In this article, we shall study resonating circuit: a) LCR Series Circuit and b) LC parallel circuit.

Series Resonating Circuit:

Consider LCR circuit containing Resistance R, Inductance L and capacitance C connected in series. Let the combination be connected to an A.C. source supplying current i = i_o Sinωt.

The instantaneous e.m.f. of the circuit is equal to the sum of voltage drops across the resistor, the capacitor, and the inductor.

Dividing both sides of the equation by C

Substituting the values of equations (2) and (3) in equation (1)

Substituting these values in equation (4) we have

Thus the e.m.f. leads the current in phase  given by

In this condition the impedance of the circuit is minimum and the circuit acts as a purely resistive circuit. The current in the circuit is maximum. This condition is called a series resonance of the circuit. At resonance

The frequency f_r is called the resonating frequency. The frequency at which the resonance takes place and the maximum current flows through the circuit is called the resonating frequency.

The variation of the current with frequency for the series LCR circuit is as shown in the graph.

Parallel Resonating Circuit:

 Consider LC circuit containing Inductance L and capacitance C connected in parallel. Let the combination be connected to an A.C. source with e.m.f.  e = e_o Sin ωt.

The current in the inductance lags behind the e.m.f. by π/ 2. Thus the current through the inductance is given by

Where,  X_L = Inductive reactance

The current in the capacitance leads the e.m.f. by π/ 2. Thus the current through the inductance is given by

Where X_C = Capacitive reactance

The total current in the circuit is given by

When X_L = X_C i.e. ωC =  1/ ωL , the current in the circuit is zero. This condition is called parallel resonance of the circuit.

At resonance

The frequency f_r is called the resonating frequency. The frequency at which the resonance takes place and the zero current flows through the circuit is called the resonating frequency. At resonating frequency the impedance is maximum.

Science > Physics > Electromagnetic Induction > Resonating Circuit

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In this article, we shall study the vibrations of air columns in a pipe closed at one end, opened at other and in a pipe open at both ends.

Vibrations of Air Columns in a Pipe Closed at One End: 

Consider a pipe closed at one end and open at the other. If a vibrating tuning fork of proper frequency is held near its open end, it sends longitudinal waves in the pipe. They are reflected from the closed end. The incident and reflected waves interfere and stationary waves are formed.

The closed-end acts as a rigid wall. The air at the closed end is not free to vibrate. Hence a node is formed at the closed end. Air at the open end is free to vibrate with maximum amplitude and antinode is formed at the open end. This is the simplest mode of vibration of an air column closed at one end and is called the fundamental mode.

If v is the velocity of sound in air at room temperature, then

v = n λ

In the fundamental mode     L = λ/4, hence λ = 4 l

∴ V = nλ = 4 n l

∴ n = v /4 l

This is the frequency of the fundamental mode. Where ‘l’ is length of vibrating air column

Now consider the following figure. The same air column can also be made to vibrate so that there is a node at the closed end, an antinode at the open end, with one more node and antinode in between. In this case, the air column is said to be emitted the first overtone. If n₁ is the frequency and λ₁ the wavelength for the first overtone,

Thus the first overtone is the third harmonic.

Now consider the following figure. In this case, the air column is shown emitting the second overtone. If n₂ is the frequency and λ₂­ the wavelength for the second overtone, then

Therefore, the second overtone is the fifth harmonic.

Thus, for an air column closed at one end, only the odd harmonics are present as overtones.

For air column in a pipe closed at one end p^th overtone is (2p + 1) th harmonic and the frequency of p^th overtone = (2p+1)n.

Vibrations of Air Column in a Pipe Open at Both Ends:  

Consider a tube open at both ends. A vibrating tuning fork held near anyone end produces longitudinal waves in the air column. They are reflected from the other end. The incident and reflected waves interface with each other and stationary waves are formed.

When an air column open at both ends is set up into vibrating, both the ends become antinodes. In the fundamental mode, as shown in the following figure, there is only one node midway between the two antinodes. Therefore, the length of the air column,

This is the fundamental frequency.

Now consider the following figure. When the air column is emitting the first overtone, there are two nodes and one antinode in between the two antinodes at the ends. Therefore, if n₁ is the frequency and λ₁ is the wavelength for the first overtone,

Hence, in this case, the first overtone is the second harmonic.

In the following figure, the air column is shown emitting the second overtone. If n₂ is the frequency and λ₂ the wavelength for the second overtone, then

The second overtone is the third harmonic.

Thus for an air column open at both ends, both odd, as well as even harmonics, are present as overtones i.e. all harmonics are present.

For air column in a pipe open at both ends p^th overtone is (p + 1)^th harmonic and the frequency of p^th overtone = (p+1)n.

Note:

The frequency of the fundamental note of an open pipe is double that for a closed pipe of the same length.

