Two rods of different materials having coefficient of thermal expansion `alpha_(1), alpha_(2)` and young's modulii `Y_(1) ,Y_(2)` respectively are fixed between two rigid massive walls. The rods are heated such that they undergo the same increase in temperature. There is no bending of rods. If `alpha_(1) :alpha_(2)=2 : 3`, the thermal stresses developed in the two rods are equal provided `Y_(1) : Y_(2)` is equal to

Options

A. ` 2 : 3`
B. `1 : 1`
C. `3 : 2`
D. `4 : 9`

Explanation

To solve the problem, we need to find the ratio of the Young's moduli \( Y_1 \) and \( Y_2 \) of the two rods given that the thermal stresses developed in both rods are equal. We are given the ratio of their coefficients of thermal expansion \( \alpha_1 : \alpha_2 = 2 : 3 \).

### Step-by-Step Solution:

1. **Understanding Thermal Stress**:
The thermal stress in a material can be expressed as:
\[ \text{Stress} = Y \times \text{Strain} \]
where \( Y \) is the Young's modulus and the strain is caused by thermal expansion.

2. **Expressing Strain**:
The strain (\( \epsilon \)) due to thermal expansion can be expressed as:
\[ \epsilon = \frac{\Delta L}{L} = \alpha \Delta T \]
where \( \Delta L \) is the change in length, \( L \) is the original length, \( \alpha \) is the coefficient of thermal expansion, and \( \Delta T \) is the change in temperature.

3. **Setting Up the Equations**:
For rod 1 (material with \( Y_1 \) and \( \alpha_1 \)):
\[ \text{Stress}_1 = Y_1 \times \epsilon_1 = Y_1 \times \alpha_1 \Delta T \]
For rod 2 (material with \( Y_2 \) and \( \alpha_2 \)):
\[ \text{Stress}_2 = Y_2 \times \epsilon_2 = Y_2 \times \alpha_2 \Delta T \]

4. **Equating the Stresses**:
Since the thermal stresses are equal:
\[ Y_1 \alpha_1 \Delta T = Y_2 \alpha_2 \Delta T \]
We can cancel \( \Delta T \) from both sides (since it is the same for both rods):
\[ Y_1 \alpha_1 = Y_2 \alpha_2 \]

5. **Finding the Ratio of Young's Moduli**:
Rearranging the equation gives:
\[ \frac{Y_1}{Y_2} = \frac{\alpha_2}{\alpha_1} \]

6. **Substituting the Given Ratio**:
We know from the problem that:
\[ \frac{\alpha_1}{\alpha_2} = \frac{2}{3} \]
Therefore:
\[ \frac{\alpha_2}{\alpha_1} = \frac{3}{2} \]

7. **Final Ratio of Young's Moduli**:
Substituting this back into our equation:
\[ \frac{Y_1}{Y_2} = \frac{3}{2} \]

Thus, the required ratio of Young's moduli \( Y_1 : Y_2 \) is \( 3 : 2 \).

### Final Answer:
\[ Y_1 : Y_2 = 3 : 2 \]