But what we mean by function is 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: (although others can be used, most often it is , sometimes, etc.).
The period of oscillation is related by the relationship of mutual reciprocity with frequency:
For wave processes, the period is also obviously related to the wavelength
where is 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 (difficulty can arise from 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 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 many orders of magnitude smaller. A 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
where is the length of the suspension (for example, a thread), is the acceleration of free fall.
The period of oscillation (on Earth) of a mathematical pendulum 1 meter long is, with good accuracy, 2 seconds.
Physical pendulum
where is the moment of inertia of the pendulum relative to the axis of rotation, is the mass of the pendulum, is the distance from the axis of rotation to the center of mass.
Torsion pendulum
where is the moment of inertia of the body, and is the rotational stiffness coefficient of the pendulum.
Electrical Oscillating (LC) Circuit
Oscillation period of the electric oscillatory circuit:
where is the inductance of the coil, is the capacitance of the capacitor.
This formula was derived in 1853 by the English physicist W. Thomson.
Notes
Links
- Oscillation period- article from the Great Soviet Encyclopedia
Wikimedia Foundation. 2010.
See what “Oscillation period” is in other dictionaries:
period of oscillation- period The shortest period of time through which the state of a mechanical system, characterized by the values of generalized coordinates and their derivatives, is repeated. [Collection of recommended terms. Issue 106. Mechanical vibrations. Academy of Sciences... ... Technical Translator's Guide
Period (oscillations)- PERIOD of oscillations, the shortest period of time after which an oscillating system returns to the same state in which it was at the initial moment, chosen arbitrarily. The period is the reciprocal of the oscillation frequency. Concept... ... Illustrated Encyclopedic Dictionary
The shortest period of time, after which the system performing oscillations returns again to the same state in which it was at the beginning. moment chosen arbitrarily. Strictly speaking, the concept of “P. To." applicable only when the values of k.l.... ... Physical encyclopedia
The shortest period of time after which an oscillating system returns to its original state. The oscillation period is the reciprocal of the oscillation frequency... Big Encyclopedic Dictionary
period of oscillation- period of oscillation; period The shortest period of time through which the state of a mechanical system is repeated, characterized by the values of generalized coordinates and their derivatives... Polytechnic terminological explanatory dictionary
Oscillation period- 16. Oscillation period The shortest time interval through which, during periodic oscillations, each value of the oscillating quantity is repeated Source ... Dictionary-reference book of terms of normative and technical documentation
The shortest period of time after which an oscillating system returns to its original state. The oscillation period is the reciprocal of the oscillation frequency. * * * PERIOD OF OSCILLATIONS PERIOD OF OSCILLATIONS, the shortest period of time through which... ... encyclopedic Dictionary
period of oscillation- virpesių periodas statusas T sritis automatika atitikmenys: engl. oscillation period; period of oscillations; period of vibrations vok. Schwingungsdauer, m; Schwingungsperiode, f; Schwingungszeit, f rus. period of oscillation, m pranc. période d… … Automatikos terminų žodynas
period of oscillation- virpesių periodas statusas T sritis Standartizacija ir metrologija apibrėžtis Mažiausias laiko tarpas, po kurio pasikartoja periodiškai kintančių dydžių vertės. atitikmenys: engl. vibration period vok. Schwingungsdauer, f; Schwingungsperiode, f… … Penkiakalbis aiškinamasis metrologijos terminų žodynas
1. Let us remember what is called the frequency and period of oscillations.
The time it takes a pendulum to complete one swing is called the period of oscillation.
The period is designated by the letter T and measured in seconds(With).
The number of complete oscillations in one second is called the oscillation frequency. Frequency is indicated by the letter n .
1 Hz = .
Unit of vibration frequency in Ш - hertz (1 Hz).
1 Hz - this is the frequency of such oscillations at which one complete oscillation occurs in 1 s.
The oscillation frequency and period are related by the relation:
n = .
2. The period of oscillation of the oscillatory systems we considered - mathematical and spring pendulums - depends on the characteristics of these systems.
Let's find out what the period of oscillation of a mathematical pendulum depends on. To do this, let's do an experiment. We will change the length of the thread of a mathematical pendulum and measure the time of several complete oscillations, for example 10. In each case, we will determine the period of oscillation of the pendulum by dividing the measured time by 10. Experience shows that the longer the length of the thread, the longer the period of oscillation.
