions, or rather by the energy imparted to the body. But the maximum values of the sine module and the cosine module are equal to one. Therefore, the solution of equation (3.11) cannot be expressed simply by sine or cosine. It should have the form of the product of the oscillation amplitude x m by a sine or cosine.
Solution of the equation describing free oscillations. We write the solution of equation (3.11) in the following form:
and the second derivative will be:
We have obtained equation (3.11). Therefore, the function (3.12) is a solution to the original equation (3.11). The solution to this equation will also be the function
According to (3.14), the graph of the dependence of the body coordinate on time is a cosine wave (see Fig. 3.6).
Period and frequency of harmonic oscillations. During vibrations, body movements are periodically repeated. The period of time T, during which the system completes one complete cycle of oscillations, is called the period of oscillations.
Knowing the period, you can determine the frequency of oscillations, that is, the number of oscillations per unit of time, for example, per second. If one oscillation occurs in time T, then the number of oscillations per second
In the International System of Units (SI), the frequency of oscillations is equal to one if one oscillation occurs per second. The unit of frequency is called hertz (abbreviated: Hz) in honor of the German physicist G. Hertz.
The number of oscillations in 2 s is:
Value - cyclic, or circular, frequency of oscillations. If in equation (3.14) time t is equal to one period, then T \u003d 2. Thus, if at time t \u003d 0 x \u003d x m, then at time t \u003d T x \u003d x m, that is, through a period of time equal to one period, the oscillations are repeated.
The frequency of free oscillations is found by the natural frequency of the oscillatory system 1.
Dependence of the frequency and period of free oscillations on the properties of the system. The natural frequency of vibrations of a body attached to a spring, according to equation (3.13), is equal to:
It is the greater, the greater the stiffness of the spring k, and the less, the greater the body mass m. This is easy to understand: a stiff spring gives the body more acceleration, changes the body's speed faster. And the more massive the body, the slower it changes speed under the influence of force. The oscillation period is:
Having a set of springs of different rigidity and bodies of different masses, it is easy to verify from experience that formulas (3.13) and (3.18) correctly describe the nature of the dependence of u T on k and m.
It is remarkable that the period of oscillation of a body on a spring and the period of oscillation of a pendulum at small deflection angles do not depend on the oscillation amplitude.
The modulus of the coefficient of proportionality between the acceleration t and the displacement x in equation (3.10), which describes the oscillations of the pendulum, is, as in equation (3.11), the square of the cyclic frequency. Consequently, the natural frequency of oscillations of a mathematical pendulum at small angles of deviation of the thread from the vertical depends on the length of the pendulum and the free fall acceleration:
This formula was first obtained and tested by the Dutch scientist G. Huygens, a contemporary of I. Newton. It is valid only for small angles of deflection of the thread.
1 Often in what follows, for brevity, we will refer to the cyclic frequency simply as the frequency. You can distinguish the cyclic frequency from the usual frequency by notation.
The period of oscillation increases with the length of the pendulum. It does not depend on the mass of the pendulum. This can be easily verified by experiment with various pendulums. The dependence of the oscillation period on the free fall acceleration can also be found. The smaller g, the longer the period of oscillation of the pendulum and, consequently, the slower the clock with the pendulum runs. Thus, a clock with a pendulum in the form of a weight on a rod will fall behind in a day by almost 3 s if it is lifted from the basement to the upper floor of Moscow University (height 200 m). And this is only due to the decrease in the acceleration of free fall with height.
The dependence of the period of oscillation of the pendulum on the value of g is used in practice. By measuring the period of oscillation, g can be determined very accurately. The acceleration due to gravity varies with geographic latitude. But even at a given latitude it is not the same everywhere. After all, the density of the earth's crust is not the same everywhere. In areas where dense rocks occur, the acceleration g is somewhat greater. This is taken into account when prospecting for minerals.
Thus, iron ore has an increased density compared to conventional rocks. Measurements of the acceleration of gravity near Kursk, carried out under the guidance of Academician A. A. Mikhailov, made it possible to clarify the location of iron ore. They were first discovered through magnetic measurements.
The properties of mechanical vibrations are used in the devices of most electronic scales. The body to be weighed is placed on a platform under which a rigid spring is installed. As a result, mechanical vibrations occur, the frequency of which is measured by a corresponding sensor. The microprocessor connected to this sensor translates the oscillation frequency into the mass of the weighed body, since this frequency depends on the mass.
The obtained formulas (3.18) and (3.20) for the oscillation period indicate that the period of harmonic oscillations depends on the parameters of the system (spring stiffness, thread length, etc.)
Myakishev G. Ya., Physics. Grade 11: textbook. for general education institutions: basic and profile. levels / G. Ya. Myakishev, B. V. Bukhovtsev, V. M. Charugin; ed. V. I. Nikolaev, N. A. Parfenteva. - 17th ed., revised. and additional - M.: Education, 2008. - 399 p.: ill.
