Podsumowanie wiadomości o ruchu drgającym i falach

Answer the following questions and test your knowledge.

1. What is the vibrating motion?

Vibrating motion is one of the most common types of movement. The body moves on the same track in both directions and the movement repeats in equal intervals of time.

2. Define the basic quantities describing the vibrating motion.

The time required to perform one full vibration is called the period of motion. The number of vibrations made per the unit of time is the frequency of vibrationsfrequency of vibrationsfrequency of vibrations. Frequency is the inverse quantity to the period.

The unit of frequency is 1 Hz (hertz).

The amplitude of vibration is the greatest displacement of the body from the equilibrium position.

3. What is the mathematical pendulumpendulumpendulum?

An example of a body performing vibrations is the mathematical pendulumpendulumpendulum. It is a very small (punctual) body suspended on a weightless and inextensible thread. The period of oscillation of the pendulum depends on its length. The longer the pendulumpendulumpendulum, the greater the period of vibrations (i.e. the lower the frequency).

4. What is natural frequency?

Natural frequency is the frequency with which the body vibrates knocking out the equilibrium position without any resistance.

5. Present on the graph the dependence of the displacement on time in the undamped and suppressed harmonic motion.

The dependence of the displacement on time can then be presented on the graph.

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Ilustracja przedstawia wykres zależności wychylenia od czasu. Oś pozioma opisana t, w nawiasie kwadratowym s. Oś pionowa opisana x, w nawiasie kwadratowym cm, zaznaczone punkty 0 na przecięciu z osią poziomą oraz dwa punkty A znajdujące się w takiej samej odległości od osi poziomej, ale po przeciwnych jej stronach, na wysokości największego wychylenia. Narysowany wykres sinusoidy. Zaznaczona odległość od początku osi do drugiego punktu przecięcia wykresu z osią poziomą. Zaznaczono odległość 1 od osi poziomej do punktu na wykresie, który znajduje się najdalej od osi poziomej. Zaznaczono punkt na wykresie i jego odległość od osi poziomej. Z wykresu wynika, że w ruchu harmonicznym nietłumionym wartość największego wychylenia jest stała w czasie.

In real conditions, when the resistance of the motion is present, the vibrations are dumped - the amplitude decreases with time.

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Ilustracja przedstawia wykres zależności wychylenia od czasu, w rzeczywistych warunkach, gdy działają opory ruchu. Oś pozioma opisana t, w nawiasie kwadratowym s. Oś pionowa opisana x, w nawiasie kwadratowym cm, zaznaczony punkt 0. Narysowany wykres krzywej zaczynający się na osi pionowej w odległości A z indeksem dolnym 1 od osi poziomej. Następnie wykres idzie w dół, przecina oś poziomą, do punktu znajdującego się w odległości A z indeksem dolnym 2 od osi poziomej. Następnie wykres idzie w górę, przecina oś poziomą, do punktu znajdującego się w odległości A z indeksem dolnym 3 od osi poziomej. Następnie wykres idzie w dół, przecina oś poziomą, do punktu znajdującego się w odległości A z indeksem dolnym 4 od osi poziomej. Następnie wykres idzie w górę, przecina oś poziomą, do punktu znajdującego się w odległości A z indeksem dolnym 5 od osi poziomej. Następnie wykres idzie w dół, przecina oś poziomą. Obok wykresu zapis A z indeksem dolnym 1 większe od A z indeksem dolnym 2 większe od A z indeksem dolnym 3 większe od A z indeksem dolnym 4 większe od A z indeksem dolnym 5. Z wykresu wynika, że w ruchu harmonicznym tłumionym wartość największego wychylenia maleje w czasie, drgania są słabnące gdyż istnieją opory ruchu.

6. Describe the mechanical energy changes occurring during the harmonic oscillator motion.

The vibrating body has two kinds of energy: kinetic and potential energy of elasticity. During the vibrations there are cyclical mutual transformations of these energies.

The kinetic energy of the oscillating body is the greatest at the moment of the body passing through the equilibrium position (i.e. when it has the highest velocity) and equals zero at the moment of the biggest displacement.

The potential energy of the elasticity of the vibrating body is the greatest at the moment of the maximum displacement and equals zero at the moment when the body is passing through the equilibrium position.

If there were no resistance of the movement, the total mechanical energy (the sum of kinetic energy and the potential energy of elasticity) would be constant. When there is the resistance force, the body loses its mechanical energy and the vibrations are dumped.

7. What is the mechanical wave and what is the propagation of the mechanical wave?

A mechanical wave is called a disturbance that propagates in matter. This disturbance is the vibrations of the particles of the medium. The particles of the medium do not move with the wave; they only vibrate around their balance positions and stimulate further particles to vibrate. Mechanical waves can propagate only in elastic media.

8. Define the basic quantities describing the propagating wave.

The wavelength, marked with the letter $\lambda$, is the distance over which the wave travels at the time when a given medium molecule does one full vibration. The time required for the particle to perform one full vibration is called the wave period.

