Two wires of the same material and length are stretched by the same force. Their masses are in the ratio `3:2`. Their elongations are in the ratio

Options

A. `3:2`
B. `9:4`
C. `2:3`
D. `4:9`

Explanation

To solve the problem, we need to find the ratio of elongations of two wires that are made of the same material, have the same length, and are subjected to the same force. Given that their masses are in the ratio of 3:2, we can derive the ratio of their elongations step by step.

### Step-by-Step Solution:

1. **Understanding Young's Modulus**:
Young's modulus (Y) is defined as the ratio of stress to strain. Mathematically, it can be expressed as:
\[ Y = \frac{\text{Stress}}{\text{Strain}} = \frac{F/A}{\Delta L/L} \]
where \( F \) is the force applied, \( A \) is the cross-sectional area, \( \Delta L \) is the elongation, and \( L \) is the original length.

2. **Rearranging for Elongation**:
From the Young's modulus formula, we can rearrange it to find the elongation:
\[ \Delta L = \frac{F \cdot L}{A \cdot Y} \]

3. **Relating Mass to Area**:
The mass of the wire (m) can be expressed in terms of density (ρ), area (A), and length (L):
\[ m = \rho \cdot A \cdot L \]
Since both wires are made of the same material and have the same length, we can say:
\[ m \propto A \]
This means that the mass is directly proportional to the cross-sectional area.

4. **Finding the Ratio of Areas**:
Given that the masses of the wires are in the ratio \( m_1:m_2 = 3:2 \), we can express the areas in the same ratio:
\[ A_1:A_2 = 3:2 \]

5. **Finding the Ratio of Elongations**:
Since elongation is inversely proportional to the area (from the rearranged Young's modulus equation), we have:
\[ \Delta L \propto \frac{1}{A} \]
Therefore, the ratio of elongations will be:
\[ \frac{\Delta L_1}{\Delta L_2} = \frac{A_2}{A_1} \]
Substituting the area ratio:
\[ \frac{\Delta L_1}{\Delta L_2} = \frac{2}{3} \]

6. **Final Result**:
Thus, the ratio of the elongations of the two wires is:
\[ \Delta L_1 : \Delta L_2 = 2 : 3 \]

### Conclusion:
The elongations of the two wires are in the ratio \( 2:3 \).