Question

A source of frequency 10 kHz when vibrated over the mouth of a closed organ pipe is in resonance at 300 K. The beats produced per second when temperature rises by 1 K is

Answer 1

Initially frequency of vibration of closed organ pipe = 10 kHz

$f\propto \frac{v}{4l}=\frac{\sqrt{\frac{\gamma RT}{M}}}{4l}$

or $f\propto \sqrt{T}$

$\frac{f^{{}^{\prime }}}{f}=\sqrt{\frac{T^{{}^{\prime }}}{T}}=(1+\frac{1}{600})$ (using binomial expansion)

or $\Delta f=f^{{}^{\prime }}-f=\frac{f}{600}$

$=\frac{10\times {10}^3}{600}=16.67Hz$

$\therefore$ Number of beats produced = 16.67 Hz