Tag: superposition of waves-2: stationary (standing) waves: vibrations of air columns

Questions Related to superposition of waves-2: stationary (standing) waves: vibrations of air columns

In an experiment to measure the speed of sound by a resonating air column, a tuning fork of frequency $500Hz$ is used. The length of the air column is varied by changing the level of water in the resonance tube. Two successive resonancers are heard at air columns of length $50.7cm$ and $83.9cm$. Which of the following statements is (are) true?

physics superposition of waves-2: stationary (standing) waves: vibrations of air columns determining wavelength and speed of sound resonance tube resonance and sonometer

  1. The speed of sound determined from this experiment is $332m{s}^{-1}$

  2. The end correction in this experiment is $0.9cm$

  3. The wavelength of the sound wave is $66.4cm$

  4. The resonance at $50.6cm$ corresponds to the fundamental harmonic

Show answer
Correct Option: A
Explanation:

$\begin{array}{l} Let\, \, { n _{ 1 } }\, \, harmonic\, \, is\, \, corresponding\, \, to\, \, 50.7\, cm\, \, and\, \, { n _{ 2 } }\, \, harmonic\, \, is\, \, corresponding\, \, 83.9\, cm. \ { { Sin } }ce\, \, both\, \, one\, \, con\sec  utive\, \, harmonics \ \therefore their\, \, difference=\frac { \lambda  }{ 2 }  \ \therefore \frac { \lambda  }{ 2 } =\left( { 83.9-50.7 } \right) cm \ \frac { \lambda  }{ 2 } =33.2\, cm \ \lambda =66.4\, cm \ \therefore \frac { \lambda  }{ 4 } =16.6\, cm \ length\, \, corresponding\, \, to\, \, fundamental\, \, e\, \, must\, \, be\, \, close\, to\, \, \frac { \lambda  }{ 4 } \, and\, \, 50.7\, cm, \ must\, \, be\, \, and\, \, odd\, \, multiple\, \, of\, \, this\, \, length\, \, 16.6\times 3=49.8\, cm. \ Therefore,\, 50.7\, \, is\, \, { 3^{ rd } }\, \, harmonic \ e+50.7=\frac { { 3\lambda  } }{ 4 }  \ e=49.8-50.7=-0.9\, cm \ speed\, \, of\, \, sound,v=f\lambda  \ \therefore v=500\times 66.4\, cm/\sec  =332\, m/s \end{array}$

Hence,
option $(A)$ is correct answer.

An open pipe of length 33 cm resonates with frequency of 1000 Hz. If the speed of sound s 330 $ms^{-1}$, then this frequency is 

physics superposition of waves-2: stationary (standing) waves: vibrations of air columns determining wavelength and speed of sound resonance tube resonance and sonometer

  1. The fundamental frequency of the pipe'

  2. The first harmonic of the pipe

  3. The second harmonic of the pipe

  4. The forth harmonic of the pipe

Show answer
Correct Option: C
Explanation:

Fundamental frequency (first harmonic frequency) of an open organ pipe is given by ,

                  $n _{1}=v/\lambda=v/2l$ ,
where $v=$speed of sound in air ($=330m/s , given$),
           $\lambda=$ wavelength ,
           $l=$ length of organ pipe ($=33cm=0.33m$  , given) ,
therefore ,  $n _{1}=330/(2\times0.33)=500Hz$ ,
as an open pipe produces even and odd harmonics , and given frequency is 1000Hz , it is 2 times of the first harmonic frequency , therefore it is second harmonic frequency of the pipe .

In a resonance tube the first resonance with a tuning fork occurs at 16 cm and second at 49 cm. If the velocity of sound is 330 m/s, the frequency of tuning fork is :

physics superposition of waves-2: stationary (standing) waves: vibrations of air columns determining wavelength and speed of sound resonance tube resonance and sonometer

  1. 500 Hz

  2. 300 Hz

  3. 330 Hz

  4. 165 Hz

Show answer
Correct Option: A
Explanation:

For closed pipe $l _1=\dfrac{v}{4n}$
$v=2n(l _2-l _1)$
$n=\dfrac{v}{2(l _2-l _1)}$$=\dfrac{330}{2\times (0.49-0.16)}=500 Hz$

Which of the following is not correct regarding the experiment to determine the velocity of sound in laboratory by resonance tube method?

physics superposition of waves-2: stationary (standing) waves: vibrations of air columns determining wavelength and speed of sound resonance tube resonance and sonometer

  1. the resonance tube apparatus should be kept inclined

  2. the turning fork should be struck gently against a soft rubber pad

  3. the vibrating tuning fork should be held horizontally over the open end of the tube

  4. the prongs of vibrating tuning fork should not touch the tube

Show answer
Correct Option: A
Explanation:

The length of the air column inside the resonance tube is determined by the amount of water kept in the tube. 
If a water filled tube is inclined, the length of resonance column would be different at different lines parallel to the tube, and thus a number of resonating lengths would be created.

