A stationary tuning fork is in resonance with an air column in a pipe . If the tuning fork is moved with a speed of `2ms^(-1)` in front of the open end of the pipe and parallel to it , the length of the pipe should be changed for the resonance to occur with the moving tuning fork if the speed of sound in air is `320ms^(-1)`, the smallest value of the percentage change required in the length of the pipe is _________.

Explanation

To solve the problem, we need to determine the change in length of the air column in the pipe required for resonance to occur when the tuning fork is moving. Here's a step-by-step breakdown of the solution:

### Step 1: Understanding the Problem
We have a stationary tuning fork that is in resonance with an air column in a pipe. When the tuning fork is moved with a speed of \(2 \, \text{m/s}\), we need to find out how the length of the pipe should change to maintain resonance.

### Step 2: Determine the Speed of Sound
The speed of sound in air is given as \(v = 320 \, \text{m/s}\).

### Step 3: Calculate the Apparent Frequency
When the tuning fork moves towards the open end of the pipe, the apparent frequency \(f'\) can be calculated using the formula:
\[ f' = f \left(\frac{v}{v - v_s}\right) \]
where \(v_s\) is the speed of the source (the tuning fork), which is \(2 \, \text{m/s}\).

### Step 4: Calculate the Original Frequency
The original frequency \(f\) of the stationary tuning fork is given by:
\[ f = \frac{v}{\lambda} \]
where \(\lambda\) is the wavelength. For a pipe closed at one end, the fundamental frequency corresponds to:
\[ \lambda = 4L \]
Thus,
\[ f = \frac{v}{4L} \]

### Step 5: Substitute into the Apparent Frequency Equation
Substituting for \(f\) in the apparent frequency equation:
\[ f' = \frac{v}{4L} \left(\frac{v}{v - 2}\right) \]

### Step 6: Find the New Wavelength
The new wavelength \(\lambda'\) corresponding to the new frequency \(f'\) is given by:
\[ \lambda' = \frac{v}{f'} \]
Substituting \(f'\) gives:
\[ \lambda' = \frac{v(v - 2)}{v/4L} = 4L \cdot \frac{(v - 2)}{v} \]

### Step 7: Relate the New Length to the Original Length
For resonance to occur, the new length \(L'\) must satisfy:
\[ \lambda' = 4L' \]
Thus,
\[ 4L' = 4L \cdot \frac{(v - 2)}{v} \]
This simplifies to:
\[ L' = L \cdot \frac{(v - 2)}{v} \]

### Step 8: Calculate the Percentage Change in Length
The percentage change in length is given by:
\[ \text{Percentage Change} = \left(\frac{L' - L}{L}\right) \times 100 \]
Substituting \(L'\):
\[ \text{Percentage Change} = \left(\frac{L \cdot \frac{(v - 2)}{v} - L}{L}\right) \times 100 = \left(\frac{(v - 2) - v}{v}\right) \times 100 = \left(\frac{-2}{v}\right) \times 100 \]
Substituting \(v = 320 \, \text{m/s}\):
\[ \text{Percentage Change} = \left(\frac{-2}{320}\right) \times 100 = -0.625\% \]

Since we are looking for the smallest value of the percentage change required in the length of the pipe, we take the absolute value:
\[ \text{Smallest Percentage Change} = 0.625\% \]

### Final Answer
The smallest value of the percentage change required in the length of the pipe is **0.625%**.

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