Two rods of different materials having coefficients 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 the 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 the two rods are equal when they are heated.

### Step 1: Understand the relationship between thermal stress and the parameters given
The thermal stress \( \sigma \) in a rod can be expressed as:
\[ \sigma = Y \cdot \alpha \cdot \Delta T \]
where:
- \( Y \) is the Young's modulus,
- \( \alpha \) is the coefficient of thermal expansion,
- \( \Delta T \) is the change in temperature.

### Step 2: Set up the equation for thermal stress in both rods
Let the thermal stress in rod 1 be \( \sigma_1 \) and in rod 2 be \( \sigma_2 \). According to the problem, these stresses are equal:
\[ \sigma_1 = \sigma_2 \]
Substituting the expression for thermal stress, we have:
\[ Y_1 \cdot \alpha_1 \cdot \Delta T = Y_2 \cdot \alpha_2 \cdot \Delta T \]

### Step 3: Cancel out the common term
Since both rods undergo the same temperature change \( \Delta T \), we can cancel it from both sides:
\[ Y_1 \cdot \alpha_1 = Y_2 \cdot \alpha_2 \]

### Step 4: Rearrange the equation to find the ratio of Young's moduli
Rearranging the equation gives us:
\[ \frac{Y_1}{Y_2} = \frac{\alpha_2}{\alpha_1} \]

### Step 5: Substitute the given ratio of coefficients of thermal expansion
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 6: Substitute this ratio into the equation for Young's moduli
Now substituting this into our equation gives:
\[ \frac{Y_1}{Y_2} = \frac{3}{2} \]

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

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