Question

The fundamental frequency of an open pipe is 100 Hz. If the bottom end of the pipe is closed and \( \frac{1}{3} \) of the pipe is filled with water, then the fundamental frequency of the pipe is:

• A. \( 200 \) Hz
• B. \( 100 \) Hz
• C. \( 75 \) Hz
• D. \( 150 \) Hz

Hint:

When a pipe is closed at one end, its fundamental frequency is halved compared to an open pipe. If part of the pipe is filled with water, the effective length of the air column decreases, leading to an increase in frequency.

Answer 1

The Correct Option is C

Step 1: Understanding the Fundamental Frequency of an Open Pipe
The fundamental frequency of an open pipe is given by: \[ f_{\text{open}} = \frac{v}{2L} \] where \( v \) is the speed of sound and \( L \) is the length of the pipe.
Step 2: Effect of Closing One End and Partially Filling with Water
- When one end is closed, the pipe behaves as a closed pipe, where the fundamental frequency is: \[ f_{\text{closed}} = \frac{v}{4L} \] which is half of the open pipe's frequency.
- If the bottom \( \frac{1}{3} \) of the pipe is filled with water, the effective length of the vibrating column becomes \( L_{\text{eff}} = \frac{2}{3} L \).
Step 3: New Fundamental Frequency Calculation
Since the effective length is reduced, the new fundamental frequency becomes: \[ f' = \frac{v}{4L_{\text{eff}}} = \frac{v}{4 \times \frac{2}{3} L} = \frac{3}{8} \times \frac{v}{L} \] Given that \( f_{\text{open}} = \frac{v}{2L} = 100 \) Hz, we substitute: \[ f' = \frac{3}{8} \times 2 \times 100 = 75 \text{ Hz} \] Step 4: Conclusion
Thus, the new fundamental frequency of the pipe is \( 75 \) Hz.