In ,a resonance tube the first resonance occurs at 16 cm and the second resonance occurs at 49 cm. The end corrections will be :

physics superposition of waves-2: stationary (standing) waves: vibrations of air columns determining wavelength and speed of sound resonance tube resonance and sonometer

  1. 0.3 cm

  2. 0.5 cm

  3. 0.8 cm

  4. 1.0 cm

Show answer
Correct Option: B
Explanation:

Answer is B.

The end correction is given as follows.
End corrections $\displaystyle =\frac { { l } _{ 2 }-3{ l } _{ 1 } }{ 2 }$
$\displaystyle =\frac { 49cm-3\times 16cm }{ 2 } =0.5cm$.
Hence, the end corrections will be 0.5 cm.

In the experiment to determine the speed of sound using a resonance column,

physics superposition of waves-2: stationary (standing) waves: vibrations of air columns determining wavelength and speed of sound resonance tube resonance and sonometer

  1. prongs of the tuning fork are kept in a vertical plane

  2. prongs of the tuning fork are kept in a horizontal plane

  3. in one of the two resonances observed, the length of the resonating air column is close to the wavelength of sound

    in air

  4. in one of the two resonances observed, the length of the resonating air column is close to half of the wavelength of

    sound in air

Show answer
Correct Option: A
Explanation:

    In an experiment to determine the speed of sound using a resonance column , prongs of the tuning fork are kept in a vertical plane so that both the prongs can send sound waves inside the tube .

  In one resonance the length of resonating air column is , 
                        $l _{1}=\lambda/4$ ,
and in another resonance the length of resonating air column is , 
                        $l _{2}=3\lambda/4$ ,
only option B is correct .

In a resonance column experiment, the first resonance is obtained when the level of the water in tube is $20 cm$ from the open end. Resonance will also be obtained when the water level is at a distance of

physics superposition of waves-2: stationary (standing) waves: vibrations of air columns determining wavelength and speed of sound resonance tube resonance and sonometer

  1. $40 cm$ from the open end.

  2. $60$ cm from the open end.

  3. $80$ cm from the open end.

  4. data insufficient

Show answer
Correct Option: B
Explanation:

Frequency of first resonance in closed pipe    $\nu = \dfrac{v}{4L _1}$      ....(1)
where $L _1 = 20 \ cm$
Frequency of next resonance in closed pipe  $\nu = \dfrac{3v}{4L _2}$      ....(2)
Equating (1) and (2), we get
$\dfrac{v}{4L _1} = \dfrac{3v}{4L _2}$
$\implies$  $L _2 = 3L _1  = 3\times 20 = 60 \ cm$

A resonance air column of length 20 cm resonates with a tuning fork of frequency 250 Hz. The speed of sound in air is

physics superposition of waves-2: stationary (standing) waves: vibrations of air columns determining wavelength and speed of sound resonance tube resonance and sonometer

  1. 300 m/s

  2. 200 m/s

  3. 150 m/s

  4. 75 m/s

Show answer
Correct Option: B
Explanation:

Length of air column  $l = 20 \ cm = 0.2 \  m$
In resonance air column,
$\lambda = 4l  = 4\times 0.2 = 0.8 \ m$
Now, $\displaystyle \lambda =0.8m$ and $\displaystyle \nu=250Hz$
Hence, speed of sound in air  $\displaystyle v=\nu\lambda =250\times 0.8=200{ m }/{ s }$

Consider the following statements regarding the experiment to the determine the velocity of sound in laboratory by resonance tube method.
1. The first resonance is obtained for the length ${x} _{1}$ of the air column
2. The second resonance is obtained for the length ${x} _{2}$ of the air column.
If $n$ be the frequency of the tuning fork then which is the correct relation ($v$ represents the velocity of sound) ?

physics superposition of waves-2: stationary (standing) waves: vibrations of air columns determining wavelength and speed of sound resonance tube resonance and sonometer

  1. $v=2n({x} _{1}+{x} _{2})$

  2. $v=2n({x} _{1}-{x} _{2})$

  3. $v=2n({x} _{2}-{x} _{1})$

  4. $v=2n({x} _{1}{x} _{2})$

Show answer
Correct Option: C
Explanation:

Frequency in a resonant tube=$f=p\dfrac{v}{2l}$

where $p$ is for the pth harmonic.
Thus $n=\dfrac{v}{2x _1}$
and $n=2\dfrac{v}{2x _2}$
$\implies v=2n(x _2-x _1)$