First Resonance Position in Resonance Column Experiment

To determine the observed reading of the water level for the first resonance in the Resonance Column experiment, we follow these steps:

1. Given Parameters:

• Inner diameter of the tube, $D = 4\text{ cm}$
• Radius of the tube, $r = \frac{D}{2} = 2\text{ cm}$
• Frequency of the tuning fork, $f = 512\text{ Hz}$
• Speed of sound in air, $v = 336\text{ m/s} = 33600\text{ cm/s}$

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2. Wavelength Calculation:

The wavelength $\lambda$ of the sound wave emitted by the tuning fork is given by the wave relation: $\lambda = \frac{v}{f}$

Substituting the given values: $\lambda = \frac{336\text{ m/s}}{512\text{ Hz}} = \frac{21}{32}\text{ m} = 0.65625\text{ m} = 65.625\text{ cm}$

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3. End Correction Calculation:

The end correction $e$ for a cylindrical pipe open at one end is given by: $e = 0.6 \cdot r = 0.3 \cdot D$

Substituting $r = 2\text{ cm}$: $e = 0.6 \times 2\text{ cm} = 1.2\text{ cm}$

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4. First Resonance Condition:

For the first resonance mode in a closed pipe (quarter-wave resonance), the effective acoustic length of the air column is: $L_{\text{eff}} = l_1 + e = \frac{\lambda}{4}$

where $l_1$ is the measured length of the air column from the top rim (which coincides with the $0\text{ cm}$ mark on the scale) to the water surface.

Rearranging the equation to solve for $l_1$: $l_1 = \frac{\lambda}{4} - e$

Substituting the calculated values of $\lambda$ and $e$: $\frac{\lambda}{4} = \frac{65.625\text{ cm}}{4} = 16.40625\text{ cm}$

$l_1 = 16.40625\text{ cm} - 1.2\text{ cm} = 15.20625\text{ cm} \approx 15.2\text{ cm}$

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Conclusion:

The observed reading of the water level on the scale for the first resonance is $15.2\text{ cm}$.

Correct Option: B