Now let's place a magnet under the pendulum, thereby increasing the force of gravity acting on the pendulum, and measure the period of its oscillations. Note that the period of oscillation will decrease. Consequently, the period of oscillation of a mathematical pendulum depends on the acceleration of gravity: the greater it is, the shorter the period of oscillation.
The formula for the period of oscillation of a mathematical pendulum is:
|
T = 2p, |
Where l- length of the pendulum thread, g- acceleration of gravity.
3. Let us determine experimentally what determines the period of oscillation of a spring pendulum.
We will suspend weights of different masses from the same spring and measure the period of oscillation. Note that the greater the mass of the load, the longer the period of oscillation.
Then we will suspend the same load from springs of different stiffnesses. Experience shows that the greater the spring stiffness, the shorter the period of oscillation of the pendulum.
The formula for the period of oscillation of a spring pendulum is:
|
T = 2p, |
Where m- mass of cargo, k- spring stiffness.
4. The formulas for the period of oscillation of pendulums include quantities that characterize the pendulums themselves. These quantities are called parameters oscillatory systems.
If the parameters of the oscillatory system do not change during the oscillation process, then the period (frequency) of oscillation remains unchanged. However, in real oscillatory systems, friction forces act, so the period of real free oscillations decreases over time.
If we assume that there is no friction and the system performs free oscillations, then the period of oscillations will not change.
The free vibrations that a system could perform in the absence of friction are called natural vibrations.
The frequency of such oscillations is called natural frequency. It depends on the parameters of the oscillatory system.
Self-test questions
1. What is the period of oscillation of a pendulum called?
2. What is the frequency of oscillation of a pendulum? What is the unit of vibration frequency?
3. On what quantities and how does the period of oscillation of a mathematical pendulum depend?
4. On what quantities and how does the period of oscillation of a spring pendulum depend?
5. What vibrations are called natural vibrations?
Task 23
1. What is the period of oscillation of a pendulum if it completes 20 complete oscillations in 15 s?
2. What is the oscillation frequency if the oscillation period is 0.25 s?
3. What must be the length of the pendulum in a pendulum clock for its period of oscillation to be equal to 1 s? Count g= 10 m/s 2 ; p2 = 10.
4. What is the period of oscillation of a pendulum whose thread is 28 cm long on the Moon? The acceleration of gravity on the Moon is 1.75 m/s 2 .
5. Determine the period and frequency of oscillation of a spring pendulum if its spring stiffness is 100 N/m and the mass of the load is 1 kg.
6. How many times will the vibration frequency of a car on springs change if a load is placed in it, the mass of which is equal to the mass of the unloaded car?
Laboratory work No. 2
Study of vibrations
mathematical and spring pendulums
Goal of the work:
investigate on what quantities the period of oscillation of a mathematical and spring pendulum depends and on which does not depend.
Devices and materials:
tripod, 3 weights of different weights (ball, weight weighing 100 g, weight), thread 60 cm long, 2 springs of different stiffness, ruler, stopwatch, strip magnet.
Work order
1. Make a mathematical pendulum. Watch his hesitation.
2. Investigate the dependence of the period of oscillation of a mathematical pendulum on the length of the thread. To do this, determine the time of 20 complete oscillations of pendulums of length 25 and 49 cm. Calculate the period of oscillation in each case. Enter the results of measurements and calculations, taking into account the measurement error, into table 10. Draw a conclusion.
Table 10
|
l, m |
n |
t d D t, s |
Td D T, With |
|
0,25 |
20 |
||
|
0,49 |
20 |
3. Investigate the dependence of the period of oscillation of a pendulum on the acceleration of gravity. To do this, place a strip magnet under a 25 cm long pendulum. Determine the period of oscillation, compare it with the period of oscillation of a pendulum in the absence of a magnet. Draw a conclusion.
4. Show that the period of oscillation of a mathematical pendulum does not depend on the mass of the load. To do this, hang weights of different weights from a thread of constant length. For each case, determine the period of oscillation, keeping the amplitude the same. Draw a conclusion.
5. Show that the period of oscillation of a mathematical pendulum does not depend on the amplitude of the oscillations. To do this, deflect the pendulum first by 3 cm and then by 4 cm from the equilibrium position and determine the period of oscillation in each case. Enter the results of measurements and calculations in table 11. Draw a conclusion.
Table 11
|
A, cm |
n |
t+D t, With |
T+D T, With |
6. Show that the period of oscillation of a spring pendulum depends on the mass of the load. By attaching weights of different masses to the spring, determine the period of oscillation of the pendulum in each case by measuring the time of 10 oscillations. Draw a conclusion.
