Wave Motion 2 Question 28
28. 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.
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Answer:
Correct Answer: 28. $(a, c)$
Solution:
- Let $n$th harmonic is corresponding to $50.7 cm$ and $(n+1)$ th harmonic is corresponding $83.9 cm$.
$\therefore$ Their difference is $\frac{\lambda}{2}$.
$$ \begin{aligned} & \therefore \quad \frac{\lambda}{2}=(83.9-50.7) cm \\ & \text { or } \quad \lambda=66.4 cm \\ & \therefore \quad \frac{\lambda}{4}=16.6 cm \end{aligned} $$
Length corresponding to fundamental mode must be close to $\frac{\lambda}{4}$ and $50.7 cm$ 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
$$ \begin{aligned} e+50.7= & \frac{3 \lambda}{4} \\ & e=49.8-50.7=-0.9 cm \end{aligned} $$
$\therefore$ Speed of sound, $v=f \lambda$
$$ \Rightarrow \quad v=500 \times 66.4 cm / s=332 m / s $$
