Question

The vibrations of four air columns are represented in the figure. The ratio of frequencies np : nq : nr : ns is

A. 12 : 6 : 3 : 5
B. 1 : 2 : 4 : 3
C. 4 : 2 : 3 : 1
D. 6 : 2 : 3 : 4

Answer 1

Hint: In the case of the closed end air column, the wavelength to be considered will be 4 times the length of the column, whereas, in the case of the open end air column, the wavelength to be considered will be 2 times the length of the column, while calculating the frequencies.

Formula used:
$$n={\displaystyle \frac{v}{4l}}$$
$$n={\displaystyle \frac{v}{2l}}$$

Complete step by step answer:
The air column p represents the fundamental closed end.
Thus the frequency of the air column p is given as,
$$n_p={\displaystyle \frac{v}{4l}}$$

The air column q represents the fundamental open end.
Thus the frequency of the air column q is given as,
$$n_q={\displaystyle \frac{v}{2l}}$$

The air column r represents the second overtone open end.
Thus the frequency of the air column r is given as,
$$\begin{array}{rl}& n_r={\displaystyle \frac{2v}{2l}}\\ & n_r={\displaystyle \frac{v}{l}}\end{array}$$

The air column s represents the second overtone closed end.
Thus the frequency of the air column s is given as,
$$n_s={\displaystyle \frac{3v}{4l}}$$

The ratio of the frequencies of the air columns p, q, r and s is given as follows:
$$n_p:n_q:n_r:n_s={\displaystyle \frac{1}{4}}:{\displaystyle \frac{1}{2}}:1:{\displaystyle \frac{3}{4}}$$
Multiply by 4 to take the LCM.
Therefore, the ratio of the frequencies is,
$$n_p:n_q:n_r:n_s=1:2:4:3$$

As, the ratio of frequencies np : nq : nr : ns of four air columns is 1 : 2 : 4 : 3, thus option (B) 1 : 2 : 4 : 3 is correct.

Note:
The things to be on your finger-tips for further information on solving these types of problems are: The frequencies should be calculated considering open end and the closed end concepts.