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 step by step, we will analyze the thermal stresses developed in two rods made of different materials when they are subjected to the same increase in temperature.

### Step 1: Understand the Problem
We have two rods fixed between rigid walls, and they are heated such that they undergo the same increase in temperature (ΔT). The coefficients of thermal expansion for the two rods are given as α₁ and α₂, and their Young's moduli are Y₁ and Y₂. We need to find the ratio of Young's moduli (Y₁ : Y₂) under the condition that the thermal stresses in both rods are equal.

### Step 2: Write the Expression for Thermal Stress
The thermal stress (σ) in a rod can be expressed using the formula:
\[ \sigma = Y \cdot \text{strain} \]
Where strain (ε) is given by:
\[ \text{strain} = \frac{\Delta L}{L_0} = \alpha \cdot \Delta T \]
Here, ΔL is the change in length, L₀ is the original length, α is the coefficient of thermal expansion, and ΔT is the change in temperature.

### Step 3: Express Thermal Stress for Both Rods
For rod 1:
\[ \sigma_1 = Y_1 \cdot \alpha_1 \cdot \Delta T \]

For rod 2:
\[ \sigma_2 = Y_2 \cdot \alpha_2 \cdot \Delta T \]

### Step 4: Set the Thermal Stresses Equal
Since the thermal stresses in both rods are equal, we can set the two expressions for stress equal to each other:
\[ Y_1 \cdot \alpha_1 \cdot \Delta T = Y_2 \cdot \alpha_2 \cdot \Delta T \]

### Step 5: Cancel Out ΔT
Since ΔT is the same for both rods and is not equal to zero, we can cancel it from both sides:
\[ Y_1 \cdot \alpha_1 = Y_2 \cdot \alpha_2 \]

### Step 6: Rearrange the Equation
Rearranging the equation gives us:
\[ \frac{Y_1}{Y_2} = \frac{\alpha_2}{\alpha_1} \]

### Step 7: Substitute the Given Ratio of α
We are given that the ratio of the coefficients of thermal expansion is:
\[ \frac{\alpha_1}{\alpha_2} = \frac{2}{3} \]
This implies:
\[ \frac{\alpha_2}{\alpha_1} = \frac{3}{2} \]

### Step 8: Substitute into the Young's Moduli Ratio
Now substituting this ratio into our rearranged equation:
\[ \frac{Y_1}{Y_2} = \frac{3}{2} \]

### Conclusion
Thus, the ratio of Young's moduli is:
\[ Y_1 : Y_2 = 3 : 2 \]