wave_mechanics critical_thinking Question 6
Question: Two identical flutes produce fundamental notes of frequency 300 Hz at $ 27^{o} $ C. If the temperature of air in one flute is increased to $ 31^{o} $ C, the number of the beats heard per second will be
[UPSEAT 2002]
Options:
A) 1
B) 2
C) 3
D) 4
Show Answer
Answer:
Correct Answer: B
Solution:
Velocity of sound increases if the temperature increases.
So with $ v=n\lambda $ , if $ v $ increases $ n $ will increase at $ 27^{o}C,v _{1}=n\lambda $ , at $ 31^{o}C,v _{2}=(n+x)\lambda $ Now using $ v\propto \sqrt{T} $
$ ( \because v=\sqrt{\frac{\gamma RT}{M}} ) $
$ \frac{v _{2}}{v _{1}}=\sqrt{\frac{T _{2}}{T _{1}}}=\frac{n+x}{n} $
Therefore $ \frac{300+x}{300}=\sqrt{\frac{(273+31)}{(273+27)}}=\sqrt{\frac{304}{300}}=\sqrt{\frac{300+4}{300}} $
Therefore $ 1+\frac{x}{300}={{( 1+\frac{4}{300} )}^{1/2}}=( 1+\frac{1}{2}\times \frac{4}{300} ) $
Therefore x = 2.
$ [\because {{(1+x)}^{n}}=1+nx] $
BETA