End Correction:

It was shown by Regnault, that the antinode is not formed exactly at the open end but at a distance 0.3 d above the open end where d is the internal diameter of the tube. This additional distance called the end correction,

End correction is necessary, as the wave spreads out slightly, just above the open end, and the air particles just outside the open end are also set into vibrations. For a pipe open at one end and closed at the other

Corrected length = l + 0.3d

Hence, the velocity of sound in air at room temperature is

∴   v = nλ = 4n (l + 0.3d)

Hence, knowing n, l and d, the velocity of sound in air is determined.

For a pipe open at both ends, end correction = 2 × 0.3 d = 0.6 d

Corrected length = l + 0.6 d

Hence, the velocity of sound in air at room temperature is

∴   v = nλ = 2n ( l  + 0.6d)

Elimination of End Correction:

By the usual method, we find the first resonating length (l₁). Similarly, we find the second resonating length (l₂) for which a loud sound is heard once again. If “e” is the end correction, then

l₁ + e = λ/4    …………. (1)

l₂ + e = 3λ/4    …………. (1)

Subtracting equation (1) from (2)

l₂ –  l₁ = 3λ/4 – λ/4

∴    l₂ –  l₁ = λ/2

∴  λ = 2( l₂ –  l₁ )

Now, v = n λ = 2n( l₂ –  l₁ )

Using this formula velocity of sound can be found without considering end correction.

End Correction for a Pipe Closed at One End:

If v is the velocity of sound in air, λ is the wavelength of sound. ‘l’  is the vibrating length of a pipe closed at one end, n is the frequency of vibration in the fundamental mode which is equal to the frequency of the tuning fork at time of resonance. Let ‘e’ be the end correction. This end correction depends on the diameter of the tube. we have

V = nλ=  4n ( l + e)

For the first case

V = 4n₁ ( l₁+ e)    ………… (1)

Where l₁ is a length of vibrating air column vibrating in unison with a tuning fork of frequency n₁.

For the second case

V = 4n₂ ( l₂+ e)    ………… (2)

Where l₂ is a length of vibrating air column vibrating in unison with a tuning fork of frequency n₂.

From (1) and (2)

4n₁ ( l₁+ e) =  4n₂ ( l₂+ e)

∴    l₁n₁  +  en₁ =  l₂n₂  +  en₂

∴    l₁n₁  – l₂n₂ =  en₂ – en₁

∴    l₁n₁  – l₂n₂ =  e(n₂ – n₁)

End Correction for a Pipe Open at Both Ends:

If v is the velocity of sound in air, λ is the wavelength of sound.  ‘l’ is the vibrating length of a pipe open at both the ends, n is the frequency of vibration in the fundamental mode which is equal to the frequency of the tuning fork at time of resonance. Let ‘e’ be the end correction. This end correction depends on the diameter of the tube. we have

V = nλ=  2n ( l + 2e)

For the first case

V = 2n₁ ( l₁+ 2e)    ………… (1)

Where l1 is a length of vibrating air column vibrating in unison with a tuning fork of frequency n₁.

For the second case

V = 2n₂ ( l₂+ 2e)    ………… (2)

Where l2 is a  of vibrating air column vibrating in unison with a tuning fork of frequency n₂.

From (1) and (2)

2n₁ ( l₁+ 2e) =  2n₂ ( l₂+ 2e)

∴    l₁n₁ +  2en₁ =  l₂n₂  +  2en₂

∴    l₁n₁  – l₂n₂ =  2en₂ – 2en₁

∴    l₁n₁  – l₂n₂ =  2e(n₂ – n₁)

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Science > Physics > Stationary Waves > Vibrations of Air Columns

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In this article, we shall study the phenomenon of resonance, its characteristics, advantages, and disadvantages.

Free Vibrations:

A body or a system capable of vibrating, when displaced from its position of rest, vibrates with a certain definite frequency. This frequency is characteristic of the body or the system. Such oscillations are called free oscillations or free vibrations and the frequency of such oscillations is called the natural frequency of the body or the system.

• Example – 1: When a wire under tension, which is fixed at its ends, is plucked and released, it vibrates with a frequency which depends on the length of the string, its mass per unit length and tension in the string.
• Example – 2: When the bob of a simple pendulum oscillates, its frequency of oscillation depends on the length of the pendulum.

Due to frictional force, the amplitude of oscillation decreases continuously and finally, the body stops vibrating.

Characteristics of Free Vibrations:

• They are produced when a body capable of vibrating is disturbed from its normal equilibrium position and then released.
• The frequency of vibration depends on the body and is called natural frequency.
• The frequency of vibration is the same as the natural frequency of the body.
• The amplitude of vibration is large.
• Vibration continues for a little more time after the external force is removed.
• Example: Oscillations of bob of the pendulum

Forced Vibrations:

Forced vibrations are the vibrations produced in a body by applying an external periodic force having a frequency, normally different from the natural frequency of the body.

