A sonometer wire of length `l` vibrates in fundamental mode when excited by a tunning fork of frequency 416 Hz. If the length is double keeping other things same the string will

Options

A. vibrates with a frequency of 416 Hz
B. vibrate with a frequency of 208 Hz
C. vibrate with a frequency of 832 Hz
D. stop vibrating

Answer

To solve the problem, we need to analyze how the frequency of a sonometer wire changes when its length is altered. Let's break down the solution step by step.

### Step 1: Understanding the Fundamental Frequency Formula
The fundamental frequency \( f \) of a vibrating string (or sonometer wire) is given by the formula:

\[ f = \frac{1}{2L} \sqrt{\frac{T}{\mu}} \]

where:
- \( f \) is the fundamental frequency,
- \( L \) is the length of the wire,
- \( T \) is the tension in the wire,
- \( \mu \) is the mass per unit length of the wire.

### Step 2: Initial Conditions
From the problem, we know:
- The initial length of the wire is \( L \).
- The initial frequency when the wire vibrates in the fundamental mode is \( f = 416 \, \text{Hz} \).

### Step 3: Doubling the Length
Now, if the length of the wire is doubled, the new length becomes \( L' = 2L \).

### Step 4: Calculating the New Frequency
Substituting the new length into the frequency formula, we have:

\[ f' = \frac{1}{2L'} \sqrt{\frac{T}{\mu}} = \frac{1}{2(2L)} \sqrt{\frac{T}{\mu}} = \frac{1}{4L} \sqrt{\frac{T}{\mu}} \]

### Step 5: Relating New Frequency to Old Frequency
We can see that:

\[ f' = \frac{1}{2} \left( \frac{1}{2L} \sqrt{\frac{T}{\mu}} \right) = \frac{1}{2} f \]

This means that the new frequency \( f' \) is half of the original frequency \( f \).

### Step 6: Conclusion
Since the original frequency \( f \) is 416 Hz, the new frequency when the length is doubled will be:

\[ f' = \frac{1}{2} \times 416 \, \text{Hz} = 208 \, \text{Hz} \]

However, the question asks about the frequency with which the string vibrates when excited by the tuning fork, which remains unchanged at 416 Hz.

### Final Answer
The string will vibrate with a frequency of 416 Hz.

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