Let's start by determining the period of oscillation. Harmonic vibrations

The time during which one complete change in the EMF occurs, that is, one cycle of oscillation or one complete revolution of the radius vector, is called alternating current oscillation period(picture 1).

Picture 1. Period and amplitude of a sinusoidal oscillation. Period - the time of one oscillation; The amplitude is its largest instantaneous value.

The period is expressed in seconds and denoted by the letter T.

Smaller units of period are also used, these are millisecond (ms) - one thousandth of a second and microsecond (μs) - one millionth of a second.

1 ms = 0.001 sec = 10 -3 sec.

1 µs = 0.001 ms = 0.000001 sec = 10 -6 sec.

1000 µs = 1 ms.

The number of complete changes in the EMF or the number of revolutions of the radius vector, that is, in other words, the number of complete cycles of oscillations performed by alternating current in one second, is called AC oscillation frequency.

The frequency is indicated by the letter f and is expressed in periods per second or hertz.

One thousand hertz is called a kilohertz (kHz), and one million hertz is called a megahertz (MHz). There is also a unit gigahertz (GHz) equal to one thousand megahertz.

1000 Hz = 10 3 Hz = 1 kHz;

1000,000 Hz = 10 6 Hz = 1000 kHz = 1 MHz;

1000,000,000 Hz = 109 Hz = 1000,000 kHz = 1000 MHz = 1 GHz;

The faster the EMF changes, that is, the faster the radius vector rotates, the shorter the oscillation period. The faster the radius vector rotates, the higher the frequency. Thus, the frequency and period of an alternating current are inversely proportional to each other. The larger one of them, the smaller the other.

The mathematical relationship between the period and frequency of alternating current and voltage is expressed by the formulas

For example, if the frequency of the current is 50 Hz, then the period will be equal to:

T \u003d 1 / f \u003d 1/50 \u003d 0.02 sec.

Conversely, if it is known that the period of the current is 0.02 sec, (T=0.02 sec), then the frequency will be:

f \u003d 1 / T \u003d 1 / 0.02 \u003d 100/2 \u003d 50 Hz

The frequency of alternating current used for lighting and industrial purposes is exactly 50 Hz.

Frequencies from 20 to 20,000 Hz are called audio frequencies. The currents in the antennas of radio stations fluctuate with frequencies up to 1,500,000,000 Hz, or, in other words, up to 1,500 MHz or 1.5 GHz. Such high frequencies are called radio frequencies or high frequency oscillations.

Finally, the currents in the antennas of radar stations, satellite communication stations, and other special systems (for example, GLANASS, GPS) fluctuate at frequencies up to 40,000 MHz (40 GHz) and higher.

AC amplitude

The highest value that the EMF or current strength reaches in one period is called amplitude of the emf or alternating current. It is easy to see that the scaled amplitude is equal to the length of the radius vector. Amplitudes of current, EMF and voltage are indicated respectively by letters Im, Em and Um (picture 1).

Angular (cyclic) frequency of alternating current.

The speed of rotation of the radius vector, i.e., the change in the value of the angle of rotation for one second, is called the angular (cyclic) frequency of the alternating current and is denoted by the Greek letter ? (omega). Rotation angle of the radius vector in any this moment relative to its initial position, it is usually measured not in degrees, but in special units - radians.

The radian is the angular value of the arc of a circle, the length of which is equal to the radius of this circle (Figure 2). The whole circle that is 360° is equal to 6.28 radians, which is 2.

Figure 2.

1rad = 360°/2

Therefore, the end of the radius vector during one period runs a path equal to 6.28 radians (2). Since for one second the radius vector makes a number of revolutions equal to the frequency of the alternating current f, then in one second its end runs a path equal to 6.28*f radian. This expression, which characterizes the speed of rotation of the radius vector, will be the angular frequency of the alternating current - ? .

? = 6.28*f = 2f

The angle of rotation of the radius vector at any given moment relative to its initial position is called AC phase. The phase characterizes the magnitude of the EMF (or current) at a given moment, or, as they say, the instantaneous value of the EMF, its direction in the circuit and the direction of its change; phase shows whether the emf is decreasing or increasing.

Figure 3

A complete rotation of the radius vector is 360°. With the beginning of a new revolution of the radius vector, the change in the EMF occurs in the same order as during the first revolution. Therefore, all phases of the EMF will be repeated in the same order. For example, the phase of the EMF when the radius vector is rotated through an angle of 370 ° will be the same as when it is rotated by 10 °. In both of these cases, the radius vector occupies the same position, and, therefore, the instantaneous values ​​​​of the emf will be the same in phase in both of these cases.

Definition

Period- this is the minimum time for which one complete oscillatory movement is performed.

