A student is performing the experiment of resonance column. The diameter of the column tube is 6 cm. The frequency of the tuning fork is 504 Hz. Speed of the sound at the given temperature is 336 m/s. The zero of the metre scale coincides with the top end of the resonance column tube. The reading of the water level in the column when the first resonance occurs is :

Waves PYQ — JEE Main 2021 · 25 Feb · Shift 1 · Q60 with Solution

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Correct answer: D

Given data
- Diameter of tube: $d = 6\text{ cm}$
- Frequency of tuning fork: $f = 504\text{ Hz}$
- Speed of sound: $v = 336\text{ m/s}$
Find the wavelength $\lambda = \frac{v}{f} = \frac{336}{504} = \frac{2}{3}\text{ m} = 66.67\text{ cm}$
Condition for first resonance in a resonance column For a closed pipe, first resonance occurs when the effective length is $L_{\text{eff}} = \frac{\lambda}{4}$ So, $L_{\text{eff}} = \frac{66.67}{4} = 16.67\text{ cm}$
Apply end correction Since the tube has diameter $d = 6\text{ cm}$ , radius is $r = 3\text{ cm}$ End correction for a resonance tube is approximately $e = 0.6r = 0.6 \times 3 = 1.8\text{ cm}$

Now, $L_{\text{eff}} = L + e$ where $L$ is the actual air column length.

Hence, $L = L_{\text{eff}} - e = 16.67 - 1.8 = 14.87\text{ cm}$
Interpretation of metre scale reading The zero of the metre scale is at the top end of the tube, so the reading of water level directly gives the air column length.

Therefore, the reading at first resonance is $\boxed{14.8\text{ cm}}$
Check options
- A: $13\text{ cm}$ ❌
- B: $18.4\text{ cm}$ ❌
- C: $16.6\text{ cm}$ ❌
- D: $14.8\text{ cm}$ ✅

Thus, the correct option is D.