Question

5th harmonic of a closed organ pipe matches with 1st harmonic of an open organ pipe. Find ratio of their lengths.

• A. $5$
• B. $2$
• C. $\dfrac{5}{2}$
• D. $\dfrac{2}{5}$

Hint:

Closed organ pipes support only odd harmonics, while open organ pipes support all harmonics.

Answer 1

The Correct Option is C

Step 1: Frequency of harmonics in organ pipes.
For a closed organ pipe:
\[ f_n = \dfrac{n v}{4L_{\text{closed}}} \quad (n = 1,3,5,\dots) \]
For an open organ pipe:
\[ f_1 = \dfrac{v}{2L_{\text{open}}} \]
Step 2: Applying the given matching condition.
\[ f_{5,\text{closed}} = f_{1,\text{open}} \]
\[ \dfrac{5v}{4L_{\text{closed}}} = \dfrac{v}{2L_{\text{open}}} \]
Step 3: Simplifying the equation.
\[ \dfrac{L_{\text{closed}}}{L_{\text{open}}} = \dfrac{5}{2} \]