Question

An air column in a pipe, which is closed at one end will be in resonance with a vibrating tuning fork of frequency 264 Hz for various lengths. Which one of the following lengths is not possible? (V = 330 m/s)

• A. 62.50 cm
• B. 93.75 cm
• C. 156.25 cm
• D. 31.25 cm

Hint:

Remember that closed pipes behave like odd numbers (1, 3, 5...), while open pipes include all numbers (1, 2, 3...). If a length is an even multiple of the fundamental length, it belongs to an open pipe, not a closed one!

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We need to identify which resonant length is not possible for an air column closed at one end, given the tuning fork frequency and the speed of sound.

Step 2: Key Formula or Approach:
For a pipe closed at one end, the fundamental length is $l = \frac{V}{4n}$. The pipe will resonate exclusively at odd harmonics, so the possible resonating lengths are $l, 3l, 5l, 7l \dots$

Step 3: Detailed Explanation:
Given frequency $n = \text{264 Hz}$ and speed of sound $V = \text{330 m/s}$.
Calculate the fundamental length ($l$):
$$l = \frac{330}{4 \times 264} = 0.3125\ \text{m} = \text{31.25 cm}$$
Now, calculate the subsequent possible lengths by multiplying with odd integers:
First harmonic length $= 1 \times 31.25 = \text{31.25 cm}$
Third harmonic length $= 3 \times 31.25 = \text{93.75 cm}$
Fifth harmonic length $= 5 \times 31.25 = \text{156.25 cm}$
The value 62.50 cm is exactly $2 \times 31.25$, which corresponds to an even harmonic. Closed pipes do not produce even harmonics.

Step 4: Final Answer:
The length 62.50 cm is not possible, matching option (A).