Question 6

Solution: 1. Calculation of Wavelength ($\lambda$) In a resonance column experiment, the difference between any two successive resonance lengths ($\ell_1$ and $\ell_2$) is always equal to half of the wavelength ($\frac{\lambda}{2}$), regardless of which harmonics they represent.

• Given first resonance length, $\ell_1 = 50.7 \text{ cm}$
• Given second successive resonance length, $\ell_2 = 83.9 \text{ cm}$

$\ell_2 - \ell_1 = \frac{\lambda}{2}$ $83.9 - 50.7 = \frac{\lambda}{2}$ $33.2 = \frac{\lambda}{2}$ $\mathbf{\lambda = 66.4 \text{ cm}}$

• Conclusion: Statement (C) is true.

2. Calculation of the Speed of Sound ($v$)

The relation between the speed of sound, frequency, and wavelength is given by: $v = f \times \lambda$ Given frequency ($f$) $= 500 \text{ Hz}$ and wavelength ($\lambda$) $= 66.4 \text{ cm} = 0.664 \text{ m}$: $v = 500 \times 0.664$ $\mathbf{v = 332 \text{ ms}^{-1}}$

• Conclusion: Statement (A) is true.

3. Identification of the Harmonic

For a closed organ pipe (resonance tube), the theoretical length for the fundamental harmonic (1st harmonic) is approximately $\frac{\lambda}{4}$: $\frac{\lambda}{4} = \frac{66.4}{4} = 16.6 \text{ cm}$ The first resonance length given in the problem is $\ell_1 = 50.7 \text{ cm}$. Let's check which odd multiple of $\frac{\lambda}{4}$ aligns with this length: $3 \times \left(\frac{\lambda}{4}\right) = 3 \times 16.6 = 49.8 \text{ cm}$ Since $50.7 \text{ cm}$ is close to $49.8 \text{ cm}$, the resonance at $50.7 \text{ cm}$ corresponds to the 3rd harmonic (1st overtone), not the fundamental harmonic.

• Conclusion: Statement (D) is false.

4. Calculation of End Correction ($e$)

Using the equation for the 3rd harmonic including the end correction ($e$): $\ell_1 + e = 3\left(\frac{\lambda}{4}\right)$ $50.7 + e = 3 \times 16.6$ $50.7 + e = 49.8$ $e = 49.8 - 50.7$ $\mathbf{e = -0.9 \text{ cm}}$ Mathematically, the value is $-0.9 \text{ cm}$ (which indicates experimental error in the question's data since physical end correction cannot be negative). Statement (B) claims it is $+0.9 \text{ cm}$.

• Conclusion: Statement (B) is false.

Final Answer: (A) & (C)