Two pipes have each of length 2 m, one is closed at on end and the other is open at both ends. The speed of sound in air is 340 m/s . The frequency at which both can resonate is

Options

A. 340 Hz
B. 510 Hz
C. 42.5 Hz
D. none of these

Answer

To solve the problem of finding the frequency at which both pipes can resonate, we need to analyze the characteristics of each pipe based on their configurations.

### Step-by-Step Solution:

1. **Identify the properties of the pipes**:
- Pipe 1: Closed at one end (length = 2 m)
- Pipe 2: Open at both ends (length = 2 m)
- Speed of sound in air (v) = 340 m/s

2. **Determine the fundamental frequency of Pipe 1 (closed at one end)**:
- The formula for the fundamental frequency (f1) of a closed pipe is given by:
\[ f_1 = \frac{v}{4L} \]
- Here, L = 2 m, so:
\[ f_1 = \frac{340 \, \text{m/s}}{4 \times 2 \, \text{m}} = \frac{340}{8} = 42.5 \, \text{Hz} \]
- The harmonics for this pipe will be odd multiples of the fundamental frequency: \(f_1, 3f_1, 5f_1, \ldots\)

3. **Determine the fundamental frequency of Pipe 2 (open at both ends)**:
- The formula for the fundamental frequency (f2) of an open pipe is given by:
\[ f_2 = \frac{v}{2L} \]
- Again, L = 2 m, so:
\[ f_2 = \frac{340 \, \text{m/s}}{2 \times 2 \, \text{m}} = \frac{340}{4} = 85 \, \text{Hz} \]
- The harmonics for this pipe will be all integer multiples of the fundamental frequency: \(f_2, 2f_2, 3f_2, \ldots\)

4. **Set the frequencies equal to find resonance**:
- For resonance, we need to find integers \(n\) and \(m\) such that:
\[ n \cdot f_1 = m \cdot f_2 \]
- Substituting the expressions for \(f_1\) and \(f_2\):
\[ n \cdot 42.5 = m \cdot 85 \]
- Rearranging gives:
\[ \frac{n}{m} = \frac{85}{42.5} = 2 \]
- This means \(n = 2m\).

5. **Check for integer solutions**:
- The smallest integer solution is \(m = 1\) and \(n = 2\).
- This means that the first resonance occurs at:
\[ f = 2 \cdot f_1 = 2 \cdot 42.5 = 85 \, \text{Hz} \]
- Therefore, both pipes can resonate at 85 Hz.

### Final Answer:
Both pipes can resonate at a frequency of **85 Hz**.

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