The number of possible natural oscillations of air column in a pipe closed at one end of length 85 cm whose frequencies lie below 1250 Hz are (velocity of sound `=340ms^(-1)`).

Options

A. `4`
B. `5`
C. `7`
D. `6`

Explanation

To find the number of possible natural oscillations of an air column in a pipe closed at one end, we can follow these steps:

### Step 1: Understand the Harmonics in a Closed Pipe
For a pipe closed at one end, the natural frequencies of oscillation are given by:
\[ f_n = \frac{(2n - 1)v}{4L} \]
where:
- \( f_n \) is the frequency of the nth harmonic,
- \( n \) is the harmonic number (1, 2, 3, ...),
- \( v \) is the velocity of sound in air (given as 340 m/s),
- \( L \) is the length of the pipe (given as 85 cm or 0.85 m).

### Step 2: Calculate the Frequencies
We need to calculate the frequencies for different values of \( n \) until we exceed 1250 Hz.

1. **For \( n = 1 \)**:
\[ f_1 = \frac{(2 \cdot 1 - 1) \cdot 340}{4 \cdot 0.85} = \frac{1 \cdot 340}{3.4} = 100 \text{ Hz} \]

2. **For \( n = 2 \)**:
\[ f_2 = \frac{(2 \cdot 2 - 1) \cdot 340}{4 \cdot 0.85} = \frac{3 \cdot 340}{3.4} = 300 \text{ Hz} \]

3. **For \( n = 3 \)**:
\[ f_3 = \frac{(2 \cdot 3 - 1) \cdot 340}{4 \cdot 0.85} = \frac{5 \cdot 340}{3.4} = 500 \text{ Hz} \]

4. **For \( n = 4 \)**:
\[ f_4 = \frac{(2 \cdot 4 - 1) \cdot 340}{4 \cdot 0.85} = \frac{7 \cdot 340}{3.4} = 700 \text{ Hz} \]

5. **For \( n = 5 \)**:
\[ f_5 = \frac{(2 \cdot 5 - 1) \cdot 340}{4 \cdot 0.85} = \frac{9 \cdot 340}{3.4} = 900 \text{ Hz} \]

6. **For \( n = 6 \)**:
\[ f_6 = \frac{(2 \cdot 6 - 1) \cdot 340}{4 \cdot 0.85} = \frac{11 \cdot 340}{3.4} = 1100 \text{ Hz} \]

7. **For \( n = 7 \)**:
\[ f_7 = \frac{(2 \cdot 7 - 1) \cdot 340}{4 \cdot 0.85} = \frac{13 \cdot 340}{3.4} = 1300 \text{ Hz} \]

### Step 3: Count the Valid Frequencies
Now we count the frequencies that are below 1250 Hz:
- \( f_1 = 100 \) Hz
- \( f_2 = 300 \) Hz
- \( f_3 = 500 \) Hz
- \( f_4 = 700 \) Hz
- \( f_5 = 900 \) Hz
- \( f_6 = 1100 \) Hz

The frequency \( f_7 = 1300 \) Hz exceeds 1250 Hz, so we do not count it.

### Conclusion
The number of possible natural oscillations of the air column in the pipe closed at one end, whose frequencies lie below 1250 Hz, is **6**.

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