Period and frequency of oscillations. What is oscillation frequency? What is the period of oscillation
§ 1.6. Mechanical vibrations
Oscillatory processes are very common in the nature and technology around us. A significant part of mechanical movements is the movement of machines operating cyclically; almost all acoustic phenomena; alternating current, used in everyday life and in various technical devices; radio engineering and part of electronics; all wave optics; wave properties of particles - this is not a complete list of phenomena and technical applications described in the language of oscillatory and wave processes. After all, our hearts beat; our lungs fluctuate when we breathe; we shiver when we are cold; We can hear and speak thanks to the vibrations of the eardrums and vocal cords. The light waves that allow us to see are oscillatory in nature. When we walk, our legs oscillate. Even the atoms that make us up vibrate. If we broadly interpret the term “oscillations,” it immediately becomes obvious that many events in everyday life have an extraordinary cyclical nature. The world we live in is surprisingly prone to fluctuation. That is why oscillatory motion is given special attention in physics and technology.
In addition, periodic non-harmonic motion can be reduced to the sum of harmonic motions, and these component motions are accessible to direct observation using modern equipment. Moreover, there is equipment that allows you to add specified harmonic movements and thus obtain periodic movements of a complex nature.
In the process of development of science, a powerful and convenient mathematical apparatus has been created for describing and studying periodic movements of various physical natures.
Oscillations are movements, processes, changes in state that are characterized by a certain repeatability in time of the values of physical quantities that determine this movement, process or state.
Oscillation is called periodic, if the values of quantities changing during the oscillation process are repeated at regular intervals. Oscillation period T- this is the minimum period of time through which certain states of the system are repeated (the time during which one complete oscillation occurs). The period is measured in seconds.
Oscillation frequency(linear frequency) is a scalar physical quantity equal to the number of oscillations performed by the system per unit time. The vibration frequency is measured in Hertz (Hz).
If over some time t the system performs N oscillations, then 
And 
. It follows that 
And 
.
Harmonic vibrations
Among the various periodic movements, harmonic oscillatory movement occupies a special place. Harmonic are called oscillations in which the quantity x of interest to us (for example, linear or angular displacement from an equilibrium position, speed, acceleration, charge, voltage, etc.) changes with time t according to the law of cosine or sine, that is
The cyclic frequency is related to the linear frequency and period by the following relations 
.
WITH 
the speed and acceleration of the body also change according to the harmonic law. Having differentiated equation (1) with respect to time, we find the rate of change of the value x - 
and acceleration 
:
In this case, the maximum value of the speed of the oscillating body is V max = Aω 0, the maximum value of the acceleration module is a max = Aω 0 2.
Kinetic energy of an oscillating body W k = ½mv 2 = ½mA 2 ω 0 2 sin 2 (ω 0 t+φ).
Potential energy (taking into account that the force is quasi-elastic) W p = ½ kx 2 = ½ kA 2 cos 2 (ω 0 t+φ).
The total energy of the system during harmonic oscillations is W= W k + W p =½ kA 2 = ½ mω 0 2 A 2.
The figure shows graphs of the time dependence of displacement x, speed V, acceleration a, kinetic W k and potential W p energy of harmonic oscillations at the initial phase φ = 0. The figure shows the frequency of change of kinetic W k and potential W p energy during harmonic oscillations twice the frequency of displacement, velocity and acceleration changes.
Comparing equations (1) and (2), we see that 
, or
. (3)
This linear homogeneous differential equation of the second order is called harmonic equation.
An oscillatory system that performs harmonic oscillations is called a harmonic oscillator. If an oscillatory system performing harmonic oscillations has one degree of freedom (one coordinate is sufficient to characterize the position), then such a system is called a linear harmonic oscillator.
To determine the nature of the motion of a mechanical system, an equation of motion of the system is drawn up (based on the laws of dynamics or the law of conservation of energy). If the equation is reduced to the form (3), then we can unambiguously state that this system performs a harmonic oscillation, the natural frequency 
which is equal to the square root of the coefficient of x(t). Let us use this method to determine the cyclic frequencies and periods of oscillation of spring and mathematical pendulums.
