Oscillation period notation. Oscillation period: experiments, formulas, tasks. Definition and physical meaning
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 full cycles oscillations made by alternating current for 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.
What is the period of oscillation? What is this value, what physical meaning does it have and how to calculate it? In this article, we will deal with these issues, consider various formulas by which the period of oscillations can be calculated, and also find out what relationship exists between such physical quantities as the period and frequency of oscillations of a body / system.
Definition and physical meaning
The period of oscillation is such a period of time in which the body or system makes one oscillation (necessarily complete). In parallel, we can note the parameter at which the oscillation can be considered complete. The role of such a condition is the return of the body to its original state (to the original coordinate). The analogy with the period of a function is very well drawn. Incidentally, it is a mistake to think that it takes place exclusively in ordinary and higher mathematics. As you know, these two sciences are inextricably linked. And the period of functions can be encountered not only when solving trigonometric equations, but also in various branches of physics, namely, we are talking about mechanics, optics and others. When transferring the period of oscillations from mathematics to physics, it should be understood simply as a physical quantity (and not a function), which has a direct dependence on the passing time.
What are the fluctuations?
Oscillations are divided into harmonic and anharmonic, as well as periodic and non-periodic. It would be logical to assume that in the case of harmonic oscillations, they occur according to some harmonic function. It can be either sine or cosine. In this case, the coefficients of compression-stretching and increase-decrease may also turn out to be in the case. Also, vibrations are damped. That is, when a certain force acts on the system, which gradually “slows down” the oscillations themselves. In this case, the period becomes shorter, while the frequency of oscillations invariably increases. The simplest experiment using a pendulum demonstrates such a physical axiom very well. It can be spring type, as well as mathematical. It does not matter. By the way, the oscillation period in such systems will be determined by different formulas. But more on that later. Now let's give examples.
Experience with pendulums
You can take any pendulum first, there will be no difference. The laws of physics are the laws of physics, that they are respected in any case. But for some reason, the mathematical pendulum is more to my liking. If someone does not know what it is: it is a ball on an inextensible thread that is attached to a horizontal bar attached to the legs (or the elements that play their role - to keep the system in balance). The ball is best taken from metal, so that the experience is clearer.
So, if you take such a system out of balance, apply some force to the ball (in other words, push it), then the ball will begin to swing on the thread, following a certain trajectory. Over time, you can notice that the trajectory along which the ball passes is reduced. At the same time, the ball begins to scurry back and forth faster and faster. This indicates that the oscillation frequency is increasing. But the time it takes for the ball to return to its original position decreases. But the time of one complete oscillation, as we found out earlier, is called a period. If one value decreases and the other increases, then they speak of inverse proportionality. So we got to the first moment, on the basis of which formulas are built to determine the period of oscillations. If we take a spring pendulum for testing, then the law will be observed there in a slightly different form. In order for it to be most clearly represented, we set the system in motion in a vertical plane. To make it clearer, it was first worth saying what a spring pendulum is. From the name it is clear that a spring must be present in its design. And indeed it is. Again, we have a horizontal plane on supports, to which a spring of a certain length and stiffness is suspended. To it, in turn, a weight is suspended. It can be a cylinder, a cube or another figure. It may even be some third-party item. In any case, when the system is taken out of equilibrium, it will begin to perform damped oscillations. The increase in frequency is most clearly seen in the vertical plane, without any deviation. On this experience, you can finish.
So, in their course, we found out that the period and frequency of oscillations are two physical quantities that have an inverse relationship.
Designation of quantities and dimensions
Usually, the oscillation period is denoted by the Latin letter T. Much less often, it can be denoted differently. The frequency is denoted by the letter µ (“Mu”). As we said at the very beginning, a period is nothing more than the time during which a complete oscillation occurs in the system. Then the dimension of the period will be a second. And since the period and frequency are inversely proportional, the frequency dimension will be unit divided by a second. In the record of tasks, everything will look like this: T (s), µ (1/s).
