Question

The fifth harmonic of a closed organ pipe is found to be in unison with the first harmonic of an open pipe. The ratio of lengths of closed pipe to that of the open pipe is \( \frac{5}{x} \). The value of \( x \) is ________.

• A. 2
• B. 3
• C. 4
• D. 1

Hint:

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

Answer 1

The Correct Option is A

Step 1: Write frequency formula for closed organ pipe.
For a closed pipe, frequency of the \(n\)-th harmonic is given by \[ f_n = \frac{n v}{4L_c} \quad (n = 1,3,5,\dots) \] For the fifth harmonic,
\[ f_5 = \frac{5v}{4L_c} \]
Step 2: Write frequency formula for open organ pipe.
For an open pipe, frequency of the first harmonic is \[ f_1 = \frac{v}{2L_o} \]
Step 3: Use condition of unison.
Since both frequencies are equal,
\[ \frac{5v}{4L_c} = \frac{v}{2L_o} \]
Step 4: Simplify and find ratio of lengths.
\[ \frac{5}{4L_c} = \frac{1}{2L_o} \Rightarrow \frac{L_c}{L_o} = \frac{5}{2} \]
Comparing with given ratio \( \frac{5}{x} \), we get
\[ x = 2 \]