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), (B), (C)

The wavelength of the sound wave is $66.4cm$


Let $n_1$ be the harmonic corresponding to $50.7cm$ & $n_2$ the harmonic corresponding to $83.9cm.$
Since both are consecutive harmonics, their difference = $\frac{\lambda }{2}$
$\therefore \frac{\lambda }{2}=(83.9–50.7)cm$
$\frac{\lambda }{2}=33.2cm$
$\lambda =66.4cm$
$\frac{\lambda }{4}=16.6cm$
Hence option C is correct.

Length corresponding to the fundamental mode, must be close to $\frac{\lambda }{4}$ & $50.7cm$ must be closed to an odd multiple of this length
so, $16.6\times 3=49.8cm$, therefore $50.7$ is $3^{rd}$ harmonic
Now, $e+50.7=\frac{3\lambda }{4}$
$e=49.8–50.7=-0.9cm$
Option B is correct.

Speed of sound, $v=f\lambda$
$\therefore v=500\times 66.4cm/sec=332m/s$.

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