Formula for a mathematical pendulum. Task #1
As in the case with the experiments, I decided first of all to deal with the mathematical pendulum. We will not go into the derivation of the formula in detail, since such a task was not originally set. Yes, and the conclusion itself is cumbersome. But let's get acquainted with the formulas themselves, find out what kind of quantities they include. So, the formula for the period of oscillation for a mathematical pendulum is as follows:
Where l is the length of the thread, n \u003d 3.14, and g is the acceleration of gravity (9.8 m / s ^ 2). The formula should not cause any difficulties. Therefore, without additional questions Let's move on to solving the problem of determining the period of oscillation of a mathematical pendulum. A metal ball weighing 10 grams is suspended from an inextensible thread 20 centimeters long. Calculate the period of oscillation of the system, taking it for a mathematical pendulum. The solution is very simple. As in all problems in physics, it is necessary to simplify it as much as possible by discarding unnecessary words. They are included in the context in order to confuse the decisive one, but in fact they have absolutely no weight. In most cases, of course. Here it is possible to exclude the moment with “inextensible thread”. This phrase should not lead to a stupor. And since we have a mathematical pendulum, we should not be interested in the mass of the load. That is, the words about 10 grams are also simply designed to confuse the student. But we know that there is no mass in the formula, so with a clear conscience we can proceed to the solution. So, we take the formula and simply substitute the values \u200b\u200binto it, since it is necessary to determine the period of the system. Since no additional conditions were specified, we will round the values to the 3rd decimal place, as is customary. Multiplying and dividing the values, we get that the period of oscillation is 0.886 seconds. Problem solved.
Formula for a spring pendulum. Task #2
Pendulum formulas have a common part, namely 2n. This value is present in two formulas at once, but they differ in the root expression. If in the problem concerning the period of a spring pendulum, the mass of the load is indicated, then it is impossible to avoid calculations with its use, as was the case with the mathematical pendulum. But you should not be afraid. This is how the period formula for a spring pendulum looks like:
In it, m is the mass of the load suspended from the spring, k is the coefficient of spring stiffness. In the problem, the value of the coefficient can be given. But if in the formula of a mathematical pendulum you don’t really clear up - after all, 2 out of 4 values are constants - then a 3rd parameter is added here, which can change. And at the output we have 3 variables: the period (frequency) of oscillations, the coefficient of spring stiffness, the mass of the suspended load. The task can be oriented towards finding any of these parameters. Searching for a period again would be too easy, so we'll change the condition a bit. Find the stiffness of the spring if the full swing time is 4 seconds and the weight of the spring pendulum is 200 grams.
To solve any physical problem, it would be good to first make a drawing and write formulas. They are half the battle here. Having written the formula, it is necessary to express the stiffness coefficient. It is under our root, so we square both sides of the equation. To get rid of the fraction, multiply the parts by k. Now let's leave only the coefficient on the left side of the equation, that is, we divide the parts by T^2. In principle, the problem could be a little more complicated by setting not a period in numbers, but a frequency. In any case, when calculating and rounding (we agreed to round up to the 3rd decimal place), it turns out that k = 0.157 N/m.
The period of free oscillations. Free period formula
The formula for the period of free oscillations is understood to mean those formulas that we examined in the two previously given problems. They also make up an equation of free oscillations, but there we are already talking about displacements and coordinates, and this question belongs to another article.
1) Before taking on a task, write down the formula that is associated with it.
2) The simplest tasks do not require drawings, but in exceptional cases they will need to be done.
3) Try to get rid of roots and denominators if possible. An equation written in a line that does not have a denominator is much more convenient and easier to solve.
Everything on the planet has its frequency. According to one version, it is even the basis of our world. Alas, the theory is very complicated to present it within the framework of one publication, so we will consider only the frequency of oscillations as an independent action. Within the framework of the article, this physical process, its units of measurement and the metrological component will be defined. And in the end, an example of the importance of an ordinary sound in ordinary life will be considered. We learn what it is and what its nature is.
What is the oscillation frequency?
By this is meant a physical quantity that is used to characterize a periodic process, which is equal to the number of repetitions or occurrences of certain events in one unit of time. This indicator is calculated as the ratio of the number of these incidents to the period of time during which they were committed. Each element of the world has its own oscillation frequency. A body, an atom, a road bridge, a train, an airplane - they all make certain movements, which are called so. Let these processes are not visible to the eye, they are. The units of measurement in which the oscillation frequency is considered are hertz. They got their name in honor of the physicist German descent Heinrich Hertz.
Instantaneous frequency
A periodic signal can be characterized by an instantaneous frequency, which, up to a factor, is the rate of phase change. It can be represented as the sum of harmonic spectral components that have their own constant oscillations.
Cyclic oscillation frequency
It is convenient to apply it in theoretical physics, especially in the section on electromagnetism. Cyclic frequency (also called radial, circular, angular) is a physical quantity that is used to indicate the intensity of the origin of an oscillatory or rotational motion. The first is expressed in revolutions or oscillations per second. During rotational motion, the frequency is equal to the modulus of the angular velocity vector.