The wave in a given medium propagates at a constant speed, which can be calculated with the following formula.

Because $T=\frac{1}{f}$, we can also write that $v=\lambda \cdot f$.

The amplitude of the wave is the maximum displacement from the equilibrium position of the particles of the medium in which the wave propagates.

9. Classify the waves according to the direction of wave propagationwave propagationwave propagation and the direction of vibrations.

We classify waves into longitudinal and transverse ones.

The transverse wave is a wave in which the particles of the medium vibrate in the direction perpendicular to the direction of the wave propagation.

The longitudinal wave is a wave in which the particles of the medium vibrate in the same direction as the wave propagationwave propagationwave propagation.

10. What is a sound?

Sound is an example of a mechanical wave; it is based on propagation of vibrations of the medium particles. It is a spherical wave which spreads in all directions. Like all mechanical waves, it cannot propagate in vacuum. The speed of the sound varies in different media. It is the greatest largest in solids, the smallest in gases.

Bodies vibrating at the frequency of 16 Hz to 20000 Hz are the source of sounds. Strings (guitar, piano), diaphragms (drums) or air columns (flute, organ) can produce vibrations in musical instruments.

11. List and describe the properties of the sound.

Individual sounds differ in timbre, volume and altitude.

The timbre depends on kind of the sound source.

The pitch of the sound depends on its frequency - the higher the frequency, the higher the sound.

The volume depends on the amplitude - the higher the amplitude of the wave, the louder the sound. We measure sound intensity in decibels [dB]. 0 dB is called audibility threshold, 120 dB is called pain threshold.

12. What is echo and reverbreverbreverb?

Echo and reverbreverbreverb are phenomena related to sound propagation and its reflectionreflectionreflection. The sound after the reflection from the barrier returns to the source and if the delay between the wave sent and reflected is large enough, we hear an echo; if it is small, we hear reverb.

13. What are ultrasoundsultrasoundsultrasounds and infrasounds?

Waves with frequencies below 16 Hz are called infrasounds and, respectively, those above 20000 Hz are called ultrasoundsultrasoundsultrasounds. Ultrasounds are used in many technologies including medicine, echolocation, defectoscopy, cosmetic procedures, etc. Many animals hear ultrasoundsultrasoundsultrasounds.

Click on the tag and you will get information about the pendulumpendulumpendulum clock.

Summary

• Vibrating motion is one of the most common types of movement.

• The time required to perform one full vibration is called the period of movement. The number of vibrations made per the unit of time is the frequency of vibrationsfrequency of vibrationsfrequency of vibrations.

• An example of a body performing vibrations is the mathematical pendulumpendulumpendulum. It is a very small (punctual) body suspended on a weightless and inextensible thread.

• A vibrating body has two kinds of energy: kinetic energy and potential energy of elasticity. During vibrations there are cyclical conversions between these energies.

• A mechanical wave is called a disorder that propagates in the medium. The particles of the medium vibrate in this disorder.

• The wave in a given medium propagates at a constant speed.

• Sound is an example of a mechanical wave; it is based on propagation of vibrations of the medium particles. It is a spherical wave which spreads in all directions. Like all mechanical waves, it cannot propagate in vacuum. The speed of the sound varies in different media.

• Individual sounds differ in timbre, volume and altitude.

Exercises

Exercise 2

The whistle, i.e. a pipe closed at one end, emits a basic tone of 2750 Hz. The sound speed in the air is 330 $\frac{m}{s}$.

a) Calculate the length of the whistle.

a) Calculate the length of the whistle.

a) Make an appropriate drawing and use it to calculate the length of the whistle.

b) What would be the frequency of the basic tone, if the whistle was shortened by $\frac{1}{6}$ of the length?

Show solution

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Na ilustracji po lewej stronie narysowana jest inia pionowa, od jej końców narysowane linie poziome długości L. Między nimi narysowane są dwie krzywe, których lewy koniec jest na linii pionowej, w środku odległości między liniami poziomymi. Na końcu linii poziomych krzywe stykają się z nimi.

a)

$\lambda =\frac{v}{f}$

$\lambda =\frac{330\frac{m}{s}}{2750Hz}=0,12m$

$L=\frac{\lambda }{4}$

$L=\frac{0,12m}{4}=0,03m=3cm$

b)

$L_1=L-\frac{1}{6}·L=\frac{5}{6}·L$

$L_1=\frac{5·0,03m}{6}=0,025m$

$L_1=\frac{{\lambda }_1}{4}$

${\lambda }_1=4·L_1$

${\lambda }_1=4\cdot 0,025m=0,1m$

${\lambda }_1=\frac{v}{f_1}$

$f_1=\frac{v}{{\lambda }_1}$

$f_1=\frac{330\frac{m}{s}}{0,1m}=3300Hz$

Glossary

Keywords

frequency of vibrationsfrequency of vibrationsfrequency of vibrations

harmonic oscillationsharmonic oscillationsharmonic oscillations

harmonic waveharmonic waveharmonic wave

vibration periodvibration periodvibration period

wave propagationwave propagationwave propagation