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 : 1`
D. `4 : 9`

Explanation

To solve the problem, we need to establish the relationship between the thermal expansions of the two rods and the resulting thermal stresses. Here’s the step-by-step solution:

### Step 1: Understand the Problem
We have two rods made of different materials, fixed between two rigid walls, and subjected to the same temperature increase. The coefficients of thermal expansion for the rods are given as \( \alpha_1 \) and \( \alpha_2 \), and their Young's moduli are \( Y_1 \) and \( Y_2 \). The ratio of the coefficients of thermal expansion is given as \( \frac{\alpha_1}{\alpha_2} = \frac{2}{3} \).

### Step 2: Write the Equation for Thermal Expansion
The linear expansion of a rod due to a temperature change \( \Delta T \) is given by:
\[ \Delta L = L \cdot \alpha \cdot \Delta T \]
For rod 1:
\[ \Delta L_1 = L_1 \cdot \alpha_1 \cdot \Delta T \]
For rod 2:
\[ \Delta L_2 = L_2 \cdot \alpha_2 \cdot \Delta T \]

### Step 3: Relate Thermal Expansion to Elastic Compression
Since the rods are fixed and cannot expand freely, the thermal expansion will be countered by the elastic compression. The compressive force \( F \) in each rod can be expressed in terms of Young's modulus \( Y \) and the strain (which is the change in length divided by the original length):
\[ \text{Stress} = \frac{F}{A} = Y \cdot \text{Strain} \]
The strain for each rod is given by:
\[ \text{Strain} = \frac{\Delta L}{L} \]
Thus, for rod 1:
\[ \frac{F_1}{A} = Y_1 \cdot \frac{\Delta L_1}{L_1} \]
And for rod 2:
\[ \frac{F_2}{A} = Y_2 \cdot \frac{\Delta L_2}{L_2} \]

### Step 4: Set the Expansions Equal to the Compressions
Since the rods are in equilibrium and there is no bending, we can equate the expansions to the compressions:
\[ L_1 \cdot \alpha_1 \cdot \Delta T = \frac{F_1 L_1}{A Y_1} \]
\[ L_2 \cdot \alpha_2 \cdot \Delta T = \frac{F_2 L_2}{A Y_2} \]

### Step 5: Divide the Two Equations
Dividing the first equation by the second:
\[ \frac{L_1 \cdot \alpha_1 \cdot \Delta T}{L_2 \cdot \alpha_2 \cdot \Delta T} = \frac{F_1 L_1}{A Y_1} \cdot \frac{A Y_2}{F_2 L_2} \]
This simplifies to:
\[ \frac{\alpha_1}{\alpha_2} = \frac{Y_2}{Y_1} \]

### Step 6: Substitute the Given Ratio
We know that:
\[ \frac{\alpha_1}{\alpha_2} = \frac{2}{3} \]
Thus, we can write:
\[ \frac{2}{3} = \frac{Y_2}{Y_1} \]

### Step 7: Find the Required Ratio
To find the ratio \( \frac{Y_1}{Y_2} \):
\[ \frac{Y_1}{Y_2} = \frac{3}{2} \]

### Final Answer
The ratio of the Young's moduli is:
\[ Y_1 : Y_2 = 3 : 2 \]