Question

In an experiment to measure the speed of sound by a resonating air column, a tuning fork of frequency $500 \, Hz$ 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 $50.7 \, cm$ and $83.9 \, cm$. Which of the following statements is (are) true?

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

Answer 1

Answer: (C)

The wavelength of the sound wave is $66.4 \, cm$

Answer 2

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