Of the following the graph that better represents the variation of frequency 'v' of a vibrating string with the square root of tension `sqrt(T)` is

Options

A.
B.
C.
D.

Explanation

To solve the problem of how the frequency \( \nu \) of a vibrating string varies with the square root of tension \( \sqrt{T} \), we can follow these steps:

### Step 1: Understand the formula for frequency
The frequency \( \nu \) of a vibrating string is given by the formula:
\[ \nu = \frac{1}{2L} \sqrt{\frac{T}{\mu}} \]
where:
- \( L \) is the length of the string,
- \( T \) is the tension in the string,
- \( \mu \) is the mass per unit length of the string.

### Step 2: Isolate the variables
If we keep \( L \) and \( \mu \) constant, we can rewrite the equation as:
\[ \nu = C \sqrt{T} \]
where \( C = \frac{1}{2L} \sqrt{\frac{1}{\mu}} \) is a constant.

### Step 3: Relate frequency to square root of tension
From the equation \( \nu = C \sqrt{T} \), we see that frequency \( \nu \) is directly proportional to the square root of tension \( \sqrt{T} \). This means that if we plot \( \nu \) on the y-axis and \( \sqrt{T} \) on the x-axis, we should get a straight line passing through the origin.

### Step 4: Identify the graph
The graph that represents this relationship will be a straight line with a positive slope, indicating that as the square root of tension increases, the frequency also increases linearly.

### Conclusion
The correct graph that represents the variation of frequency \( \nu \) with the square root of tension \( \sqrt{T} \) is a straight line starting from the origin.

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