A tuning fork of frequency `340 Hz` is excited and held above a cylindrical tube of length `120 cm`. It is slowly filled with water. The minimum height of water column required for resonance to be first heard( Velocity of sound `=340 ms^-1`) is.

Options

A. (a)25 cm
B. (b)75 cm
C. (c)45 cm
D. (d)105 cm

Explanation

To solve the problem, we need to determine the minimum height of the water column required for the first resonance to be heard when a tuning fork of frequency 340 Hz is excited above a cylindrical tube of length 120 cm. The speed of sound is given as 340 m/s.

### Step-by-Step Solution:

1. **Determine the Wavelength (λ)**:
The relationship between velocity (v), frequency (f), and wavelength (λ) is given by the formula:
\[ v = f \cdot \lambda \]
Rearranging this formula to solve for wavelength gives:
\[ \lambda = \frac{v}{f} \]
Substituting the given values:
\[ \lambda = \frac{340 \, \text{m/s}}{340 \, \text{Hz}} = 1 \, \text{m} \]
Converting to centimeters:
\[ \lambda = 100 \, \text{cm} \]

2. **Identify the Resonance Condition**:
For a tube open at one end and closed at the other (the end filled with water), the resonance occurs at odd multiples of \( \frac{\lambda}{4} \):
- First resonance: \( \frac{\lambda}{4} \)
- Second resonance: \( \frac{3\lambda}{4} \)
- Third resonance: \( \frac{5\lambda}{4} \), and so on.

The first resonance occurs at:
\[ L_1 = \frac{\lambda}{4} = \frac{100 \, \text{cm}}{4} = 25 \, \text{cm} \]

3. **Calculate the Length of the Air Column**:
The length of the air column at which the first resonance is heard is 25 cm. Since the total length of the tube is 120 cm, the height of the water column can be calculated as:
\[ \text{Height of water column} = \text{Total length of tube} - \text{Length of air column} \]
\[ \text{Height of water column} = 120 \, \text{cm} - 25 \, \text{cm} = 95 \, \text{cm} \]

4. **Conclusion**:
The minimum height of the water column required for the first resonance to be heard is:
\[ \text{Height of water column} = 95 \, \text{cm} \]

### Final Answer:
The minimum height of the water column required for resonance to be first heard is **95 cm**.