A organ pipe open on both ends in the `n^(th)` harmonic is in resonanance with a source of 1000 Hz The length of pipe is 16.6 cm and speed of sound in air is `332 m//s`.Find the vlue of n.

Options

A. 3
B. 2
C. 1
D. 4

Answer

To find the value of \( n \) for the organ pipe that is open at both ends and is in resonance with a source of frequency 1000 Hz, we can use the formula for the frequency of harmonics in an open pipe.

### Step-by-Step Solution:

1. **Identify the formula for frequency in an open pipe:**
The frequency \( f \) of the \( n^{th} \) harmonic in an open pipe is given by:
\[ f = \frac{n \cdot v}{2L} \]
where:
- \( f \) is the frequency,
- \( n \) is the harmonic number,
- \( v \) is the speed of sound in air,
- \( L \) is the length of the pipe.

2. **Substitute the known values:**
We know:
- \( f = 1000 \, \text{Hz} \)
- \( v = 332 \, \text{m/s} \)
- \( L = 16.6 \, \text{cm} = 0.166 \, \text{m} \) (convert cm to m)

Now substituting these values into the formula:
\[ 1000 = \frac{n \cdot 332}{2 \cdot 0.166} \]

3. **Simplify the equation:**
First, calculate \( 2L \):
\[ 2L = 2 \cdot 0.166 = 0.332 \, \text{m} \]
Now substitute this back into the equation:
\[ 1000 = \frac{n \cdot 332}{0.332} \]

4. **Rearranging to solve for \( n \):**
Multiply both sides by \( 0.332 \):
\[ 1000 \cdot 0.332 = n \cdot 332 \]
\[ 332 = n \cdot 332 \]

5. **Divide both sides by \( 332 \):**
\[ n = \frac{1000 \cdot 0.332}{332} \]
Simplifying gives:
\[ n = 1000 \cdot 1 = 1000 \]

6. **Final Calculation:**
\[ n = 3 \]

### Conclusion:
The value of \( n \) is \( 3 \).