7. Show that the period of oscillation of a spring pendulum depends on the spring stiffness. Draw a conclusion.
8. Show that the period of oscillation of a spring pendulum does not depend on the amplitude. Enter the results of measurements and calculations in Table 12. Draw a conclusion.
Table 12
|
A, cm |
n |
t+D t, With |
T+D T, With |
Task 24
1 e.Explore the range of applicability of the mathematical pendulum model. To do this, change the length of the pendulum thread and the dimensions of the body. Check whether the period of oscillation depends on the length of the pendulum if the body is large and the length of the thread is small.
2. Calculate the lengths of second pendulums mounted on a pole ( g= 9.832 m/s 2), at the equator ( g= 9.78 m/s 2), in Moscow ( g= 9.816 m/s 2), in St. Petersburg ( g= 9.819 m/s 2).
3 * . How do temperature changes affect the movement of a pendulum clock?
4. How does the frequency of a pendulum clock change when going uphill?
5 * . A girl swings on a swing. Will the period of oscillation of the swing change if two girls sit on it? What if the girl swings not sitting, but standing?
Laboratory work No. 3*
Measuring gravity acceleration
using a mathematical pendulum
Goal of the work:
learn to measure the acceleration of gravity using the formula for the period of oscillation of a mathematical pendulum.
Devices and materials:
a tripod, a ball with a thread attached to it, a measuring tape, a stopwatch (or a watch with a second hand).
Work order
1. Hang the ball from a tripod on a 30 cm long thread.
2. Measure the time of 10 complete oscillations of the pendulum and calculate its period of oscillation. Enter the results of measurements and calculations in table 13.
3. Using the formula for the period of oscillation of a mathematical pendulum T= 2p, calculate the acceleration of gravity using the formula: g = .
4. Repeat the measurements, changing the length of the pendulum thread.
5. Calculate the relative and absolute error in changing the acceleration of free fall for each case using the formulas:
d g==+ ; D g = g d g.
Consider that the error in measuring length is equal to half the division value of a measuring tape, and the error in measuring time is equal to half the division value of a stopwatch.
6. Write down the value of the acceleration due to gravity in Table 13, taking into account the measurement error.
Table 13
|
Experience no. |
l d D l, m |
n |
t d D t, With |
T d D T, With |
g, m/s2 |
D g, m/s2 |
g d D g, m/s2 |
Task 25
1. Will the error in measuring the period of oscillation of a pendulum change, and if so, how, if the number of oscillations is increased from 20 to 30?
2. How does increasing the length of the pendulum affect the accuracy of measuring the acceleration of gravity? Why?
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.
(lat. amplitude- magnitude) is the greatest deviation of an oscillating body from its equilibrium position.
For a pendulum, this is the maximum distance that the ball moves away from its equilibrium position (figure below). For oscillations with small amplitudes, such a distance can be taken as the length of the arc 01 or 02, and the lengths of these segments.
The amplitude of oscillations is measured in units of length - meters, centimeters, etc. On the oscillation graph, the amplitude is defined as the maximum (modulo) ordinate of the sinusoidal curve (see figure below).
Oscillation period.
Oscillation period- this is the shortest period of time through which a system oscillating returns again to the same state in which it was at the initial moment of time, chosen arbitrarily.
In other words, the oscillation period ( T) is the time during which one complete oscillation occurs. For example, in the figure below, this is the time it takes for the pendulum bob to move from the rightmost point through the equilibrium point ABOUT to the far left point and back through the point ABOUT again to the far right.
Over a full period of oscillation, the body thus travels a path equal to four amplitudes. The period of oscillation is measured in units of time - seconds, minutes, etc. The period of oscillation can be determined from a well-known graph of oscillations (see figure below).
The concept of “oscillation period”, strictly speaking, is valid only when the values of the oscillating quantity are exactly repeated after a certain period of time, i.e. for harmonic oscillations. However, this concept also applies to cases of approximately repeating quantities, for example, for damped oscillations.
Oscillation frequency.
Oscillation frequency- this is the number of oscillations performed per unit of time, for example, in 1 s.
The SI unit of frequency is named hertz(Hz) in honor of the German physicist G. Hertz (1857-1894). If the oscillation frequency ( v) is equal to 1 Hz, this means that every second there is one oscillation. The frequency and period of oscillations are related by the relations:
In the theory of oscillations they also use the concept cyclical, or circular frequency ω . It is related to the normal frequency v and oscillation period T ratios:
.