Let's first consider a spring pendulum (Figure 1 b). Let a body suspended from a spring be pulled away from the equilibrium position by a distance x (Fig. 1.c), and then left to its own devices. The force of gravity and elasticity act on the body. Under the influence of these forces, the body moves with acceleration. Let's write the equation of Newton's second law for this case (Fig. 1.c)
.
This is an equation in projection onto the OX axis and taking into account the fact that for one-dimensional motion acceleration is the second derivative of the coordinate with respect to time, that is 
, will sign up
The magnitude of the elastic force 
, acting on the body by mass m, we find using the formula of Hooke’s law
After substituting (5) into (4) we get
The amount of tension of the spring in the equilibrium position 
(Fig. 1.a and 1.b) we find from the equation of Newton’s second law for a stationary body suspended from a spring 
,
, (7)
,
(8)
From (7) and (8) it follows that
.
(9)
After substituting (9) into (6) and bringing similar terms, we obtain: 
, or
(10)
Comparing equations (3) and (10), we find that for a spring pendulum 
.
.
(11)
Similar reasoning can be carried out for a mathematical pendulum (Fig. 2) and show that 
.
.
(12)
A mathematical pendulum is a material point on a weightless and inextensible thread of length. In harmonic oscillations, the displacement of the pendulum from the equilibrium position x is much less than the length of the thread x<<
, therefore, for the angle of deviation of the thread from the vertical 
there is a relation
Consequently, Newton’s second law for a material point of mass m: ma = F can be written in the form 
, where is the acceleration of the point, F = mg sin 
=mg - restoring force. The minus sign on the right side means that the restoring force is directed opposite to the displacement x.
Thus, the differential equation of harmonic oscillations of a mathematical pendulum 
Comparing this equation with equation (3), we obtain formulas for the natural frequency and period of oscillation of a mathematical pendulum 
And 
.
A physical pendulum is an absolutely rigid body that oscillates about the horizontal axis O, which does not pass through the center of mass of the pendulum C. The basic equation for the dynamics of rotational motion for a pendulum is Jε = M, where J is the moment of inertia of the pendulum about the horizontal axis passing through point O. Angular acceleration pendulum ε 
. The moment of gravity of the pendulum relative to the horizontal axis passing through the point OM = mgd sinφ, where m is the mass of the pendulum, d = CO is the distance from the axis to the center of mass of the pendulum C. At small angles of deviation of the pendulum from the vertical, we can assume that 
Substituting everything into the equation of Newton's second law, we get 
.
A minus sign means that the moment of the restoring force is opposite to the angular displacement. From here we get 
This is the differential equation of harmonic oscillations of a physical pendulum. From a comparison of this equation with equation (3), we find the oscillation period of the physical pendulum 
Reduced length of a physical pendulum 
- this is the length of the thread of a mathematical pendulum, whose period of oscillation coincides with the period of a given physical pendulum. 
Oscillations are called own, if they occur due to the initially imparted energy in the subsequent absence of external influences on the oscillatory system.
Damped oscillations
All real oscillatory systems are dissipative 1. The energy of mechanical oscillations of the system is spent over time on work against friction forces, so natural oscillations always dampen - their amplitude gradually decreases. Energy loss also occurs during deformations of bodies, since completely elastic bodies do not exist, and deformations of not completely elastic bodies are accompanied by a partial transition of mechanical energy into the energy of chaotic thermal motion of particles of these bodies.
In many cases, as a first approximation, we can assume that at low speeds of movement, the forces causing the damping of mechanical vibrations are proportional to the magnitude of the speed. We will call these forces, regardless of their origin, the forces of friction or resistance and calculate them using the following formula: 
. Here r is the resistance coefficient of the medium, 
– speed of body movement. The minus sign indicates that friction forces are always directed in the direction opposite to the direction of movement of the body.
Let us write down the equation of Newton's second law for damped rectilinear oscillations of a spring pendulum
Here: m is the mass of the load, k is the spring stiffness, 
– projection of velocity onto the OX axis, 
– projection of acceleration onto the OX axis. Let's divide both sides of equation (13) by massm and rewrite it in the form:
.
