Question

In an experiment to measure the speed of sound by a resonating air column, a tuning fork of frequency $500Hz$ is used. The length of the air column is varied by changing the level of water in the resonance tube. Two successive resonances are heard at air columns of length $50.7cm$ and $83.9cm$. Which of the following statements is (are) true?

• A. The speed of sound determined from this experiment is $332m{s}^{-1}$
• B. The end correction in this experiment is $0.9cm$
• C. The wavelength of the sound wave is $66.4cm$
• D. The resonance at $50.7cm$ corresponds to the fundamental harmonic

Answer 1

Answer: (A), (C)

Let - $n_1$ harmonic is corresponding to 50.7 cm & $n_2$ harmonic is corresponding 83.9 cm.
sinc both one consecutive harmonics.
$\therefore$ their difference $=\frac{\lambda }{2}$
$\therefore \frac{\lambda }{2}=(83.9-50.7)cm$
$\frac{\lambda }{2}=33.2cm.$
$\lambda =66.4cm$
$\therefore \frac{\lambda }{4}=16.6cm$
length corresponding to fundamental mode must be close to $\frac{\lambda }{4}\&50.7cm$ must be an odd multiple of this length $16.6\times 3=49.8cm$. therefore 50.7 is 3rd harmonic
If end correction is e, then
$e+50.7=\frac{3\lambda }{4}$
$e=49.8-50.7=-0.9cm$
speed of sound, $v=f\lambda$
$\therefore v=500\times 66.4cm/\sec =332.000m/s$