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 resonant frequencies of two pipes (one closed at one end and the other open at both ends), we can follow these steps:

### Step 1: Understand the properties of the pipes
- **Pipe 1** (closed at one end): The fundamental frequency (first harmonic) is given by the formula:
\[ f_1 = \frac{v}{4L} \]
where \( v \) is the speed of sound and \( L \) is the length of the pipe. The harmonics for this pipe will be odd multiples of the fundamental frequency: \( f_1, 3f_1, 5f_1, \ldots \).

- **Pipe 2** (open at both ends): The fundamental frequency is given by:
\[ f_2 = \frac{v}{2L} \]
The harmonics for this pipe will include all integer multiples of the fundamental frequency: \( f_2, 2f_2, 3f_2, \ldots \).

### Step 2: Calculate the fundamental frequencies
Given:
- Length of each pipe, \( L = 2 \, \text{m} \)
- Speed of sound, \( v = 340 \, \text{m/s} \)

For **Pipe 1**:
\[ f_1 = \frac{340}{4 \times 2} = \frac{340}{8} = 42.5 \, \text{Hz} \]

For **Pipe 2**:
\[ f_2 = \frac{340}{2 \times 2} = \frac{340}{4} = 85 \, \text{Hz} \]

### Step 3: Determine the resonant frequencies
The resonant frequencies for Pipe 1 (closed at one end) are:
\[ f_{1n} = (2n - 1)f_1 = (2n - 1) \times 42.5 \, \text{Hz} \quad (n = 1, 2, 3, \ldots) \]

The resonant frequencies for Pipe 2 (open at both ends) are:
\[ f_{2m} = mf_2 = m \times 85 \, \text{Hz} \quad (m = 1, 2, 3, \ldots) \]

### Step 4: Set the equations equal to find common frequencies
To find the resonant frequencies where both pipes resonate, we set:
\[ (2n - 1) \times 42.5 = m \times 85 \]

### Step 5: Solve for integers \( n \) and \( m \)
Rearranging gives:
\[ 2n - 1 = \frac{m \times 85}{42.5} = 2m \]
This implies:
\[ 2n - 1 = 2m \implies 2n = 2m + 1 \implies n = m + \frac{1}{2} \]
Since \( n \) and \( m \) must be natural numbers, this equation cannot hold true. Therefore, there are no integer solutions for \( n \) and \( m \).

### Conclusion
The two pipes do not resonate at any common frequency.

### Final Answer
The answer is: **They do not resonate at any frequency.**

---