(14)
Let us introduce the following notation:
,
(15)
.
(16)
Let's call 
attenuation coefficient, and 
we previously called the natural cyclic frequency. Taking into account the introduced notations (15 and 16), equation (14) will be written
.
(17)
This is a differential equation of damped oscillations of any nature. The form of the solution to this second-order linear differential equation depends on the relationship between the quantity 
– natural frequency of undamped oscillations and damping coefficient 
.
If the friction is very high (in this case 
), then the system, removed from the equilibrium position, returns to it without oscillating (“crawls”). This movement (curve 2 in Fig. 3) is called aperiodic.
If at the initial moment the system with high friction 
is in an equilibrium position and is given a certain initial speed 
, then the system reaches the greatest deviation from the equilibrium position 
, stops and after that the displacement asymptotically tends to zero (Fig. 4).
Fig.3 Fig.4
If the system is taken out of equilibrium under the condition 
and released without an initial velocity, then the system also does not pass the equilibrium position. But in this case, the time of practical approach to it turns out to be less than in the case of a large 
friction (curve 1 in Fig. 3). This mode is called critical and is sought after when using various measuring instruments (for the fastest reading).
at low friction (in this case 
) the movement is oscillatory in nature (Fig. 5) and the solution to equation (17) has the form:
(19)
describes a change amplitudes of damped oscillations with time. The amplitude of damped oscillations decreases over time (Fig. 5) and the faster, the higher the drag coefficient and the lower the mass of the oscillating body, that is, the lower the inertia of the system.
R 
is.5
Size
(20)
called the cyclic frequency of damped oscillations. Damped oscillations are non-periodic oscillations, since they never repeat, for example, the maximum values of displacement, speed and acceleration. Therefore call 
frequency can only be conditional in the sense that it shows how many times per 
seconds, the oscillating system passes through the equilibrium position. For the same reason, the value
(21)
can be called conditional period of damped oscillations.
To characterize the attenuation, we introduce the following quantities:
logarithmic damping decrement;
relaxation time;
quality factor
The ratio of any two consecutive displacements separated in time by one period is called damping decrement.
Logarithmic damping decrement 
is the natural logarithm of the ratio of the amplitude values of damped oscillations at times t and t+T (the natural logarithm of the ratio of any two consecutive displacements separated in time by one period):
Because the 
and then 
.
Let us use the formula for the dependence of amplitude on time (19) and obtain
Let's find out the physical meaning of quantities 
And 
. Let us denote by 
the period of time during which the amplitude of damped oscillations decreases by a factor of e and we will call it relaxation time. Then 
.it follows that
Basic provisions:
Oscillatory motion- a movement that repeats exactly or approximately at regular intervals.
Oscillations in which the fluctuating quantity changes over time according to the law of sine or cosine are harmonic.
Period oscillation T is the shortest period of time after which the values of all quantities characterizing oscillatory motion are repeated. During this period of time, one complete oscillation occurs.
Frequency Periodic oscillations are the number of complete oscillations that occur per unit time. .
Cyclic(circular) frequency of oscillations is the number of complete oscillations that occur in 2π units of time.
Harmonic oscillations are oscillations in which the oscillating quantity x changes over time according to the law:
where A, ω, φ 0 are constant values.
A > 0 – a value equal to the largest absolute value of the fluctuating quantity x and is called amplitude hesitation.
The expression determines the value of x at a given time and is called phase hesitation.
At the moment the time count begins (t = 0), the oscillation phase is equal to the initial phase φ 0.
Math pendulum- this is an idealized system, which is a material point suspended on a thin, weightless and inextensible thread.
Period of free oscillation of a mathematical pendulum: .
Spring pendulum- a material point attached to a spring and capable of oscillating under the influence of elastic force.
Period of free oscillation of a spring pendulum: .
Physical pendulum is a rigid body capable of rotating around a horizontal axis under the influence of gravity.
Period of oscillation of a physical pendulum: .
