Question

Fifth overtone of a closed pipe is found to be in unison with the third overtone of an open organ pipe. Find ratio of lengths of open organ pipe to that of closed organ pipe.

• A. $\frac{5}{11}$
• B. $\frac{7}{11}$
• C. $\frac{4}{11}$
• D. $\frac{8}{11}$

Answer 1

The correct option is D $\frac{8}{11}$

For the closed pipe, frequency of fifth overtone
$f_5^c=\left(2\right(5)+1)f_0^c$
$f_5^c=\frac{11V}{4L_c}$ $(\because f_0^c=\frac{V}{4L_c})$
For the open pipe, frequency of third overtone
$f_3^o=(3+1)f_0^o$
$=\frac{4V}{2L_o}$ $(\because f_0^o=\frac{V}{2L_o})$
$=\frac{2V}{L_o}$
Given fifth overtone of a closed pipe is in unison with the third overtone of an open organ pipe
$\Rightarrow f_5^c=f_3^o$
$\Rightarrow \frac{11V}{4L_c}=\frac{2V}{L_o}$
$\Rightarrow \frac{L_o}{L_c}=\frac{8}{11}$