A pipe's lower end is immersed in water such that the length of air column from the top open end has a certain length 25 cm. The speed of sound in air is 350 m/s. The air column is found to resonate with a tuning fork of frequency 1750 Hz. By what minimum distance should the pipe be raised in order to make the air column resonate again with the same tuning fork?
(a) 7cm
(b) 5cm
(c) 35cm
(d) 10cm

Options

A. 7cm
B. 5cm
C. 35cm
D. 10cm

Answer

To solve the problem, we need to determine the minimum distance by which the pipe should be raised to achieve resonance again with the same tuning fork frequency. Let's break down the solution step by step.

### Step 1: Identify the parameters given in the problem
- Length of the air column (L) = 25 cm = 0.25 m
- Speed of sound in air (v) = 350 m/s
- Frequency of the tuning fork (f) = 1750 Hz

### Step 2: Use the formula for the frequency of a one-sided closed pipe
The formula for the frequency of a one-sided closed pipe is given by:
\[ f_n = \frac{(2n - 1)v}{4L} \]
where \( n \) is the mode number (1, 2, 3,...).

### Step 3: Calculate the mode number (n) for the first resonance
We can rearrange the formula to find \( n \):
\[ n = \frac{4Lf}{v} + 1 \]
Substituting the values:
\[ n = \frac{4 \times 0.25 \times 1750}{350} + 1 \]
Calculating:
\[ n = \frac{1750}{350} + 1 = 5 + 1 = 6 \]
Since \( n \) must be an integer, we can see that the first resonance occurs at \( n = 3 \) (the first three odd harmonics).

### Step 4: Determine the next mode number for resonance
To find the next resonance, we increase \( n \) by 1:
\[ n_{next} = 3 + 1 = 4 \]

### Step 5: Calculate the new length of the air column (L')
Using the frequency formula again for \( n = 4 \):
\[ f = \frac{(2n - 1)v}{4L'} \]
Rearranging gives:
\[ L' = \frac{(2n - 1)v}{4f} \]
Substituting \( n = 4 \):
\[ L' = \frac{(2 \cdot 4 - 1) \cdot 350}{4 \cdot 1750} \]
Calculating:
\[ L' = \frac{(8 - 1) \cdot 350}{7000} = \frac{7 \cdot 350}{7000} = \frac{2450}{7000} = 0.35 \text{ m} \]

### Step 6: Calculate the minimum distance to raise the pipe
The increase in length is given by:
\[ \Delta L = L' - L = 0.35 - 0.25 = 0.10 \text{ m} = 10 \text{ cm} \]

### Conclusion
The minimum distance by which the pipe should be raised is **10 cm**.

### Final Answer
(d) 10 cm

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