Fourier's theorem: any real periodic signal can be represented as a sum of harmonic oscillations with different amplitudes and frequencies. This sum is called the harmonic spectrum of a given signal.
Forced are called oscillations that are caused by the action of external forces F(t) on the system, periodically changing over time.
The force F(t) is called the disturbing force.
Fading oscillations are vibrations whose energy decreases over time, which is associated with a decrease in the mechanical energy of the oscillating system due to the action of friction and other resistance forces.
If the frequency of oscillations of the system coincides with the frequency of the disturbing force, then the amplitude of oscillations of the system increases sharply. This phenomenon is called resonance.
The propagation of oscillations in a medium is called a wave process, or wave.
The wave is called transverse, if the particles of the medium oscillate in a direction perpendicular to the direction of propagation of the wave.
The wave is called longitudinal, if the oscillating particles move in the direction of wave propagation. Longitudinal waves propagate in any medium (solid, liquid, gaseous).
Propagation of transverse waves is possible only in solids. In gases and liquids that do not have an elastic shape, the propagation of transverse waves is impossible.
Wavelength is the distance between the nearest points oscillating in the same phase, i.e. the distance a wave travels in one period.
Wave speed V is the speed of propagation of vibrations in the medium.
Period and frequency of a wave - the period and frequency of oscillations of particles of the medium.
Wavelengthλ – the distance over which the wave propagates in one period: .
Sound– an elastic longitudinal wave propagating from a sound source in a medium.
The perception of sound waves by a person depends on the frequency; audible sounds range from 16 Hz to 20,000 Hz.
Sound in air is a longitudinal wave.
Pitch determined by the frequency of sound vibrations, volume sound - its amplitude.
Control questions:
1. What motion is called harmonic oscillation?
2. Give definitions of quantities characterizing harmonic oscillations.
3. What is the physical meaning of the oscillation phase?
4. What is called a mathematical pendulum? What is its period?
5. What is called a physical pendulum?
6. What is resonance?
7. What is called a wave? Define transverse and longitudinal waves.
8. What is wavelength called?
9. What is the frequency range of sound waves? Can sound travel in a vacuum?
Complete the tasks:
Harmonic oscillations are oscillations performed according to the laws of sine and cosine. The following figure shows a graph of changes in the coordinates of a point over time according to the cosine law.
picture
Oscillation amplitude
The amplitude of a harmonic vibration is the greatest value of the displacement of a body from its 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 sine and cosine can take values in the range from -1 to 1, the equation must contain a factor Xm, expressing 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 to complete one complete oscillation. The period of oscillation is designated by the letter T. The units of measurement of the period correspond to the units of time. That is, in SI these are seconds.
Oscillation frequency is the number of oscillations performed per unit of time. The oscillation frequency is designated by the letter ν. The oscillation frequency can be expressed in terms of the oscillation period.
ν = 1/T.
Frequency units are 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 quantity is called the cyclic frequency of oscillations. In some literature the name circular frequency appears. The natural frequency of an oscillatory system is the frequency of free oscillations.
The frequency of natural oscillations is calculated using the formula:
The frequency of natural vibrations depends on the properties of the material and the mass of the load. The greater the spring stiffness, the greater the frequency of its own vibrations. 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 thrown out of balance. The greater the mass of a body, the slower the speed of this body will change.
Free oscillation period:
T = 2*pi/ ω0 = 2*pi*√(m/k)
It is noteworthy that at small angles of deflection 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 equal
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 increasing length of the pendulum thread. The longer the length, the slower the body will vibrate.
The period of oscillation does not depend at all on the mass of the load. But it depends on the acceleration of free fall. 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.
In principle, it coincides with the mathematical concept of the period of a function, but by means of a function we mean the dependence of a physical quantity that oscillates on time.
This concept in this form is applicable to both harmonic and anharmonic strictly periodic oscillations (and approximately - with varying degrees of success - and non-periodic oscillations, at least those 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, with greater or less accuracy and usefulness, 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)- the speed of wave propagation (more precisely, the phase speed).
In quantum physics the period of oscillation is directly related to energy (since in quantum physics the energy of an object - for example, a particle - is the frequency of oscillation of its wave function).
