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 $332m{s}^{-1}$
• 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: (A), (B), (C)

The correct options are
A The speed of sound determined from this experiment is $332m{s}^{-1}$
B The end correction in this experiment is $0.9$ cm
C The wavelength of the sound wave is $66.4$ cm

Let $n_1$ be the harmonic corresponding to $50.7$cm & $n_2$ the harmonic corresponding to $83.9$ cm.

Since both are consecutive harmonics, their difference = $\frac{\lambda }{2}$

$\therefore \frac{\lambda }{2}=(83.9–50.7)$cm

$\frac{\lambda }{2}=33.2cm$

$\lambda =66.4cm$

$\frac{\lambda }{4}=16.6$cm

Length corresponding to fundamental mode must be close to $\frac{\lambda }{4}$ & $50.7$cm must be closed to an odd multiple of this length as $16.6\times 3=49.8$cm. therefore $50.7$ is $3^{rd}$ harmonic

$e+50.7=\frac{3\lambda }{4}$

$e=49.8–50.7=-0.9cm$

Speed of sound, v = $f\lambda$

$\therefore v=500\times 66.4cm/sec=332.000m/s$.

Hence, Option (A), (B) and (C) are correct