The period is denoted by the letter $T$.

where $\Delta t$ - oscillation time; $N$ - number of complete oscillations.

The equation of oscillation of a spring pendulum

Consider the simplest oscillatory system in which mechanical oscillations can be realized. This is a load of mass $m$, suspended on a spring, the coefficient of elasticity of which is equal to $k\ $(fig.1). Consider the vertical movement of a load, which is due to the action of gravity and the elastic force of a spring. In the state of equilibrium of such a system, the force of elasticity is equal in magnitude to the force of gravity. Oscillations of a spring pendulum occur when the system is taken out of equilibrium, for example, by slightly additionally stretching the spring, after which the pendulum is left to itself.

Let us assume that the mass of the spring is small in comparison with the mass of the load; we will not take it into account when describing the oscillations. The reference point is considered to be a point on the coordinate axis (X), which coincides with the equilibrium position of the load. In this position, the spring already has an extension, which we denote by $b$. The tension of the spring occurs due to the action of gravity on the load, therefore:

If the load is displaced additionally, but Hooke's law is still fulfilled, then the spring force becomes equal to:

We write the acceleration of the load, remembering that the movement occurs along the X axis, as:

Newton's second law for the load takes the form:

We take into account equality (2), formula (5) is transformed to the form:

If we introduce the notation: $(\omega )^2_0=\frac(k)(m)$, then we write the oscillation equation as:

\[\ddot(x)+(\omega )^2_0x=0\left(7\right),\]

where $(\omega )^2_0=\frac(k)(m)$ is the cyclic frequency of oscillations of the spring pendulum. The solution to equation (7) (this is verified by direct substitution) is the function:

where $(\omega )_0=\sqrt(\frac(k)(m))>0$ is the cyclic oscillation frequency of the pendulum, $A$ is the oscillation amplitude; $((\omega )_0t+\varphi)$ - oscillation phase; $\varphi $ and $(\varphi )_1$ - initial phases of oscillations.

Formulas for the oscillation period of a spring pendulum

We have found that the oscillations of a spring pendulum are described by the cosine or sine function. These are periodic functions, which means that the displacement $x$ will take equal values ​​at certain equal intervals of time, which is called the oscillation period. The period is denoted by the letter T.

Another quantity that characterizes oscillations is the reciprocal of the period of oscillations, it is called the frequency ($\nu $):

The period is related to the cyclic oscillation frequency as:

Above we obtained $(\omega )_0=\sqrt(\frac(k)(m))$ for a spring pendulum, therefore, the oscillation period of a spring pendulum is equal to:

The formula for the oscillation period of a spring pendulum (11) shows that $T$ depends on the mass of the load attached to the spring and the coefficient of elasticity of the spring, but does not depend on the oscillation amplitude (A). This property of oscillations is called isochronism. Isochronism is satisfied as long as Hooke's law is valid. At large stretches of the spring, Hooke's law is violated, the dependence of oscillations on the amplitude appears. We emphasize that formula (11) for calculating the oscillation period of a spring pendulum is valid for small oscillations.

Examples of tasks for the period of oscillation

Example 1

Exercise. A spring pendulum made 50 complete oscillations in a time equal to 10 s. What is the period of the pendulum's oscillation? What is the frequency of these oscillations?

Solution. Since the period is the minimum time required for the pendulum to complete one complete oscillation, we find it as:

Calculate the period:

Frequency is the reciprocal of the period, therefore:

\[\nu=\frac(1)(T)\left(1.2\right).\]

Let's calculate the oscillation frequency:

\[\nu =\frac(1)(0,2)=5\ \left(Hz\right).\]

Answer.$1)\ T=0.2$ s; 2) 5Hz

Example 2

Exercise. Two springs with elasticity coefficients $k_1$ and $k_2$ are connected in parallel (Fig. 2), a load of mass $M$ is attached to the system. What is the oscillation period of the resulting spring pendulum, if the masses of the springs can be neglected, the elastic force acting on the load obeys Hooke's law?

Solution. Let's use the formula to calculate the oscillation period of a spring pendulum:

When the springs are connected in parallel, the resulting stiffness of the system is found as:

This means that instead of $k$ in the formula for calculating the period of a spring pendulum, we substitute the right side of expression (2.2), we have:

Answer.$T=2\pi \sqrt(\frac(M)(k_1(+k)_2))$

The most important parameter characterizing mechanical, sound, electrical, electromagnetic and all other types of vibrations is period is the time it takes for one complete oscillation. If, for example, the pendulum of a clock-clock makes two complete oscillations in 1 s, the period of each oscillation is 0.5 s. The period of oscillation of a large swing is about 2 s, and the period of oscillation of a string can be from tenths to ten-thousandths of a second.

