A tunning fork of frequency 340 Hz resonates in the fundamental mode with an air column of length 125 cm in a cylindrical tube closed at one end. When water is slowly poured in it, the minimum height of water required for observing resonance once again is ________ cm.
(Velocity of sound in air is `340ms^(-1)`)

Answer

To solve the problem, we need to determine the minimum height of water required to observe resonance again in a cylindrical tube closed at one end when a tuning fork of frequency 340 Hz is used.

### Step-by-Step Solution:

1. **Understanding the Fundamental Frequency**:
The tube is closed at one end, which means it supports a fundamental frequency with a quarter wavelength fitting in the length of the air column. The fundamental frequency (first harmonic) can be described by the formula:
\[ L_1 = \frac{\lambda}{4} \]
where \(L_1\) is the length of the air column and \(\lambda\) is the wavelength.

2. **Calculating the Wavelength**:
Given that the frequency \(f\) is 340 Hz and the speed of sound \(v\) is 340 m/s, we can calculate the wavelength \(\lambda\) using the formula:
\[ v = f \cdot \lambda \implies \lambda = \frac{v}{f} = \frac{340 \, \text{m/s}}{340 \, \text{Hz}} = 1 \, \text{m} = 100 \, \text{cm} \]

3. **Finding the First Resonating Length**:
From the fundamental mode, we know:
\[ L_1 = \frac{\lambda}{4} = \frac{100 \, \text{cm}}{4} = 25 \, \text{cm} \]
This means that when the air column is 125 cm long, it has a fundamental frequency resonance at 25 cm.

4. **Determining the Second Resonating Length**:
The second resonance (first overtone) for a closed tube occurs at:
\[ L_2 = \frac{3\lambda}{4} = \frac{3 \times 100 \, \text{cm}}{4} = 75 \, \text{cm} \]

5. **Finding the Third Resonating Length**:
The third resonance (second overtone) occurs at:
\[ L_3 = \frac{5\lambda}{4} = \frac{5 \times 100 \, \text{cm}}{4} = 125 \, \text{cm} \]

6. **Calculating the Height of Water**:
Since the total length of the tube is 125 cm, when water is poured into the tube, the effective length of the air column decreases. To maintain resonance, we need to ensure that the first resonating length (25 cm) remains unchanged. Thus, the maximum height of water that can be poured in without changing the first resonating length is:
\[ \text{Height of water} = L_3 - L_2 = 125 \, \text{cm} - 75 \, \text{cm} = 50 \, \text{cm} \]

### Final Answer:
The minimum height of water required for observing resonance once again is **50 cm**.