Two strings of copper are stretched to the same tension. If their cross-section area are in the ratio 1 : 4 , then the respective wave velocities will be

Options

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

Answer

To solve the problem, we need to understand the relationship between the wave velocity in a string, the tension applied, and the mass per unit length (linear density) of the string.

### Step-by-Step Solution:

1. **Understanding Wave Velocity**:
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. **Mass per Unit Length**:
The mass per unit length \( \mu \) can be expressed as:
\[ \mu = \frac{m}{L} \]
where \( m \) is the mass of the string and \( L \) is its length.

3. **Relating Mass to Density**:
The mass \( m \) can also be expressed in terms of the density \( \rho \) and the volume \( V \):
\[ m = \rho \cdot V \]
For a string, the volume can be expressed as:
\[ V = A \cdot L \]
where \( A \) is the cross-sectional area of the string. Thus:
\[ m = \rho \cdot A \cdot L \]

4. **Substituting into Mass per Unit Length**:
Substituting the expression for mass into the formula for \( \mu \):
\[ \mu = \frac{\rho \cdot A \cdot L}{L} = \rho \cdot A \]

5. **Wave Velocity in Terms of Area**:
Now substituting \( \mu \) back into the wave velocity formula:
\[ v = \sqrt{\frac{T}{\rho \cdot A}} \]
From this, we can see that the wave velocity is inversely proportional to the square root of the area:
\[ v \propto \frac{1}{\sqrt{A}} \]

6. **Finding the Ratio of Velocities**:
Let \( A_1 \) and \( A_2 \) be the cross-sectional areas of the two strings. Given that the ratio of their areas is:
\[ \frac{A_1}{A_2} = \frac{1}{4} \]
Therefore, we can express \( A_2 \) in terms of \( A_1 \):
\[ A_2 = 4A_1 \]

7. **Calculating the Ratio of Velocities**:
Using the relationship derived earlier:
\[ \frac{v_1}{v_2} = \sqrt{\frac{A_2}{A_1}} = \sqrt{\frac{4A_1}{A_1}} = \sqrt{4} = 2 \]
Thus, the ratio of the velocities \( v_1 : v_2 = 2 : 1 \).

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