Theoretical finding Determining the period of oscillation of a particular physical system comes down, as a rule, to finding a solution to the dynamic equations (equations) that describe this system. For the category of linear systems (and approximately for linearizable systems in the 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, strobotachometers, and oscilloscopes are used. Also used are beats, heterodyning method in different types, and 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 required, specially developed for a specific difficult case (the difficulty can be both the measurement of time itself, especially if we are talking about extremely short or, conversely, very large times, and the difficulty of observing a fluctuating value).
Periods of oscillations in nature
An idea of the periods of oscillations of various physical processes is given by the article Frequency Intervals (considering that the period in seconds is the reciprocal of the frequency in hertz).
Some idea of the magnitude 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 sound audible by humans 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 at extremely large and extremely small periods of oscillation, measurement methods tend to become increasingly indirect (even to the point of smoothly flowing into theoretical extrapolations), it is difficult to name clear upper and lower limits for the period of oscillation measured directly. Some estimate for the upper limit can be given by the lifetime of modern science (hundreds of years), and for the lower limit - the period of oscillations of the wave function of the heaviest currently known particle ().
Anyway border below can serve as the Planck time, which is so small that, according to modern concepts, not only can it hardly be physically measured at all, but it is also unlikely that in the more or less foreseeable future it will be possible to get closer to measuring quantities even of much greater orders of magnitude, and border on top- the existence of the Universe is more than ten billion years.
Periods of oscillations of the simplest physical systems
Spring pendulum
Math pendulum
T = 2 π l g (\displaystyle T=2\pi (\sqrt (\frac (l)(g))))
Where l (\displaystyle l)- length of suspension (for example, thread), g (\displaystyle g)- acceleration of gravity .
The period of small oscillations (on Earth) of a mathematical pendulum 1 meter long with good accuracy is 2 seconds.
Physical pendulum
T = 2 π J m g l (\displaystyle T=2\pi (\sqrt (\frac (J)(mgl))))
Where J (\displaystyle J)- moment of inertia of the pendulum relative to the axis of rotation, m (\displaystyle m)- 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 processes characterized by a certain repeatability over time. The process of propagation of vibrations in space is called a wave. It is no exaggeration to say 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 vibrate when breathing. A person hears and speaks due to vibrations of his eardrums and vocal cords. Light waves (oscillations of electric and magnetic fields) allow us to see. Modern technology also makes extremely extensive use of oscillatory processes. Suffice it to say that many engines are associated with vibrations: periodic movement of pistons in internal combustion engines, movement of valves, etc. Other important examples are alternating current, electromagnetic oscillations in an oscillating circuit, radio waves, etc. As can be seen from the above examples, the nature of the oscillations is different. However, they come down to two types - mechanical and electromagnetic vibrations. It turned out that, despite the difference in the physical nature of the vibrations, they are described by the same mathematical equations. This allows us to single out the study of oscillations and waves as one of the branches of physics, which implements a unified approach to the study of oscillations of various physical natures.
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 left to itself are called free oscillations. Free oscillations are damped, since the energy imparted to the oscillatory system constantly decreases.
Harmonic oscillations are those in which any physical quantity describing the process changes over time according to the law of cosine or sine:
Let us find out the physical meaning of the constants A, w, a included in this equation.
The constant A is called the amplitude of the oscillation. Amplitude is the largest value that an oscillating quantity can take. By definition, it is always positive. The expression wt+a 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 represents the phase value 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 start of the time count. The quantity w is called cyclic frequency, the physical meaning of which is associated with the concepts of period and frequency of oscillations. The period of undamped oscillations is the shortest period of time after which the oscillating quantity takes on its previous value, or in short - the time of one complete oscillation. The number of oscillations performed per unit time is called the oscillation frequency. Frequency v is related to the period T of oscillations by the ratio v=1/T
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 connection between frequency and cyclic frequency of oscillation. Using the formula, we find the values of the oscillating quantity at times 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 , since cosine is a periodic function with a period of 2p radians. From here. We get. From this relationship follows the physical meaning of cyclic frequency. It shows how many oscillations occur in 2p seconds.
