An open pipe is in resonance in `2nd` harmonic with frequency `f_(1)`. Now one end of the tube is closed and frequency is increased to `f_(2)` such that the resonance again ocuurs in `nth` harmonic. Choose the correct option

Options

A. `n = 3, f_2 = (3)/(4) f_1`
B. `n = 3, f_2 = (5)/(4) f_1`
C. `n = 5, f_2 = (3)/(4) f_1`
D. `n = 5, f_2 = (5)/(4) f_1`

Explanation

To solve the problem, we need to analyze the situation step by step:

### Step 1: Understanding the Open Pipe
An open pipe can resonate in harmonics. The frequency of the nth harmonic in an open pipe is given by the formula:
\[ f_n = n \frac{v}{2L} \]
Where:
- \( f_n \) is the frequency of the nth harmonic,
- \( v \) is the speed of sound in air,
- \( L \) is the length of the pipe,
- \( n \) is the harmonic number (1, 2, 3,...).

In this case, the pipe is in resonance at the 2nd harmonic with frequency \( f_1 \):
\[ f_1 = 2 \frac{v}{2L} = \frac{v}{L} \]

### Step 2: Understanding the Closed Pipe
When one end of the pipe is closed, it behaves as a closed pipe. The frequency of the nth harmonic in a closed pipe is given by:
\[ f_n = n \frac{v}{4L} \]
Where \( n \) must be an odd number (1, 3, 5,...).

### Step 3: Relating Frequencies
We need to find the new frequency \( f_2 \) when the pipe is closed and resonates at the nth harmonic:
\[ f_2 = n \frac{v}{4L} \]

### Step 4: Expressing \( f_2 \) in terms of \( f_1 \)
From the expression for \( f_1 \):
\[ f_1 = \frac{v}{L} \]
We can express \( v \) in terms of \( f_1 \):
\[ v = f_1 \cdot L \]

Now, substituting \( v \) into the expression for \( f_2 \):
\[ f_2 = n \frac{f_1 \cdot L}{4L} = \frac{n}{4} f_1 \]

### Step 5: Analyzing the Harmonic Numbers
Since \( f_2 \) must be greater than \( f_1 \) (as stated in the problem), we can set up the inequality:
\[ \frac{n}{4} f_1 > f_1 \]
This simplifies to:
\[ \frac{n}{4} > 1 \]
\[ n > 4 \]

### Step 6: Finding Possible Values of \( n \)
Since \( n \) must be an odd integer greater than 4, the smallest possible value for \( n \) is 5.

### Conclusion
Thus, the correct option for \( n \) is 5, which means:
\[ f_2 = \frac{5}{4} f_1 \]