AP EAMCET · PHYSICS · Waves and Sound

An organ pipe \(P_1\), closed at one end and containing a gas of density \(\rho_1\) is vibrating in its first harmonic. Another organ pipe \(P_2\), open at both ends and containing a gas of density \(\rho_2\) is vibrating in its third harmonic. Both the pipes are in resonance with a given tuning fork. If the compressibility of gases is equal in both pipes, the ratio of the lengths of \(P_1\) and \(P_2\) is (assume the given gases to be monoatomic)

A \(\frac{1}{3}\)
B 3
C \(\frac{1}{6} \sqrt{\frac{\rho_1}{\rho_2}}\)
D \(\frac{1}{6} \sqrt{\frac{\rho_2}{\rho_1}}\)

Verified Solution

Answer & Solution

Correct Answer

(D) \(\frac{1}{6} \sqrt{\frac{\rho_2}{\rho_1}}\)

Step-by-step Solution

Detailed explanation

Frequency of closed organ pipe for first harmonic \(n_1=\frac{v_1}{4 l_1}\). Frequency of open organ pipe for third harmonic \(n_3=\frac{3 v_2}{2 l_2}\) At resonance, \(\quad n_1=n_3\) or \(\quad \frac{v_1}{4 l_1}=\frac{3 v_2}{2 l_2}\) or…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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