A string of length 2x is stretched by 0.1 x and the velocity of a transverse wave along it is v. When it is stretched by 0.4x , the velocity of the wave is

Options

A. `sqrt(5/6)v`
B. `sqrt(11/7)v`
C. `sqrt(32/7)v`
D. `sqrt(27/6)v`

Answer

To solve the problem, we need to analyze how the velocity of a transverse wave along a string changes when the string is stretched by different amounts. The velocity of a wave on a string is given by the formula:

\[ v = \sqrt{\frac{T}{\mu}} \]

where:
- \( T \) is the tension in the string,
- \( \mu \) is the mass per unit length of the string.

### Step 1: Understand the relationship between tension and stretch
The tension \( T \) in the string is proportional to the amount it is stretched. If the string is stretched by an amount \( \Delta x \), we can express the tension as:

\[ T = k \cdot \Delta x \]

where \( k \) is a constant of proportionality.

### Step 2: Calculate the initial conditions
In the first case, the string of length \( 2x \) is stretched by \( 0.1x \). Therefore, the tension can be expressed as:

\[ T_1 = k \cdot 0.1x \]

The effective length of the string when stretched is:

\[ L_1 = 2x + 0.1x = 2.1x \]

Now, substituting into the velocity formula:

\[ v = \sqrt{\frac{T_1}{\mu}} = \sqrt{\frac{k \cdot 0.1x}{\mu}} \]

### Step 3: Calculate the second case
In the second case, the string is stretched by \( 0.4x \). Thus, the tension becomes:

\[ T_2 = k \cdot 0.4x \]

The effective length of the string when stretched is:

\[ L_2 = 2x + 0.4x = 2.4x \]

Now, substituting into the velocity formula for the second case:

\[ v_2 = \sqrt{\frac{T_2}{\mu}} = \sqrt{\frac{k \cdot 0.4x}{\mu}} \]

### Step 4: Compare the two cases
To find the ratio of the velocities \( \frac{v}{v_2} \):

\[ \frac{v}{v_2} = \frac{\sqrt{k \cdot 0.1x}}{\sqrt{k \cdot 0.4x}} = \sqrt{\frac{0.1}{0.4}} = \sqrt{\frac{1}{4}} = \frac{1}{2} \]

### Step 5: Express \( v_2 \) in terms of \( v \)
From the ratio, we can express \( v_2 \):

\[ v_2 = 2v \]

### Step 6: Final calculation
Now substituting the expressions back, we can find \( v_2 \):

\[ v_2 = \sqrt{\frac{0.4 \cdot k}{\mu}} = \sqrt{\frac{0.4}{2.4}} \cdot v \]

### Conclusion
Thus, the velocity of the wave when the string is stretched by \( 0.4x \) is:

\[ v_2 = \sqrt{\frac{0.4}{2.4}} \cdot v = \sqrt{\frac{1}{6}} \cdot v \]