Two steel rods and an aluminium rod of equal length `l_0` and equal cross- section are joined rigidly at their ends as shown in the figure below. All the rods are in a state of zero tension at `0^@ C`. Find the length of the system when the temperature is raised to `theta`. Coefficient of linear expansion of aluminium and steel are `alpha_(a)` and `alpha_(s)` respectively. Young's modulus of aluminium is `Y_(a)` and of steel is `Y_(s)`.

Explanation

To find the length of the system when the temperature is raised to θ, we will follow these steps:

### Step 1: Understand the Initial Conditions
At 0°C, the initial length of each rod is \( L_0 \), and they are in a state of zero tension. This means that the rods are not experiencing any stress at this temperature.

### Step 2: Define the Change in Length
When the temperature is raised to θ, the change in length for each rod can be expressed using the formula for linear expansion:
\[ \Delta L = L_0 \cdot \alpha \cdot \Delta T \]
where \( \Delta T = \theta - 0 = \theta \).

### Step 3: Calculate the Change in Length for Each Rod
- For the two steel rods, the change in length will be:
\[ \Delta L_s = L_0 \cdot \alpha_s \cdot \theta \]
Thus, for two steel rods, the total change in length is:
\[ \Delta L_{total, s} = 2 \cdot \Delta L_s = 2 \cdot L_0 \cdot \alpha_s \cdot \theta \]

- For the aluminum rod, the change in length will be:
\[ \Delta L_a = L_0 \cdot \alpha_a \cdot \theta \]

### Step 4: Total Change in Length
The total change in length of the system when the temperature is raised to θ is the sum of the changes in length of all three rods:
\[ \Delta L_{total} = \Delta L_{total, s} + \Delta L_a = 2 \cdot L_0 \cdot \alpha_s \cdot \theta + L_0 \cdot \alpha_a \cdot \theta \]

### Step 5: Express the Final Length
The final length \( L \) of the system at temperature θ can be expressed as:
\[ L = L_0 + \Delta L_{total} \]
Substituting the expression for \( \Delta L_{total} \):
\[ L = L_0 + (2 \cdot L_0 \cdot \alpha_s \cdot \theta + L_0 \cdot \alpha_a \cdot \theta) \]
Factoring out \( L_0 \):
\[ L = L_0 \left(1 + 2 \alpha_s \theta + \alpha_a \theta\right) \]

### Step 6: Simplify the Expression
This can be further simplified to:
\[ L = L_0 \left(1 + \theta (2 \alpha_s + \alpha_a)\right) \]

### Final Expression
Thus, the final length of the system when the temperature is raised to θ is:
\[ L = L_0 \left(1 + \theta (2 \alpha_s + \alpha_a)\right) \]