This indicator is expressed in radians per second. The dimension of cyclic frequency is the reciprocal of time. In numerical terms, it is equal to the number of oscillations or revolutions that occurred in the number of seconds 2π. Its introduction for use makes it possible to significantly simplify the various range of formulas in electronics and theoretical physics. The most popular use case is the calculation of the resonant cyclic frequency of an oscillating LC circuit. Other formulas can become much more complicated.
Discrete event frequency
This value means the value, which is equal to the number of discrete events that occur in one unit of time. In theory, the indicator is usually used - a second to the minus first degree. In practice, hertz is usually used to express the frequency of pulses.
Rotation frequency
It is understood as a physical quantity, which is equal to the number of complete revolutions that occur in one unit of time. The indicator is also used here - a second to the minus first degree. To indicate the work done, phrases such as revolution per minute, hour, day, month, year and others can be used.
Units
What is the frequency of oscillations measured in? If we take into account the SI system, then here the unit of measurement is hertz. It was originally introduced by the International Electrotechnical Commission back in 1930. And the 11th General Conference on Weights and Measures in 1960 consolidated the use of this indicator as a unit of SI. What was put forward as the "ideal"? They were the frequency when one cycle is completed in one second.
But what about production? Arbitrary values were fixed for them: kilocycle, megacycle per second, and so on. Therefore, picking up a device that works with an indicator in GHz (like a computer processor), you can roughly imagine how many actions it performs. It would seem how slowly time passes for a person. But technology over the same period manages to perform millions and even billions of operations per second. In one hour, the computer is already doing so many things that most people can't even imagine them in numerical terms.
Metrological aspects
The oscillation frequency has found its application even in metrology. Various devices have many functions:
- Measure the pulse frequency. They are represented by electronic counting and capacitor types.
- Determine the frequency of the spectral components. There are heterodyne and resonant types.
- Perform spectrum analysis.
- Reproduce the required frequency with a given accuracy. In this case, various measures can be applied: standards, synthesizers, signal generators and other equipment in this area.
- The indicators of the received oscillations are compared; for this purpose, a comparator or an oscilloscope is used.
Work example: sound
Everything written above can be quite difficult to understand, since we used the dry language of physics. To understand the above information, you can give an example. Everything will be detailed in it, based on the analysis of cases from modern life. To do this, consider the most famous example of vibrations - sound. Its properties, as well as the features of the implementation of mechanical elastic oscillations in a medium, are directly dependent on frequency.
Human hearing organs can pick up vibrations that are in the range from 20 Hz to 20 kHz. Moreover, with age, the upper limit will gradually decrease. If the frequency of sound oscillations falls below 20 Hz (which corresponds to mi subcontra-octave), then infrasound will be created. This type, which in most cases is not audible to us, people can still feel tactilely. When the limit of 20 kilohertz is exceeded, oscillations are generated, which are called ultrasound. If the frequency exceeds 1 GHz, then in this case we will be dealing with hypersound. If we consider such musical instrument like a piano, it can create vibrations in the range from 27.5 Hz to 4186 Hz. At the same time, it should be borne in mind that the musical sound does not consist only of the fundamental frequency - overtones and harmonics are also added to it. It all together determines the timbre.
Conclusion
As you have had the opportunity to learn, the frequency of oscillation is an extremely important component that allows our world to function. Thanks to her, we can hear, with her assistance computers work and many other useful things are carried out. But if the oscillation frequency exceeds the optimal limit, then certain destruction may begin. So, if you influence the processor so that its crystal works with twice as much performance, then it will quickly fail.
The same can be said about human life, when at a high frequency, his eardrums burst. Other negative changes will also occur with the body, which will entail certain problems, up to and including death. Moreover, due to the peculiarities of the physical nature, this process will stretch for a rather long period of time. By the way, taking this factor into account, the military is considering new opportunities for developing weapons of the future.
In principle, it coincides with the mathematical concept of the period of the function, but meaning by the function the dependence of the physical quantity that oscillates on time.
This concept in this form is applicable to both harmonic and anharmonic strictly periodic oscillations (and approximately - with one success or another - and non-periodic oscillations, at least to those close to periodicity).
In the case when we are talking about vibrations 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.).
T = 1 ν , ν = 1 T . (\displaystyle T=(\frac (1)(\nu )),\ \ \ \nu =(\frac (1)(T)).)For wave processes, the period is also obviously related to the wavelength λ (\displaystyle \lambda )
v = λ ν , T = λ v , (\displaystyle v=\lambda \nu ,\ \ \ T=(\frac (\lambda )(v)),)where v (\displaystyle v) is the wave propagation velocity (more precisely, the 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 also for linearizable systems in a linear approximation, which is often very good), there are standard relatively simple mathematical methods, allowing 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. Also applied are beats, a method of heterodyning in different types, 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 quantity).