A body or a system, capable of vibrating can also be made to vibrate at any desired frequency. The body can be made to vibrate with the same frequency as the frequency of the applied periodic force. Suppose that the natural frequency of a metal vessel is 200 Hz.  If a tuning fork of frequency 256 Hz is set up into vibrations and its stem is placed in contact with the vessel, then the vessel will be forced to vibrate at a frequency of 256 Hz.  In such a case, the vessel is said to perform forced vibrations.

Initially, the body tends to vibrate with its natural frequency. But very soon, the natural vibrations die out and it begins to vibrate with the frequency of the applied periodic force.

The amplitude of the forced vibrations depends on:

• The difference in frequencies of the external force and the natural frequency of the body.
  • The amplitude of the applied force.
  • damping.

Characteristics of  Forced Vibrations:

• They are produced when an external periodic force acts on the body.
• The frequency of vibration is the same as the frequency of external periodic force.
• The frequency of vibration is different from the natural frequency of the body.
• The amplitude of vibration is small.
• Amplitude becomes zero as soon as the external force is removed.
• Example: A vibrating tuning fork on a wooden box, a musical instrument having a soundboard or box.

Resonance:

The amplitude of the forced vibrations depends on the difference between the natural frequency of the body and the frequency of the applied periodic force. When the difference between the two frequencies is large, the response of the body is poor or the forced vibrations are of small amplitude. When the frequency difference becomes smaller, the body vibrates more readily or the amplitude of the forced vibrations increases. Finally, when the frequency (f) of the applied periodic force becomes the same as the natural frequency fo of the body, the amplitude of the forced vibrations becomes maximum and the phenomenon is known as resonance.

If anybody is made to vibrate, by an external periodic force, with a frequency which is the same as the natural frequency of the body, the body begins to vibrate with a very large amplitude. This phenomenon is called resonance.

Distinguishing Between Forced Vibrations and Resonance:

Advantages of Resonance:

• Resonance is useful to determine an unknown frequency.
• Resonance is useful to increase the intensity of sound in musical instruments.
• Resonance is useful to tune a radio receiver to any desired frequency.
• Resonance is useful to analyze musical notes.

Disadvantages of Resonance:

Soldiers are asked to break steps when crossing a bridge. It can be explained as follows

Soldiers marching on a bridge take steps with definite frequency and force the bridge to vibrate with the frequency of the steps.

If the forced frequency on the bridge is equal to the natural frequency of vibration of the bridge, the bridge is set into resonant vibrations.

Due to the resonance, the bridge vibrates with higher amplitude and due to this, it may collapse.

Due to the rhythmic clapping of the audience, the roof of the stadium may collapse. It can be explained as follows

When the audience claps rhythmically they do so with a certain frequency and force the roof of a stadium to vibrate with the frequency of the clap.

If the forced frequency on the roof of a stadium is equal to the natural frequency of vibration of the roof of a stadium, the roof of a stadium is set into resonant vibrations.

Due to the resonance, the roof of a stadium vibrate with higher amplitude and due to this, it may collapse.

When the speed of an aircraft increases, different parts are forced to vibrate. which is dangerous for the structure of the aircraft.

Resonance Tube Experiment to Determine Velocity of Sound in Air:

A metal tube, open at both the ends is immersed in a tall glass jar filled with water. An air column is thus formed between the open end of the metal tube and the surface of the water. The length of the air column, which is closed by the water surface at the lower end, can be varied by raising or lowering the metal tube. A tuning fork is set up into vibrations and held near the mouth of the tube so that its arms vibrate parallel to the axis of the tube. The longitudinal wave, starting from the tuning fork, travels along the length of the air column and is reflected back from the surface of the water. The incident and reflected waves interfere to produce a stationary wave.

The molecules of air, in contact with the water surface, remain at rest. Therefore, the closed-end becomes a node. The molecules of air, near the mouth of the tube, vibrate with maximum amplitude. Therefore, the open end becomes an antinode. The frequency of the air column can be changed by adjusting its length. When its frequency becomes the same as the frequency of the fork, resonance takes place, and a loud sound is heard.

If the length of the air column is increased from a small value, the first resonance occurs when there is a node at the closed end and an antinode at the open end, with no other nodes or antinodes in between. Therefore, the length of the air column  l = λ/4

∴   λ = 4 l

Due to resonance, the frequency of the air column is the same as that of the fork. Now velocity of sound is given by v = nλ

∴  v =   4 n l

End correction :

It was shown by Regnault, that the antinode is not formed exactly at the open end but at a distance 0.3 d above the open end where d is the internal diameter of the tube. This additional distance, called the end correction, is necessary, as the wave spreads out slightly, just above the open end, and the air particles just outside the open end are also set into vibrations.

∴ corrected length = l + 0.3d

Hence, the velocity of sound in air at room temperature is

V = nλ = 4n (l + 0.3d)

Hence, knowing n, l, and d, the velocity of sound in air is determined.

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