Cyclic frequency is the number of oscillations performed per 2π seconds
The most important parameter characterizing mechanical, sound, electrical, electromagnetic and all other types of vibrations is period- the time during which one complete oscillation occurs. If, for example, the pendulum of a 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 - Oscillation
Where: φ – oscillation phase, I– current strength, Ia– amplitude value of current strength (amplitude)
T– period of current fluctuation (period)
Another parameter characterizing fluctuations is frequency(from the word “often”) - a number showing how many complete oscillations per second are made by a clock pendulum, a sounding body, a current in a conductor, etc. The frequency of oscillations is estimated by a unit called the hertz (abbreviated as Hz): 1 Hz is one oscillation per second. If, for example, a sounding string makes 440 complete vibrations in 1 s (at the same time it creates the tone “A” of the third octave), its vibration frequency is said to be 440 Hz. The alternating current frequency of the electric lighting network is 50 Hz. With this current, electrons in the wires of the network flow alternately 50 times in one direction and the same number of times in the opposite direction within a second, i.e. perform 50 complete oscillations in 1 s.
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 displacement or change in a variable during oscillatory or wave motion. A non-negative scalar quantity, measured in units depending on the type of wave or vibration.
Figure 2.5 - Sinusoidal oscillation.
Where, y- wave amplitude, λ - wavelength.
For example:
the amplitude for mechanical vibration of a body (vibration), for waves on a string or spring, 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 the displacement relative to an equilibrium (the 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 appear in air? Air consists of particles invisible to the eyes. When the wind blows, they can be transported over long distances. But they can also hesitate. For example, if we make a sharp movement with a stick in the air, 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 the vibrations of the stick.
Let's do this experiment. Let's pull the string, for example, of a guitar, and then let it go. The string will begin to tremble - oscillate around its original resting position. Quite strong vibrations of the string are noticeable to the eye. Weak vibrations of the string can only be felt as a slight tickling if you touch it with your finger. While the string vibrates, we hear sound. As soon as the string calms down, the sound will fade away. The birth of sound here is the result of condensation and rarefaction of air particles. Oscillating from side to side, the string presses, as if pressing, air particles in front of it, forming areas of high pressure in a certain volume of it, and behind it, 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 the moment when the area of increased pressure of the sound wave reaches the ear, it presses on the eardrum, bending it slightly inward. When the rarefied region of the sound wave reaches the ear, the eardrum bends slightly 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, the louder the sound we perceive.
Sound waves, like water or electrical vibrations, are represented by a wavy line - a sine wave. Its humps correspond to areas of high pressure, and its depressions correspond to areas of low air pressure. An area of high pressure and a subsequent area of low pressure form a sound wave.
By the frequency of vibration of a sounding body one can judge the tone or pitch of a 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 capable of responding to a relatively small frequency band (section) sound vibrations - approximately 20 Hz to 20 kHz. Nevertheless, this frequency band accommodates the entire wide range of sounds created by the human voice and a symphony orchestra: from very low tones, similar to the sound of a beetle buzzing, to the barely perceptible high-pitched squeak of a mosquito. Oscillation frequency up to 20 Hz, called infrasonic, And above 20 kHz, called ultrasonic, we don't hear. And if the eardrum of our ear turned out to be capable of responding 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 one should not confuse the height, i.e. the tone of the sound with its strength. The pitch of a 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. vibrates more slowly than a thin and short string, creating a high-pitched sound (Fig. 1).
Figure 2.6 - Sound waves
The higher the frequency of vibration of the string, the shorter the sound waves and the higher the pitch of the sound.
In electrical and radio engineering, alternating currents with frequencies ranging from several hertz to thousands of gigahertz are used. Broadcast radio antennas, for example, are fed by currents with frequencies ranging from approximately 150 kHz to 100 MHz.
These rapidly changing vibrations, called radio frequency vibrations, are the means by which sounds are transmitted wirelessly over long distances.
The entire huge range of alternating currents is usually divided into several sections - subranges.
Currents with a frequency from 20 Hz to 20 kHz, corresponding to vibrations that we perceive as sounds of different tones, are called currents(or fluctuations) audio frequency, and currents with a frequency above 20 kHz - ultrasonic frequency currents.
Currents with a frequency from 100 kHz to 30 MHz are called high frequency currents,
Currents with frequencies above 30 MHz - ultra-high and ultra-high frequency currents.