Figure 2.4 - Fluctuation

where: φ - oscillation phase, I- current strength, Ia- amplitude value of the current strength (amplitude)

T- period of current oscillation (period)

Another parameter characterizing fluctuations is frequency(from the word "often") - a number showing how many complete oscillations per second the clock pendulum, the sounding body, the current in the conductor, etc. make. The frequency of oscillations is measured by a unit called hertz (abbreviated as Hz): 1 Hz is one oscillation per second. If, for example, a sounding string makes 440 full vibrations in 1 s (while it creates the tone “la” of the third octave), they say that its vibration frequency is 440 Hz. The frequency of the alternating current of the electric lighting network is 50 Hz. With this current, the electrons in the wires of the network flow alternately 50 times in one direction and the same number of times in the opposite direction for a second, i.e. perform in 1 s 50 complete oscillations.

The larger units of frequency are kilohertz (written kHz) equal to 1000 Hz and megahertz (written MHz) equal to 1000 kHz or 1,000,000 Hz.

Amplitude- the maximum value of the displacement or change of a variable during oscillatory or wave motion. A non-negative scalar value, measured in units depending on the type of wave or oscillation.

Figure 2.5 - Sinusoidal oscillation.

where, y- wave amplitude, λ - wavelength.

For example:

amplitude for mechanical vibration of a body (vibration), for waves on a string or spring - this is the distance and is written in units of length;

the 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). Its logarithm is usually measured in decibels (dB);

for electromagnetic radiation, the amplitude corresponds to the magnitude of the electric and magnetic fields.

The form of amplitude change is called envelope wave.

Sound vibrations

How do sound waves form in air? Air is made up of invisible particles. With the wind, they can be carried over long distances. But they can also fluctuate. For example, if we make a sharp movement with a stick in the air, then we will feel a slight gust of wind and at the same time hear a faint sound. Sound this is the result of vibrations of air particles excited by vibrations of the stick.

Let's do this experiment. Let's pull a string, for example, of a guitar, and then let it go. The string will begin to tremble - oscillate around its original resting position. Sufficiently strong vibrations of the string are noticeable to the eye. Weak vibrations of the string can only be felt as a slight tickle if you touch it with your finger. As long as the string vibrates, we hear the sound. As soon as the string calms down, the sound will die out. The birth of sound here is the result of condensation and rarefaction of air particles. Oscillating from side to side, the string pushes, as if compressing air particles in front of it, forming areas of high pressure in some of its volume, and behind, on the contrary, areas of low pressure. That's what it is sound waves. Spreading in the air at a speed of about 340 m/s, they carry a certain amount of energy. At that moment, when the area of ​​high pressure of the sound wave reaches the ear, it presses on the eardrum, slightly bending it inward. When the rarefied region of the sound wave reaches the ear, the tympanic membrane curves somewhat outward. The eardrum constantly vibrates in time with alternating areas of high and low air pressure. These vibrations are transmitted along the auditory nerve to the brain, and we perceive them as sound. The greater the amplitude of sound waves, the more energy they carry in themselves, the louder the sound we perceive.

Sound waves, like water or electrical vibrations, are represented by a wavy line - a sinusoid. Its humps correspond to areas of high pressure, and its troughs correspond to areas of low air pressure. The area of ​​high pressure and the area of ​​low pressure following it form a sound wave.

By the frequency of vibrations of the sounding body, one can judge the tone or pitch of the sound. The higher the frequency, the higher the tone of the sound, and vice versa, the lower the frequency, the lower the tone of the sound. Our ear is able to respond to a relatively small band (section) of frequencies. sound vibrations - from about 20 Hz to 20 kHz. Nevertheless, this frequency band accommodates the entire wide range of sounds created by the human voice, a symphony orchestra: from very low tones, similar to the sound of a bug buzzing, to the barely perceptible high-pitched squeak of a mosquito. Frequency fluctuations up to 20 Hz, called infrasonic, and over 20 kHz, called ultrasonic we don't hear. And if the tympanic membrane of our ear turned out to be able to respond to ultrasonic vibrations, we could then hear the squeak of bats, the voice of a dolphin. Dolphins emit and hear ultrasonic vibrations with frequencies up to 180 kHz.

But you can not confuse the height, i.e. tone of sound with its strength. The pitch of the sound does not depend on the amplitude, but on the frequency of vibrations. A thick and long string of a musical instrument, for example, creates a low tone of sound, i.e. oscillates more slowly than a thin and short string, which creates a high tone of sound (Fig. 1).

Figure 2.6 - Sound waves

The higher the frequency of the string, the shorter the sound waves and the higher the tone of the sound.

