question_answer

Two organ pipes sounded together give 5 beats per second. If their lengths are in the ratio of 50 to 51, then the frequency of shorter and longer pipes in Hz are respectively :

A) 250,245                               

B) 245,250

C) 250,255                

D) 255,250

Correct Answer: D

Solution :

Key Idea: Number of beats per second is equal to the difference of the frequencies of sound sources. When air is blown from one end of an open pipe, a longitudinal wave travels from this end to the other. Since, pipe is open at both ends there is an antinode at each end. The distance between two consecutive antinodes is equal to half the wavelength. If \[l\] be length of pipe and \[\lambda \] be the wavelength, then \[l=\frac{\lambda }{2}\]or \[\lambda =2l\] If \[{{n}_{1}}\]be the freqency of note emitted from  pipe, then \[{{n}_{1}}=\frac{v}{\lambda }=\frac{v}{2l}\] Also number of beats per second (beat  frequency) \[={{n}_{1}}-{{n}_{2}}=\]difference of the frequencies of sound sources. Given, \[{{n}_{1}}-{{n}_{2}}=5\] \[\Rightarrow \]       \[{{n}_{1}}=5+{{n}_{2}}\] \[\therefore \]              \[\frac{5+{{n}_{2}}}{{{n}_{2}}}=\frac{51}{50}\] \[\Rightarrow \]                 \[{{n}_{2}}=250\,\text{Hz}\] and \[{{n}_{1}}=250+5=255\,Hz\] Since frequency \[\propto \frac{1}{\text{length}}\] so, frequency of shorter pipe is 255 Hz and  that  of longer pipe is 250 Hz.