If the tension of sonometer’s wire increases four times then the fundamental frequency of the wire will increase by

Options

A. 2 times
B. 4 times
C. 1/2 times
D. None of the above

Answer

To solve the problem, we need to understand the relationship between the tension in the wire and its fundamental frequency. The fundamental frequency of a vibrating wire is given by the formula:

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

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

### Step 1: Understand the initial conditions
Let the initial tension be \( T \). According to the problem, the new tension \( T' \) is four times the initial tension:

\[ T' = 4T \]

### Step 2: Write the formula for the new frequency
Using the formula for the fundamental frequency, we can express the new frequency \( f' \) when the tension is increased:

\[ f' = \frac{1}{2L} \sqrt{\frac{T'}{M}} \]

Substituting \( T' \):

\[ f' = \frac{1}{2L} \sqrt{\frac{4T}{M}} \]

### Step 3: Simplify the expression
We can simplify the square root:

\[ f' = \frac{1}{2L} \sqrt{4} \sqrt{\frac{T}{M}} \]
\[ f' = \frac{1}{2L} \cdot 2 \cdot \sqrt{\frac{T}{M}} \]
\[ f' = 2 \cdot \frac{1}{2L} \sqrt{\frac{T}{M}} \]
\[ f' = 2f \]

### Step 4: Conclusion
Thus, the new fundamental frequency \( f' \) is twice the initial fundamental frequency \( f \):

\[ f' = 2f \]

This means that if the tension of the sonometer's wire increases four times, the fundamental frequency of the wire will increase by a factor of 2.

### Final Answer
The fundamental frequency of the wire will increase by **2 times**.

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