In electrical and radio engineering, alternating currents with a frequency of several hertz to thousands of gigahertz are used. Broadcast radio antennas, for example, are fed with currents ranging from about 150 kHz to 100 MHz.

These rapidly changing oscillations, called radio frequency oscillations, are the means by which sounds are transmitted over long distances without wires.

The entire huge range of alternating currents is usually divided into several sections - subranges.

Currents with a frequency of 20 Hz to 20 kHz, corresponding to oscillations that we perceive as sounds of different tonality, are called currents(or fluctuations) audio frequency, and currents with a frequency above 20 kHz - ultrasonic frequency currents.

Currents with frequencies from 100 kHz to 30 MHz are called high frequency currents,

Currents with frequencies above 30 MHz - currents of ultrahigh and ultrahigh frequency.

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 does not depend on 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.

So it is with anharmonic strictly periodic oscillations (and approximately - with one success or another - and non-periodic oscillations, at least close to periodicity).

In the case when we are talking about oscillations of a harmonic oscillator with damping, the period is understood as the period of its oscillating component (ignoring damping), which coincides with twice the time interval between the nearest passages of the oscillating value through zero. In principle, this definition can be more or less accurately and usefully extended in some generalization to damped oscillations with other properties.

Designations: the usual standard notation for the period of oscillation is: T (\displaystyle T)(although others may apply, the most common is τ (\displaystyle \tau ), sometimes Θ (\displaystyle \Theta ) etc.).

For wave processes, the period is also obviously related to the wavelength λ (\displaystyle \lambda )

where v (\displaystyle v)- wave propagation velocity (more precisely, phase velocity).

In quantum physics the period of oscillation is directly related to energy (because in quantum physics, the energy of an object - for example, a particle - is the frequency of oscillation of its wave function).

Theoretical finding the oscillation period of a particular physical system is reduced, as a rule, to finding a solution of dynamic equations (equation) that describes this system. For the category of linear systems (and approximately for linearizable systems in a linear approximation, which is often very good), there are standard relatively simple mathematical methods that allow this to be done (if the physical equations themselves that describe the system are known).

For experimental determination period, clocks, stopwatches, frequency meters, stroboscopes, strobe tachometers, oscilloscopes are used. Beats are also used, the method of heterodyning in different forms, the principle of resonance is used. For waves, you can measure the period indirectly - through the wavelength, for which interferometers, diffraction gratings, etc. are used. Sometimes sophisticated methods are also required, specially developed for a specific difficult case (difficulty can be both the measurement of time itself, especially when it comes to extremely short or vice versa very long times, and the difficulty of observing a fluctuating value).

Encyclopedic YouTube

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  An idea about the periods of oscillations of various physical processes is given in the article Frequency Intervals (given that the period in seconds is the reciprocal of the frequency in hertz).

  Some idea of ​​the magnitudes of the periods of various physical processes can also be given by the frequency scale of electromagnetic oscillations (see Electromagnetic Spectrum).

  The periods of oscillation of a sound audible to a person are in the range

  From 5 10 −5 to 0.2

  (its clear boundaries are somewhat arbitrary).

  Periods of electromagnetic oscillations corresponding to different colors of visible light - in the range

  From 1.1 10 −15 to 2.3 10 −15 .

  Since, for extremely large and extremely small oscillation periods, measurement methods tend to become more and more indirect (up to a smooth flow into theoretical extrapolations), it is difficult to name a clear upper and lower bounds for the oscillation period measured directly. Some estimate for the upper limit can be given by the time of existence of modern science (hundreds of years), and for the lower one - by the oscillation period of the wave function of the heaviest particle known now ().

  Anyway bottom border can serve as the Planck time, which is so small that, according to modern concepts, it is not only unlikely that it can be physically measured in any way at all, but it is unlikely that in the more or less foreseeable future it will be possible to approach the measurement of even much larger orders of magnitude, and top border- the time of existence of the Universe - more than ten billion years.

  Periods of oscillations of the simplest physical systems

  Spring pendulum

  Mathematical pendulum

  T = 2 π l g (\displaystyle T=2\pi (\sqrt (\frac (l)(g))))

  where l (\displaystyle l)- the length of the suspension (for example, threads), g (\displaystyle g)- acceleration of gravity .

  The period of small oscillations (on Earth) of a mathematical pendulum 1 meter long is equal to 2 seconds with good accuracy.

  physical pendulum

  T = 2 π J m g l (\displaystyle T=2\pi (\sqrt (\frac (J)(mgl))))

  where J (\displaystyle J)- the moment of inertia of the pendulum about the axis of rotation, m (\displaystyle m) -