A glass tube of `1.0 m` length is filled with water . The water can be drained out slowly at the bottom of the tube . A vibrating tuning fork of frequency `500 Hz` is brought at the upper end of the tube and the velocity of sound is `330 m//s`. Find the number of resonances that can be obtained.

Explanation

To find the number of resonances that can be obtained in a glass tube of length 1.0 m filled with water, we can follow these steps:

### Step 1: Calculate the wavelength of the sound wave
The wavelength \(\lambda\) can be calculated using the formula:
\[ \lambda = \frac{V}{f} \]
where \(V\) is the velocity of sound (330 m/s) and \(f\) is the frequency of the tuning fork (500 Hz).

Substituting the values:
\[ \lambda = \frac{330 \, \text{m/s}}{500 \, \text{Hz}} = 0.66 \, \text{m} \]

### Step 2: Determine the resonating lengths
In a tube that is closed at one end (the bottom end is closed by water), the resonating lengths can be expressed as:
\[ L_n = \frac{(2n - 1) \lambda}{4} \]
where \(n\) is the resonance mode number (1, 2, 3,...).

#### First Resonating Length (\(n=1\)):
\[ L_1 = \frac{(2 \times 1 - 1) \lambda}{4} = \frac{1 \times 0.66}{4} = 0.165 \, \text{m} \]

#### Second Resonating Length (\(n=2\)):
\[ L_2 = \frac{(2 \times 2 - 1) \lambda}{4} = \frac{3 \times 0.66}{4} = 0.495 \, \text{m} \]

#### Third Resonating Length (\(n=3\)):
\[ L_3 = \frac{(2 \times 3 - 1) \lambda}{4} = \frac{5 \times 0.66}{4} = 0.825 \, \text{m} \]

#### Fourth Resonating Length (\(n=4\)):
\[ L_4 = \frac{(2 \times 4 - 1) \lambda}{4} = \frac{7 \times 0.66}{4} = 1.155 \, \text{m} \]

### Step 3: Determine the maximum number of resonances
The maximum length of the tube is 1.0 m. We need to check how many of the calculated resonating lengths are less than or equal to 1.0 m.

- \(L_1 = 0.165 \, \text{m}\) (valid)
- \(L_2 = 0.495 \, \text{m}\) (valid)
- \(L_3 = 0.825 \, \text{m}\) (valid)
- \(L_4 = 1.155 \, \text{m}\) (not valid)

Thus, the maximum number of resonances that can be obtained is 3.

### Final Answer:
The number of resonances that can be obtained is **3**.

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