A complete list of topics by class, a calendar plan according to the school curriculum in physics online, download video material in physics for grade 11
Lesson content lesson summary support frame lesson presentation accelerative methods interactive technologies Practice tasks and exercises self-examination workshops, trainings, cases, quests homework discussion questions rhetorical questions from students Illustrations audio, video clips and multimedia photographs, pictures graphics, tables, schemes humor, anecdotes, jokes, comics parables, sayings, crossword puzzles, quotes Add-ons abstracts articles chips for inquisitive cheat sheets textbooks basic and additional glossary of terms other Improving textbooks and lessonscorrecting errors in the textbook updating a fragment in the textbook elements of innovation in the lesson replacing obsolete knowledge with new ones Only for teachers perfect lessons calendar plan for the year methodological recommendations of the discussion program Integrated Lessonsfluctuations called movements or processes that are characterized by a certain repetition in time. Fluctuations are widespread in the surrounding world and can have a very different nature. These can be mechanical (pendulum), electromagnetic (oscillatory circuit) and other types of oscillations.
Free, or own oscillations are called oscillations that occur in a system left to itself, after it has been brought out of equilibrium by an external influence. An example is the oscillation of a ball suspended on a thread.
special role in oscillatory processes has the simplest form of oscillation - harmonic vibrations. Harmonic oscillations underlie a unified approach in the study of oscillations of various nature, since oscillations occurring in nature and technology are often close to harmonic ones, and periodic processes of a different form can be represented as a superposition of harmonic oscillations.
Harmonic vibrations such oscillations are called, in which the oscillating value varies with time according to the law sinus or cosine.
Harmonic vibration equationlooks like:
where A - oscillation amplitude (the value of the greatest deviation of the system from the equilibrium position); -circular (cyclic) frequency. Periodically changing cosine argument - called oscillation phase . The oscillation phase determines the displacement of the oscillating quantity from the equilibrium position at a given time t. The constant φ is the value of the phase at time t = 0 and is called the initial phase of the oscillation . The value of the initial phase is determined by the choice of the reference point. The x value can take values ranging from -A to +A.
The time interval T, after which certain states of the oscillatory system are repeated, called the period of oscillation . Cosine is a periodic function with a period of 2π, therefore, over a period of time T, after which the oscillation phase will receive an increment equal to 2π, the state of the system performing harmonic oscillations will repeat. This period of time T is called the period of harmonic oscillations.
The period of harmonic oscillations is : T = 2π/ .
The number of oscillations per unit time is called oscillation frequency
ν.
Frequency of harmonic vibrations
is equal to: ν = 1/T. Frequency unit hertz(Hz) - one oscillation per second.
Circular frequency = 2π/T = 2πν gives the number of oscillations in 2π seconds.
Graphically, harmonic oscillations can be depicted as a dependence of x on t (Fig. 1.1.A), and rotating amplitude method (vector diagram method)(Fig.1.1.B) .
The rotating amplitude method allows you to visualize all the parameters included in the equation of harmonic oscillations. Indeed, if the amplitude vector A located at an angle φ to the x-axis (see Figure 1.1. B), then its projection on the x-axis will be equal to: x = Acos(φ). The angle φ is the initial phase. If the vector A put into rotation with an angular velocity equal to the circular frequency of oscillations, then the projection of the end of the vector will move along the x-axis and take values ranging from -A to +A, and the coordinate of this projection will change over time according to the law:
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Thus, the length of the vector is equal to the amplitude of the harmonic oscillation, the direction of the vector at the initial moment forms an angle with the x-axis equal to the initial phase of the oscillation φ, and the change in the direction angle with time is equal to the phase of the harmonic oscillations. The time for which the amplitude vector makes one complete revolution is equal to the period T of harmonic oscillations. The number of revolutions of the vector per second is equal to the oscillation frequency ν.
This is a periodic oscillation, in which the coordinate, speed, acceleration, characterizing the movement, change according to the sine or cosine law. 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.
