Question

When an open pipe is suddenly closed at one end, the frequency of the third harmonic of the closed pipe is found to be $50\text{ Hz}$ more than the fundamental frequency of the open pipe. The fundamental frequency of the open pipe is

• A. 100 Hz
• B. 50 Hz
• C. 200 Hz
• D. 300 Hz
• E. 350 Hz

Hint:

Always remember that for the same length, an open pipe has a fundamental frequency twice that of a closed pipe. Also, closed pipes only produce odd harmonics.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The fundamental frequency of an open pipe is $f_o = \frac{v}{2L}$.
The fundamental frequency of a closed pipe is $f_c = \frac{v}{4L}$.
Note that $f_o = 2f_c$.

Step 2: Detailed Explanation:
1. Harmonics of a closed pipe are odd multiples of the fundamental: $f, 3f, 5f, \dots$
The third harmonic of the closed pipe is $3f_c = 3 \left(\frac{v}{4L}\right)$.
2. Let $f_o$ be the fundamental frequency of the open pipe.
Then $f_c = \frac{f_o}{2}$.
3. According to the question:
Third harmonic of closed pipe = $f_o + 50$
\[ 3 f_c = f_o + 50 \]
Substitute $f_c = \frac{f_o}{2}$:
\[ 3 \left(\frac{f_o}{2}\right) = f_o + 50 \]
\[ 1.5 f_o - f_o = 50 \]
\[ 0.5 f_o = 50 \implies f_o = 100\text{ Hz} \]

Step 3: Final Answer:
The fundamental frequency of the open pipe is 100 Hz.