Periods of oscillation in nature
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 bound can be given by the lifetime modern science(hundreds of years), and for the lower one - 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 ideas not only can it hardly be physically measured in any way at all, but it is also unlikely that in the more or less foreseeable future it will be possible to approach the measurement of quantities even much orders of magnitude greater, but 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 relative to axes of rotation, m (\displaystyle m)- the mass of the pendulum, l (\displaystyle l)- distance from the axis of rotation to
37. Harmonic vibrations. Amplitude, period and frequency of oscillations.
Oscillations are called processes characterized by a certain repeatability over time. The process of propagation of oscillations in space is called a wave. It can be said without exaggeration that we live in a world of vibrations and waves. Indeed, a living organism exists thanks to the periodic beating of the heart, our lungs fluctuate when we breathe. A person hears and speaks due to vibrations of his eardrums and vocal cords. Light waves (fluctuations in electric and magnetic fields) allow us to see. Modern technology also makes extremely wide use of oscillatory processes. Suffice it to say that many engines are associated with oscillations: the periodic movement of pistons in internal combustion engines, the movement of valves, etc. Other important examples are alternating current, electromagnetic oscillations in an oscillatory circuit, radio waves, etc. As can be seen from the above examples, the nature of the oscillations is different. However, they are reduced to two types - mechanical and electromagnetic oscillations. It turned out that, despite the difference in the physical nature of the oscillations, they are described by the same mathematical equations. This allows us to single out as one of the branches of physics the doctrine of oscillations and waves, in which a unified approach to the study of oscillations of various physical nature is carried out.
Any system capable of oscillating or in which oscillations can occur is called oscillatory. Oscillations occurring in an oscillatory system, taken out of equilibrium and presented to itself, are called free oscillations. Free oscillations are damped, since the energy imparted to the oscillatory system is constantly decreasing.
Oscillations are called harmonic, in which any physical quantity describing the process changes with time according to the law of cosine or sine:
Let us find out the physical meaning of the constants A, w, a entering this equation.
The constant A is called the amplitude of the oscillation. The amplitude is highest value, which can take on a fluctuating value. By definition, it is always positive. The expression wt + a, which is under the cosine sign, is called the oscillation phase. It allows you to calculate the value of a fluctuating quantity at any time. The constant value a is the value of the phase at time t =0 and is therefore called the initial phase of the oscillation. The value of the initial phase is determined by the choice of the beginning of the countdown. The value of w is called the cyclic frequency, the physical meaning of which is associated with the concepts of the period and frequency of oscillations. The period of undamped oscillations is called smallest gap the time after which the oscillating value takes its previous value, or briefly - the time of one complete oscillation. The number of oscillations per unit time is called the oscillation frequency. The frequency v is related to the period T of oscillations by the relation v=1/T
The oscillation frequency is measured in hertz (Hz). 1 Hz is the frequency of a periodic process in which one oscillation occurs in 1 s. Let's find the relationship between the frequency and the cyclic frequency of oscillation. Using the formula, we find the values of the fluctuating quantity at the moments of time t=t 1 and t=t 2 =t 1 +T, where T is the oscillation period.
According to the definition of the oscillation period, This is possible if because the cosine is a periodic function with a period of 2p radians. From here. We get . The physical meaning of the cyclic frequency follows from this relation. It shows how many oscillations are made in 2p seconds.
Free vibrations of the oscillatory system are damped. However, in practice, there is a need to create undamped oscillations, when energy losses in the oscillatory system are compensated by external energy sources. In this case, forced oscillations occur in such a system. Forced oscillations are those that occur under the influence of a periodically changing influence, and aces of the influence are called forcing. Forced oscillations occur with a frequency equal to the frequency of the forcing actions. The amplitude of forced oscillations increases as the frequency of forcing actions approaches the natural frequency of the oscillatory system. She reaches maximum value when the specified frequencies are equal. The phenomenon of a sharp increase in the amplitude of forced oscillations, when the frequency of the forcing actions is equal to the natural frequency of the oscillatory system, is called resonance.
The phenomenon of resonance is widely used in technology. It can be both beneficial and harmful. So, for example, the phenomenon of electrical resonance plays a useful role in tuning a radio receiver to the desired radio station. By changing the values of inductance and capacitance, it is possible to ensure that the natural frequency of the oscillatory circuit coincides with the frequency of electromagnetic waves emitted by any radio station. As a result, resonant oscillations of a given frequency will arise in the circuit, while the amplitudes of oscillations created by other stations will be small. This will tune the radio to the desired station.
