question_answer

The frequency of vibrating air column in closed organ pipe is n. If its length be doubled and radius halved, its frequency will be nearly

A) n            

B)                        n/2                        

C) 2 n                         

D)        4 n

Correct Answer: B

Solution :

Key Idea: Closed organ pipe has an antinode at its open end, and node at closed end. The distance between a node and antinode in closed pipe, \[\lambda /4=l\] \[\Rightarrow \]                               \[\lambda =4l\] Fundamental frequency or the first harmonic of closed pipe is \[n=\frac{v}{\lambda }\]               \[\therefore \]\[n=\frac{v}{4l}\] When \[{{l}_{2}}=2{{l}_{1}},{{r}_{2}}=\frac{{{r}_{1}}}{2},{{n}_{1}}=n\] \[\Rightarrow \]               \[\frac{{{n}_{1}}}{{{n}_{2}}}=\frac{{{l}_{2}}}{{{l}_{1}}}\] \[\Rightarrow \]               \[{{n}_{2}}={{n}_{1}}\frac{{{l}_{1}}}{{{l}_{2}}}=n\frac{{{l}_{1}}}{2{{l}_{1}}}=\frac{n}{2}\] Note: A closed pipe produces only odd harmonics.