To solve the problem step by step, we will analyze the resonance tube experiment involving a closed organ pipe.
### Step 1: Determine the Wavelength
The speed of sound in air is given as \( v = 340 \, \text{m/s} \) and the frequency of the tuning fork is \( f = 340 \, \text{Hz} \). The wavelength \( \lambda \) can be calculated using the formula:
\[
\lambda = \frac{v}{f}
\]
Substituting the values:
\[
\lambda = \frac{340 \, \text{m/s}}{340 \, \text{Hz}} = 1 \, \text{m} = 100 \, \text{cm}
\]
### Step 2: Calculate the End Correction
The end correction \( E \) for a closed organ pipe can be calculated using the formula:
\[
E = 0.3 \times d
\]
where \( d \) is the diameter of the pipe. Given \( d = 10 \, \text{cm} \):
\[
E = 0.3 \times 10 \, \text{cm} = 3 \, \text{cm}
\]
### Step 3: First Harmonic Calculation
For the first harmonic in a closed organ pipe, the length of the air column \( L \) plus the end correction \( E \) is equal to one-fourth of the wavelength:
\[
L + E = \frac{\lambda}{4}
\]
Substituting the known values:
\[
L + 3 \, \text{cm} = \frac{100 \, \text{cm}}{4} = 25 \, \text{cm}
\]
Solving for \( L \):
\[
L = 25 \, \text{cm} - 3 \, \text{cm} = 22 \, \text{cm}
\]
### Step 4: Second Harmonic Calculation
For the second harmonic, the length of the air column \( L \) plus the end correction \( E \) is equal to three-fourths of the wavelength:
\[
L + E = \frac{3\lambda}{4}
\]
Substituting the known values:
\[
L + 3 \, \text{cm} = \frac{3 \times 100 \, \text{cm}}{4} = 75 \, \text{cm}
\]
Solving for \( L \):
\[
L = 75 \, \text{cm} - 3 \, \text{cm} = 72 \, \text{cm}
\]
### Step 5: Maximum Length of Water Column
The original length of the pipe is \( 120 \, \text{cm} \). To find the length of the water column when the first harmonic is set up:
\[
\text{Length of water column} = 120 \, \text{cm} - 22 \, \text{cm} = 98 \, \text{cm}
\]
For the second harmonic:
\[
\text{Length of water column} = 120 \, \text{cm} - 72 \, \text{cm} = 48 \, \text{cm}
\]
### Step 6: Conclusion
- The minimum length of the water column to achieve resonance is \( 48 \, \text{cm} \) (for the second harmonic).
- The maximum length of the water column for the first harmonic is \( 98 \, \text{cm} \).
- The distance between two successive nodes is \( 50 \, \text{cm} \) (difference between the lengths of the first and second harmonics).
