Two strings of same material are stretched to the same tension . If their radii are in the ratio 1:2 , then respective wave velocities in them will be in ratio

Options

A. `4:1`
B. `2:1`
C. `1:2`
D. `1:4`

Answer

To solve the problem, we need to find the ratio of wave velocities in two strings of the same material, stretched under the same tension, with their radii in the ratio of 1:2.

### Step-by-Step Solution:

1. **Understand the relationship for wave velocity in a string**:
The wave velocity \( V \) in a string is given by the formula:
\[ V = \sqrt{\frac{T}{\mu}} \]
where \( T \) is the tension in the string and \( \mu \) is the mass per unit length of the string.

2. **Determine the mass per unit length \( \mu \)**:
The mass per unit length \( \mu \) can be expressed as:
\[ \mu = \frac{m}{L} = \frac{\text{density} \times \text{volume}}{L} \]
The volume of the string can be expressed in terms of its cross-sectional area and length:
\[ \text{Volume} = \text{Area} \times \text{Length} = \pi r^2 L \]
Thus, we can write:
\[ \mu = \frac{\rho \cdot \pi r^2 L}{L} = \rho \cdot \pi r^2 \]
where \( \rho \) is the density of the material.

3. **Substitute \( \mu \) into the wave velocity formula**:
Now substituting \( \mu \) back into the wave velocity formula gives:
\[ V = \sqrt{\frac{T}{\rho \cdot \pi r^2}} \]

4. **Analyze the relationship of wave velocities for two strings**:
Let \( V_1 \) and \( V_2 \) be the wave velocities in the first and second strings, respectively. The radii of the strings are in the ratio \( r_1 : r_2 = 1 : 2 \). Therefore, we can express:
\[ V_1 = \sqrt{\frac{T}{\rho \cdot \pi r_1^2}} \quad \text{and} \quad V_2 = \sqrt{\frac{T}{\rho \cdot \pi r_2^2}} \]

5. **Set up the ratio of the wave velocities**:
The ratio of the wave velocities \( \frac{V_1}{V_2} \) can be expressed as:
\[ \frac{V_1}{V_2} = \frac{\sqrt{\frac{T}{\rho \cdot \pi r_1^2}}}{\sqrt{\frac{T}{\rho \cdot \pi r_2^2}}} = \sqrt{\frac{r_2^2}{r_1^2}} = \frac{r_2}{r_1} \]

6. **Substitute the values of the radii**:
Given \( r_1 : r_2 = 1 : 2 \), we can substitute:
\[ \frac{V_1}{V_2} = \frac{2}{1} = 2 \]

### Final Result:
Thus, the ratio of the wave velocities in the two strings is:
\[ V_1 : V_2 = 2 : 1 \]