Question

Two consecutive harmonics of an air column in a pipe closed at one end are of frequencies 150 Hz and 250 Hz. The fundamental frequency of an air column is

• A. $25\ \text{Hz}$
• B. $75\ \text{Hz}$
• C. $100\ \text{Hz}$
• D. $50\ \text{Hz}$

Hint:

Remember that a closed pipe only produces odd harmonics ($1f, 3f, 5f$). Therefore, the gap between consecutive harmonics is always $2f$. For an open pipe, it produces all harmonics, so the gap is just $1f$.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
We are given the frequencies of two consecutive harmonics for a pipe closed at one end. We need to find the fundamental frequency.

Step 2: Key Formula or Approach:
For a pipe closed at one end, the harmonics are odd multiples of the fundamental frequency ($f_1$). The frequencies are $f_1, 3f_1, 5f_1, \ldots$
The difference between any two consecutive harmonics is $(p+2)f_1 - pf_1 = 2f_1$.

Step 3: Detailed Explanation:
Given the two consecutive harmonic frequencies are $150\ \text{Hz}$ and $250\ \text{Hz}$.
The difference between them is:
$$2f_1 = 250 - 150$$
$$2f_1 = 100\ \text{Hz}$$
Divide by 2 to find the fundamental frequency:
$$f_1 = \frac{100}{2} = 50\ \text{Hz}$$

Step 4: Final Answer:
The fundamental frequency is $50\ \text{Hz}$, matching option (D).