First overtone frequency of a closed organ pipe is equal to the first overtone frequency of an open organ pipe. Further nth harmonic of closed organ pipe is also equal to the nth harmonic of open pipe, where n and m are

Options

A. 5,4
B. 7,5
C. 9,6
D. 7,3

Answer

To solve the problem, we need to analyze the relationship between the frequencies of closed and open organ pipes. Let's break it down step by step.

### Step 1: Understand the Frequencies of Closed and Open Organ Pipes

1. **Closed Organ Pipe Frequencies**:
- The fundamental frequency (first harmonic) of a closed organ pipe is given by:
\[ f_c = \frac{V}{4L_c} \]
- The first overtone (third harmonic) of a closed organ pipe is:
\[ f_{c1} = \frac{3V}{4L_c} \]

2. **Open Organ Pipe Frequencies**:
- The fundamental frequency (first harmonic) of an open organ pipe is given by:
\[ f_o = \frac{V}{2L_o} \]
- The first overtone (second harmonic) of an open organ pipe is:
\[ f_{o1} = \frac{2V}{2L_o} = \frac{V}{L_o} \]

### Step 2: Set the First Overtone Frequencies Equal

According to the problem, the first overtone frequency of the closed organ pipe is equal to the first overtone frequency of the open organ pipe:
\[ f_{c1} = f_{o1} \]
Substituting the expressions we derived:
\[ \frac{3V}{4L_c} = \frac{V}{L_o} \]

### Step 3: Solve for the Ratio of Lengths

To find the ratio of the lengths \( L_c \) and \( L_o \), we can rearrange the equation:
\[ 3L_o = 4L_c \quad \Rightarrow \quad \frac{L_c}{L_o} = \frac{3}{4} \]

### Step 4: nth Harmonic Condition

Now, we consider the nth harmonic of the closed organ pipe and the mth harmonic of the open organ pipe:
\[ f_{cn} = n \cdot f_c = n \cdot \frac{V}{4L_c} \]
\[ f_{om} = m \cdot f_o = m \cdot \frac{V}{2L_o} \]
Setting these equal gives:
\[ n \cdot \frac{V}{4L_c} = m \cdot \frac{V}{2L_o} \]

### Step 5: Simplify the Equation

Dividing both sides by \( V \) and rearranging:
\[ \frac{n}{m} = \frac{4L_c}{2L_o} = 2 \cdot \frac{L_c}{L_o} \]
Substituting the ratio we found earlier:
\[ \frac{n}{m} = 2 \cdot \frac{3}{4} = \frac{3}{2} \]

### Step 6: Determine Values of n and m

From \( \frac{n}{m} = \frac{3}{2} \), we can express \( n \) and \( m \) as:
\[ n = 3k \quad \text{and} \quad m = 2k \]
for some integer \( k \).

### Conclusion

Thus, the values of \( n \) and \( m \) can be expressed in terms of a common integer \( k \):
- \( n = 3k \)
- \( m = 2k \)