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 that are made of the same material, stretched to the same tension, and have radii in the ratio of 1:2.

### Step-by-Step Solution:

1. **Identify the Given Information**:
- The radii of the two strings are in the ratio \( R_1 : R_2 = 1 : 2 \).
- Both strings are made of the same material and are under the same tension \( T \).

2. **Understand the Wave Velocity Formula**:
- The velocity of a wave in a string is given by the formula:
\[ V = \sqrt{\frac{T}{\mu}} \]
where \( \mu \) is the mass per unit length of the string.

3. **Express Mass per Unit Length**:
- The mass per unit length \( \mu \) can be expressed as:
\[ \mu = \frac{M}{L} = \frac{\rho \cdot V}{L} \]
where \( \rho \) is the density of the material and \( V \) is the volume of the string.

4. **Relate Volume to Cross-Sectional Area**:
- The volume \( V \) of the string can be expressed in terms of its cross-sectional area \( A \) and length \( L \):
\[ V = A \cdot L \]
- The cross-sectional area \( A \) for a circular string is given by:
\[ A = \pi R^2 \]
- Thus, we can write:
\[ \mu = \frac{\rho \cdot (\pi R^2 \cdot L)}{L} = \rho \cdot \pi R^2 \]

5. **Substitute into the Wave Velocity Formula**:
- Now substituting \( \mu \) back into the wave velocity formula:
\[ V = \sqrt{\frac{T}{\rho \cdot \pi R^2}} \]

6. **Analyze the Ratio of Velocities**:
- For the first string:
\[ V_1 = \sqrt{\frac{T}{\rho \cdot \pi R_1^2}} \]
- For the second string:
\[ V_2 = \sqrt{\frac{T}{\rho \cdot \pi R_2^2}} \]
- Taking the ratio of the velocities:
\[ \frac{V_1}{V_2} = \frac{\sqrt{\frac{T}{\rho \cdot \pi R_1^2}}}{\sqrt{\frac{T}{\rho \cdot \pi R_2^2}}} = \sqrt{\frac{R_2^2}{R_1^2}} = \frac{R_2}{R_1} \]

7. **Substituting the Ratio of Radii**:
- Since \( \frac{R_1}{R_2} = \frac{1}{2} \), we have:
\[ \frac{R_2}{R_1} = 2 \]
- Therefore, the ratio of the velocities is:
\[ \frac{V_1}{V_2} = 2 \implies V_1 : V_2 = 2 : 1 \]

### Final Answer:
The respective wave velocities in the two strings will be in the ratio \( 2 : 1 \).

---