Two metal rods of the same length and area of cross-section are fixed end to end between rigid supports. The materials of the rods have Young moduli `Y_(1)` and `Y_(2)` , and coefficient of linear expansion `alpha_(1)` and `alpha_(2)` . The junction between the rod does not shift and the rods are cooled

Options

A. `Y_(1)alpha_(1)=Y_(2)alpha_(2)`
B. `Y_(1)alpha_(2)=Y_(2)alpha_(1)`
C. `Y_(1)alpha_(1)^(2)=Y_(2)alpha_(2)^(2)`
D. `Y_(1)^(2)alpha_(1)=Y_(2)^(2)alpha_(2)`

Answer

To solve the problem, we need to analyze the situation with the two metal rods that are fixed end to end. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the Setup
We have two rods of the same length (L) and cross-sectional area (A) made of different materials. The Young moduli of the materials are \( Y_1 \) and \( Y_2 \), and their coefficients of linear expansion are \( \alpha_1 \) and \( \alpha_2 \). The rods are cooled, which causes them to contract.

### Step 2: Condition at the Junction
Since the junction between the rods does not shift, the tension (or stress) in both rods must be the same. This means that the force exerted by both rods is equal.

### Step 3: Relate Stress and Strain
Using the definition of Young's modulus:
\[ Y = \frac{\text{Stress}}{\text{Strain}} \]
We can express the stress in terms of tension (T) and area (A):
\[ Y = \frac{T}{A \cdot \text{Strain}} \]
The strain can be expressed as:
\[ \text{Strain} = \frac{\Delta L}{L} \]
where \( \Delta L \) is the change in length due to temperature change.

### Step 4: Change in Length Due to Temperature
The change in length for each rod due to temperature change \( \Delta \theta \) can be expressed as:
\[ \Delta L = \alpha \cdot L \cdot \Delta \theta \]
Thus, the strain for each rod becomes:
\[ \text{Strain}_1 = \frac{\alpha_1 L \Delta \theta}{L} = \alpha_1 \Delta \theta \]
\[ \text{Strain}_2 = \frac{\alpha_2 L \Delta \theta}{L} = \alpha_2 \Delta \theta \]

### Step 5: Equate Tensions
From the definition of Young's modulus, we can express the tensions in both rods:
\[ T_1 = Y_1 A \cdot \text{Strain}_1 = Y_1 A \cdot \alpha_1 \Delta \theta \]
\[ T_2 = Y_2 A \cdot \text{Strain}_2 = Y_2 A \cdot \alpha_2 \Delta \theta \]

### Step 6: Set Tensions Equal
Since \( T_1 = T_2 \):
\[ Y_1 A \alpha_1 \Delta \theta = Y_2 A \alpha_2 \Delta \theta \]
We can cancel \( A \) and \( \Delta \theta \) (assuming \( \Delta \theta \neq 0 \)):
\[ Y_1 \alpha_1 = Y_2 \alpha_2 \]

### Conclusion
Thus, the relationship between the Young moduli and coefficients of linear expansion for the two rods is:
\[ Y_1 \alpha_1 = Y_2 \alpha_2 \]