Question

The vibrations of four air columns under identical conditions are represented in the figure below. The ratio of frequencies $f_p:f_q:f_r:f_s$ will be

• A. $1:1:1:1$
• B. $1:2:4:3$
• C. $4:2:3:1$
• D. $6:2:3:4$

Answer 1

The correct option is B $1:2:4:3$

For $p$,
The pipe is closed.
$f=\frac{v}{\lambda }$


Here,
$l=\frac{\lambda }{4}$
$\therefore f_p=\frac{v}{4l}$

For $s$,
The pipe is closed.
So,
$f=\frac{v}{\lambda }$


Here,
$l=\frac{3\lambda }{4}$
$\therefore f_s=\frac{3v}{4l}$

For $q$,
The pipe is open.
$f=\frac{v}{\lambda }$


Here,
$l=\frac{\lambda }{2}$
$\therefore f_q=\frac{v}{2l}$

For $r$,
The pipe is open.
So,
$f=\frac{v}{\lambda }$


Here,
$l=\lambda$
$\therefore f_r=\frac{v}{l}$
So,
$f_p:f_q:f_r:f_s=\frac{v}{4l}:\frac{v}{2l}:\frac{v}{l}:\frac{3v}{4l}$
$\Rightarrow f_p:f_q:f_r:f_s=1:2:4:3$