A tuning fork of frequency 256 Hz produces 4 beats per second with a wire of length 25 cm vibrating in its fundamental mode. The beat frequency decreases when the length is slightly shortened. What could be the minimum length by which the wire be shortened so that it produces no beats with the tuning fork ?

Explanation

To solve the problem step by step, we will follow the concepts related to sound waves, beats, and the fundamental frequency of a vibrating wire.

### Step 1: Understand the Given Information
- Frequency of the tuning fork (ν₁) = 256 Hz
- Beat frequency (n) = 4 beats per second
- Length of the wire (L₂) = 25 cm

### Step 2: Determine the Frequency of the Wire (ν₂)
The beat frequency is the absolute difference between the frequencies of the two sources. Therefore, we can express this as:
\[ |ν₁ - ν₂| = n \]

Since the beat frequency decreases when the length of the wire is shortened, it implies that the frequency of the wire (ν₂) must be less than the frequency of the tuning fork (ν₁). Thus, we can write:
\[ ν₁ - ν₂ = n \]

Substituting the known values:
\[ 256 - ν₂ = 4 \]
\[ ν₂ = 256 - 4 = 252 \text{ Hz} \]

### Step 3: Relate Frequency to Length
The frequency of a wire vibrating in its fundamental mode is given by:
\[ ν = \frac{1}{2L} \sqrt{\frac{T}{μ}} \]
Where:
- L = length of the wire
- T = tension in the wire (assumed constant)
- μ = mass per unit length (assumed constant)

From this, we can see that frequency is inversely proportional to the length of the wire:
\[ ν \propto \frac{1}{L} \]

### Step 4: Set Up the Equation for New Length
When the wire is shortened to a new length (L₂'), the new frequency (ν₂') will equal the frequency of the tuning fork (ν₁) when there are no beats:
\[ ν₂' = ν₁ \]

Using the relationship between the frequencies and lengths:
\[ \frac{ν₂'}{ν₂} = \frac{L₂}{L₂'} \]

Substituting the known values:
\[ \frac{256}{252} = \frac{25}{L₂'} \]

### Step 5: Solve for the New Length (L₂')
Cross-multiplying gives:
\[ 256 \cdot L₂' = 252 \cdot 25 \]
\[ L₂' = \frac{252 \cdot 25}{256} \]

Calculating this:
\[ L₂' = \frac{6300}{256} \approx 24.609375 \text{ cm} \]

### Step 6: Calculate the Change in Length
Now, we need to find the minimum length by which the wire must be shortened:
\[ \Delta L = L₂ - L₂' \]
\[ \Delta L = 25 - 24.609375 \approx 0.390625 \text{ cm} \]

Rounding this to two decimal places gives:
\[ \Delta L \approx 0.4 \text{ cm} \]

### Final Answer
The minimum length by which the wire must be shortened to produce no beats with the tuning fork is approximately **0.4 cm**.

---