38. Mathematical pendulum. The period of oscillation of a mathematical pendulum.
39. Fluctuation of the load on the spring. The transformation of energy during vibrations.
40. Waves. Transverse and longitudinal waves. Velocity and wavelength.
41. Free electromagnetic oscillations in the circuit. Energy conversion in an oscillatory circuit. Energy transformation.
Periodic or almost periodic changes in charge, current and voltage are called electrical oscillations.
Getting electrical vibrations is almost as simple as making a body oscillate by hanging it on a spring. But observing electrical vibrations is no longer so easy. After all, we do not directly see either the recharge of the capacitor or the current in the coil. In addition, oscillations usually occur at a very high frequency.
Observe and investigate electrical oscillations using an electronic oscilloscope. The horizontally deflecting plates of the cathode ray tube of the oscilloscope are supplied with an alternating sweep voltage Up of a “sawtooth” shape. The voltage increases relatively slowly, and then decreases very sharply. The electric field between the plates causes the electron beam to run through the screen in a horizontal direction at a constant speed and then return almost instantly. After that, the whole process is repeated. If we now attach vertical deflection plates to the condenser, then the voltage fluctuations during its discharge will cause the beam to oscillate in the vertical direction. As a result, a time “sweep” of oscillations is formed on the screen, quite similar to that drawn by a pendulum with a sandbox on a moving sheet of paper. The fluctuations decay over time
These vibrations are free. They arise after the capacitor is given a charge that brings the system out of equilibrium. The charging of the capacitor is equivalent to the deviation of the pendulum from the equilibrium position.
IN electrical circuit forced electrical oscillations can also be obtained. Such oscillations appear in the presence of a periodic electromotive force in the circuit. A variable induction emf occurs in a wire frame of several turns when it rotates in a magnetic field (Fig. 19). In this case, the magnetic flux penetrating the frame changes periodically, in accordance with the law electromagnetic induction the emerging EMF of induction also changes periodically. When the circuit is closed, an alternating current will flow through the galvanometer and the needle will begin to oscillate around the equilibrium position.
2. Oscillatory circuit. The simplest system, in which free electrical oscillations can occur, consists of a capacitor and a coil attached to the capacitor plates (Fig. 20). Such a system is called an oscillatory circuit.
Consider why oscillations occur in the circuit. We charge the capacitor by connecting it for a while to the battery using a switch. In this case, the capacitor will receive energy:
where qm is the charge of the capacitor, and C is its capacitance. Between the capacitor plates there will be a potential difference Um.
Let's move the switch to position 2. The capacitor will begin to discharge, and a electricity. The current does not immediately reach its maximum value, but increases gradually. This is due to the phenomenon of self-induction. When a current appears, an alternating magnetic field is created. This alternating magnetic field generates a vortex electric field in the conductor. Vortex electric field with increasing magnetic field directed against the current and prevents its instantaneous increase.
As the capacitor discharges, the energy of the electric field decreases, but at the same time the energy of the magnetic field of the current increases, which is determined by the formula: fig.
where i is the current strength,. L is the inductance of the coil. At the moment when the capacitor is completely discharged (q=0), the energy of the electric field will become zero. The energy of the current (the energy of the magnetic field) according to the law of conservation of energy will be maximum. Therefore, at this moment, the current will also reach its maximum value
Despite the fact that by this time the potential difference at the ends of the coil becomes equal to zero, the electric current cannot stop immediately. This is prevented by the phenomenon of self-induction. As soon as the current strength and the magnetic field created by it begin to decrease, a vortex electric field arises, which is directed along the current and supports it.
As a result, the capacitor is recharged until the current, gradually decreasing, becomes equal to zero. The energy of the magnetic field at this moment will also be equal to zero, and the energy of the electric field of the capacitor will again become maximum.
After that, the capacitor will be recharged again and the system will return to its original state. If there were no energy losses, then this process would continue indefinitely. The oscillations would be undamped. At intervals equal to the period of oscillation, the state of the system would be repeated.
But in reality, energy losses are inevitable. Thus, in particular, the coil and connecting wires have resistance R, and this leads to a gradual transformation of the energy of the electromagnetic field into the internal energy of the conductor.
With oscillations occurring in the circuit, there is a transformation of the energy of the magnetic field into the energy of the electric field and vice versa. Therefore, these vibrations are called electromagnetic. The period of the oscillatory circuit is found by the formula.