Mathematical pendulum
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Oscillations of a mathematical pendulum. |
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Mathematical pendulum is a material point suspended on a weightless inextensible thread (physical model). |
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We will consider the movement of the pendulum under the condition that the deflection angle is small, then, if we measure the angle in radians, the statement is true: . |
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The force of gravity and the tension of the thread act on the body. The resultant of these forces has two components: a tangential one, which changes the acceleration in magnitude, and a normal one, which changes the acceleration in direction (centripetal acceleration, the body moves in an arc). |
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Because the angle is small, then the tangential component is equal to the projection of gravity on the tangent to the trajectory: . The angle in radians is equal to the ratio of the arc length to the radius (filament length), and the arc length is approximately equal to the offset ( x ≈ s): | |
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Let's compare the resulting equation with the equation of oscillatory motion. It can be seen that or is a cyclic frequency during oscillations of a mathematical pendulum. | |
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Oscillation period or (Galileo's formula). |
Galileo formula |
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The most important conclusion: the period of oscillation of a mathematical pendulum does not depend on the mass of the body! | |
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Similar calculations can be done using the law of conservation of energy. We take into account that the potential energy of the body in the gravitational field is equal to , and the total mechanical energy is equal to the maximum potential or kinetic: |
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Let's write down the law of conservation of energy and take the derivative of the left and right parts of the equation: . Because the derivative of a constant value is equal to zero, then . The derivative of the sum is equal to the sum of the derivatives: and. | |
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Therefore: , which means. | |
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Ideal gas equation of state (Mendeleev-Clapeyron equation). |
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An equation of state is an equation that relates the parameters of a physical system and uniquely determines its state. In 1834 the French physicist B. Clapeyron, who worked for a long time in St. Petersburg, derived the equation of state for an ideal gas for a constant mass of gas. In 1874 D. I. Mendeleev derived an equation for an arbitrary number of molecules. | |
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In MKT and ideal gas thermodynamics macroscopic parameters are: p, V, T, m. We know that | |
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The product of constant values is a constant value, therefore: |
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Thus, we have: Equation of state (Mendeleev-Clapeyron equation). | |
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Other forms of writing the equation of state of an ideal gas. |
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1. Equation for 1 mole of a substance. If n \u003d 1 mol, then, denoting the volume of one mole V m, we get:. For normal conditions, we get: |
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2. Write the equation in terms of density: - Density depends on temperature and pressure! | |
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3. Clapeyron equation. Often it is necessary to investigate the situation when the state of the gas changes with its constant amount (m=const) and in the absence of chemical reactions (M=const). This means that the amount of substance n=const. Then: | |
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This entry means that for a given mass of a given gas equality is true: | |
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For a constant mass of an ideal gas, the ratio of the product of pressure and volume to the absolute temperature in a given state is a constant value: . | |
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gas laws. |
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1. Avogadro's law. Equal volumes of different gases under the same external conditions contain the same number of molecules (atoms). Condition: V 1 =V 2 =…=V n ; p 1 \u003d p 2 \u003d ... \u003d p n; T 1 \u003d T 2 \u003d ... \u003d T n | |
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Proof: Therefore, under the same conditions (pressure, volume, temperature), the number of molecules does not depend on the nature of the gas and is the same. | |
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2. Dalton's Law. The pressure of a mixture of gases is equal to the sum of the partial (private) pressures of each gas. Prove: p=p 1 +p 2 +…+p n Proof: | |
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3. Pascal's law. The pressure produced on a liquid or gas is transmitted in all directions without change. | |
. Hence,. Given that 
, we get:.
- universal gas constant (universal, because it is the same for all gases).
The equation of state for an ideal gas. gas laws.
Numbers of degrees of freedom: this is the number of independent variables (coordinates) that completely determine the position of the system in space. In some problems, a monatomic gas molecule (Fig. 1, a) is considered as a material point, which is given three degrees of freedom of translational motion. This does not take into account the energy of rotational motion. In mechanics, a diatomic gas molecule in the first approximation is considered to be a set of two material points, which are rigidly connected by a non-deformable bond (Fig. 1, b). This system, in addition to three degrees of freedom of translational motion, has two more degrees of freedom of rotational motion. Rotation around the third axis passing through both atoms is meaningless. This means that a diatomic gas has five degrees of freedom ( i= 5). A triatomic (Fig. 1, c) and polyatomic nonlinear molecule has six degrees of freedom: three translational and three rotational. It is natural to assume that there is no rigid bond between atoms. Therefore, for real molecules, it is also necessary to take into account the degrees of freedom of vibrational motion.
For any number of degrees of freedom of a given molecule, the three degrees of freedom are always translational. None of the translational degrees of freedom has an advantage over the others, which means that each of them has on average the same energy equal to 1/3 of the value<ε 0 >(energy of translational motion of molecules): 
In statistical physics, Boltzmann's law on the uniform distribution of energy over the degrees of freedom of molecules: for a statistical system that is in a state of thermodynamic equilibrium, each translational and rotational degree of freedom has an average kinetic energy equal to kT / 2, and each vibrational degree of freedom has an average energy equal to kT. The vibrational degree has twice as much energy, because it accounts for both kinetic energy (as in the case of translational and rotational motions) and potential energy, and the average values of potential and kinetic energy are the same. So the average energy of the molecule 
Where i- the sum of the number of translational, the number of rotational in twice the number of vibrational degrees of freedom of the molecule: i=i post + i rotation +2 i vibrations In the classical theory, molecules are considered with a rigid bond between atoms; for them i coincides with the number of degrees of freedom of the molecule. Since in an ideal gas the mutual potential energy of interaction of molecules is equal to zero (molecules do not interact with each other), then the internal energy for one mole of gas will be equal to the sum of the kinetic energies N A of molecules: (1) Internal energy for an arbitrary mass m of gas. where M is the molar mass, ν
- amount of substance.
