Question

An open organ pipe and closed organ pipe of same length produce \(2\) beats per second in fundamental mode. The length of open pipe is made half and that of closed pipe is doubled. The number of beats produced per second will be:

• A. \(4\)
• B. \(6\)
• C. \(7\)
• D. \(8\)

Hint:

Open pipe frequency is twice that of a closed pipe of the same length in fundamental mode.

Answer 1

The Correct Option is C

Step 1: Initial Frequencies
$n_o = \frac{v}{2L}$ and $n_c = \frac{v}{4L}$.
$n_o - n_c = 2 \implies \frac{v}{4L} = 2$.

Step 2: New Frequencies
New open pipe: $L'_o = L/2 \implies n'_o = \frac{v}{2(L/2)} = \frac{v}{L} = 4(\frac{v}{4L}) = 4 \times 2 = 8 \text{ Hz}$ (relative change).
Actually, if $\frac{v}{4L} = 2$, then $n_c = 2$ and $n_o = 4$.
New $n'_o = \frac{v}{L} = 8 \text{ Hz}$.
New $n'_c = \frac{v}{4(2L)} = \frac{v}{8L} = \frac{1}{2}(\frac{v}{4L}) = 1 \text{ Hz}$.

Step 3: New Beats
Beats $= |8 - 1| = 7$.
Final Answer: (C)