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 establish the relationship between the thermal expansion of the two rods and the thermal stress developed in them. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the Problem
We have two rods made of different materials, each with a coefficient of thermal expansion (α₁ for rod 1 and α₂ for rod 2) and Young's modulus (Y₁ for rod 1 and Y₂ for rod 2). Both rods are fixed between two rigid walls and are heated, causing them to expand. The problem states that the ratio of their coefficients of thermal expansion is given as α₁ : α₂ = 2 : 3.

### Step 2: Relate Thermal Expansion and Stress
When the rods are heated, they expand. The thermal expansion (ΔL) of a rod due to a temperature change (ΔT) can be expressed as:
\[ \Delta L = \alpha L_0 \Delta T \]
where \( L_0 \) is the original length of the rod.

However, since the rods are fixed between rigid walls, they cannot expand freely. This results in a thermal stress (σ) developing in each rod, which can be expressed using Young's modulus:
\[ \sigma = \frac{F}{A} = Y \cdot \frac{\Delta L}{L_0} \]
where \( F \) is the force developed due to thermal expansion, and \( A \) is the cross-sectional area of the rod.

### Step 3: Set Up the Equations
For rod 1:
\[ \alpha_1 L_1 \Delta T = \frac{F}{A} \cdot \frac{L_1}{Y_1} \]
This simplifies to:
\[ \alpha_1 \Delta T = \frac{F}{A Y_1} \tag{1} \]

For rod 2:
\[ \alpha_2 L_2 \Delta T = \frac{F}{A} \cdot \frac{L_2}{Y_2} \]
This simplifies to:
\[ \alpha_2 \Delta T = \frac{F}{A Y_2} \tag{2} \]

### Step 4: Equate the Two Equations
Since the thermal stresses developed in both rods are equal, we can set the right-hand sides of equations (1) and (2) equal to each other:
\[ \alpha_1 \Delta T = \alpha_2 \Delta T \cdot \frac{Y_1}{Y_2} \]

### Step 5: Cancel ΔT and Rearrange
Assuming ΔT is the same for both rods, we can cancel it out:
\[ \alpha_1 = \alpha_2 \cdot \frac{Y_1}{Y_2} \]
Rearranging gives us:
\[ \frac{\alpha_1}{\alpha_2} = \frac{Y_1}{Y_2} \tag{3} \]

### Step 6: Substitute the Given Ratio
We know from the problem that:
\[ \frac{\alpha_1}{\alpha_2} = \frac{2}{3} \]
Substituting this into equation (3):
\[ \frac{2}{3} = \frac{Y_1}{Y_2} \]

### Step 7: Find the Ratio of Young's Moduli
Rearranging gives us:
\[ \frac{Y_1}{Y_2} = \frac{3}{2} \]

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