Free oscillations 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 arise in such a system. Oscillations that occur under the influence of a periodically changing influence are called forced, while those of influence are called forcing. Forced oscillations occur with a frequency equal to the frequency of the forcing influences. The amplitude of forced oscillations increases as the frequency of the forcing influences approaches the natural frequency of the oscillatory system. It reaches its maximum value when the indicated frequencies are equal. The phenomenon of a sharp increase in the amplitude of forced oscillations, when the frequency of the forcing influences 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 useful and harmful. For example, the phenomenon of electrical resonance plays a useful role when 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 of this, resonant oscillations of a given frequency will appear in the circuit, while the amplitudes of the oscillations created by other stations will be small. This leads to tuning the radio to the desired station.
38. Mathematical pendulum. Period of oscillation of a mathematical pendulum.
39. Oscillation of a load on a spring. Conversion of energy during vibrations.
40. Waves. Transverse and longitudinal waves. Speed and wavelength.
41. Free electromagnetic oscillations in a circuit. Conversion of energy in an oscillatory circuit. Transformation of energy.
Periodic or almost periodic changes in charge, current and voltage are called electrical oscillations.
Producing electrical vibrations is almost as simple as making a body vibrate by hanging it on a spring. But observing electrical vibrations is no longer so easy. After all, we do not directly see either the recharging of the capacitor or the current in the coil. In addition, oscillations usually occur with a very high frequency.
Observe and study electrical vibrations using an electronic oscilloscope. An alternating sweep voltage Up of a “sawtooth” shape is supplied to the horizontal deflection plates of the cathode ray tube of the oscilloscope. The tension increases relatively slowly and then decreases very sharply. The electric field between the plates causes the electron beam to travel horizontally across the screen at a constant speed and then return almost instantly. After this, the whole process is repeated. If we now attach vertical deflection plates to the capacitor, 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 the one drawn by a pendulum with a sandbox on a moving sheet of paper. Vibrations fade over time
These vibrations are free. They arise after a charge is imparted to the capacitor, which takes the system out of equilibrium. Charging the capacitor is equivalent to the deviation of the pendulum from its equilibrium position.
Forced electrical oscillations can also be obtained in an electrical circuit. Such oscillations appear when there is a periodic electromotive force in the circuit. An alternating induced emf arises 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 of electromagnetic induction, the resulting induced emf 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 connected to the capacitor plates (Fig. 20). Such a system is called an oscillatory circuit.
Let's consider why oscillations occur in the circuit. Let's charge the capacitor by connecting it to the battery for a while using a switch. In this case, the capacitor will receive energy:
where qm is the charge of the capacitor, and C is its electrical capacity. A potential difference Um will arise between the plates of the capacitor.
Let's move the switch to position 2. The capacitor will begin to discharge, and an electric current will appear in the circuit. The current does not immediately reach its maximum value, but increases gradually. This is due to the phenomenon of self-induction. When current appears, an alternating magnetic field appears. This alternating magnetic field generates an eddy electric field in the conductor. When the magnetic field increases, the vortex electric field is 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 becomes zero. The current energy (magnetic field energy), 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 moment the potential difference at the ends of the coil becomes zero, the electric current cannot stop immediately. This is prevented by the phenomenon of self-induction. As soon as the strength of the current and the magnetic field it creates begin to decrease, an eddy electric field appears, 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 zero, and the energy of the electric field of the capacitor will again become maximum.
After this, the capacitor will be recharged again and the system will return to its original state. If there were no energy losses, this process would continue indefinitely. The oscillations would be undamped. At intervals equal to the period of oscillation, the state of the system would repeat itself.
But in reality, energy losses are inevitable. Thus, in particular, the coil and connecting wires have a resistance R, and this leads to the gradual conversion of the energy of the electromagnetic field into the internal energy of the conductor.
When oscillations occur in the circuit, the transformation of magnetic field energy into electric field energy and vice versa is observed. Therefore, these oscillations are called electromagnetic. The period of the oscillatory circuit is found by the formula.
