A tunning fork having a frequency of 170 Hz is vibrating just above a cylindrical tube. The height of the tube is 110 cm. Water is slowly poured in it. What is the minimum height (in cm) of water column required for resonance to occur? (Velocity of sound in air `=340ms^(-1)`)

Answer

To solve the problem, we need to determine the minimum height of the water column required for resonance to occur in a cylindrical tube when a tuning fork of frequency 170 Hz is vibrating above it. The speed of sound in air is given as 340 m/s.

### Step-by-Step Solution:

1. **Understanding Resonance in a Tube**:
- For resonance to occur in an open tube, the length of the air column must be a multiple of the wavelength of the sound produced by the tuning fork.
- The fundamental frequency (first harmonic) for an open tube occurs when the length of the tube is equal to half the wavelength (λ/2).

2. **Calculate Wavelength (λ)**:
- The relationship between speed (v), frequency (f), and wavelength (λ) is given by the formula:
\[ v = f \cdot \lambda \]
- Rearranging gives:
\[ \lambda = \frac{v}{f} \]
- Substituting the values:
\[ \lambda = \frac{340 \, \text{m/s}}{170 \, \text{Hz}} = 2 \, \text{m} \]

3. **Determine the Length of the Air Column for First Resonance**:
- For the first resonance (fundamental mode), the length of the air column (L1) is:
\[ L1 = \frac{\lambda}{4} = \frac{2 \, \text{m}}{4} = 0.5 \, \text{m} = 50 \, \text{cm} \]

4. **Calculate the Height of the Water Column**:
- The total height of the tube is 110 cm. If the length of the air column required for resonance is 50 cm, then the height of the water column (H) can be calculated as:
\[ H = \text{Total height of the tube} - L1 = 110 \, \text{cm} - 50 \, \text{cm} = 60 \, \text{cm} \]

5. **Final Answer**:
- The minimum height of the water column required for resonance to occur is:
\[ \boxed{60 \, \text{cm}} \]