Question

A closed pipe and an open pipe of same length produce $2$ beats, when they are set into vibrations simultaneously in their fundamental mode. If the length of the open pipe is halved, and that of closed pipe is doubled, and if they are vibrating in the fundamental mode, then the number of beats produced is:
A. $4$
B. $7$
C. $2$
D. $8$

Answer 1

As we know that formula for frequency which depends on the velocity and length of the pipe. By considering two equations, one for closed pipe and another is for open pipe this question can be solved.

Formula Used:
${f_c} = \dfrac{v}{{4{L_c}}}$
${f_o} = \dfrac{v}{{2{L_o}}}$
Here${f_c}$ is the frequency of the closed pipe and ${f_o}$ is the frequency of the open pipe and $v$ is the velocity and ${L_O}$ is the length of open pipe and${L_c}$ is the length of closed pipe.

Complete step by step answer:
Consider for a closed pipe, the frequency of fundamental mode is –
${f_c} = \dfrac{v}{{4{L_c}}}$
Where, ${f_c}$ is frequency of closed pipe, $v$is velocity of sound in air and ${L_c}$is length of closed pipe.
Consider for an open pipe, the frequency of fundamental mode is
${f_o} = \dfrac{v}{{2{L_o}}}$
Where, ${f_o}$is frequency of open pipe, $v$is velocity and${L_o}$is the length of open pipe.
Given that, ${L_c}$= ${L_o}$
Also, ${f_o} = 2{f_c}$ ...... (A)
According to given statement
${f_o} - {f_c} = 2$ ……. (B)
Solving these equations, we get
${f_o} = 4Hz$
${f_c} = 2Hz$
As given in the statement when the length of the open pipe is halved then its frequency is
$f_o^1 = \dfrac{v}{{2\left( {\dfrac{{{L_o}}}{2}} \right)}}$ =$2{f_o} = 2 \times 4Hz = 8Hz$
As given in the statement when the length of closed pipe is doubled then its frequency is
$f_c^1 = \dfrac{v}{{4\left( {2{L_c}} \right)}}$
$\Rightarrow f_c^1$ = $\dfrac{1}{2}{f_c}$
$\Rightarrow f_c^1$ = $\dfrac{1}{2} \times 2Hz$
$\therefore f_c^1$ =$1Hz$
Then, number of beats produced is =$f_o^1 - f_c^1 = 8 - 1 = 7$

So, option B is correct.

Additional information:
Air Column can be found in various musical instruments where it is enclosed with a hollow metal tube. To conserve space it is coiled upon itself several times. It is almost nearer to one meter in length. Suppose if the end of the tube is left uncovered and thus allowing the sound waves to reach it then that end is termed as an open end. Various instruments operate on the mechanism of the open-end air column that is when the end of the tube is completely uncovered. Suppose if one of the ends of the tube is left uncovered in the surrounding atmosphere and another end of the tube is covered then it is termed as a closed air column.

Note: Fundamental note is the note of the lowest frequency of the periodic waveform. Above the fundamental notes, are called overtones. Cell phones that we use in our daily life are based on radio frequencies.Some pipe organs and their columns are composed within the orchestra. It is possible to convert open tube air columns to close tube air columns.