Two rods, one of aluminium and the other made of steel, having initial length `l_1 and l_2` are connected together to from a single rod of length `l_1+l_2.` The coefficients of linear expansion for aluminium and steel are `alpha_a and alpha_s` and respectively. If the length of each rod increases by the same amount when their temperature are raised by `t^0C,` then find the ratio `l_1//(l_1+l_2)`

Options

A. `(alpha_(s))/(alpha_(a))`
B. `(alpha_(a))/(alpha_(s))`
C. `(alpha_(s))/((alpha_(a)+alpha_(s)))`
D. `(alpha_(a))/((alpha_(a)+alpha_(s)))`

Answer

To solve the problem, we need to analyze the situation involving two rods made of different materials (aluminum and steel) and their behavior under temperature changes.

### Step-by-Step Solution:

1. **Understanding Linear Expansion**:
The linear expansion of a material can be expressed as:
\[ \Delta L = L_0 \cdot \alpha \cdot \Delta T \]
where \( \Delta L \) is the change in length, \( L_0 \) is the original length, \( \alpha \) is the coefficient of linear expansion, and \( \Delta T \) is the change in temperature.

2. **Setting Up the Problem**:
Let the initial lengths of the aluminum and steel rods be \( l_1 \) and \( l_2 \) respectively. When the temperature is raised by \( t \) degrees Celsius, the change in length for each rod can be expressed as:
- For aluminum:
\[ \Delta L_a = l_1 \cdot \alpha_a \cdot t \]
- For steel:
\[ \Delta L_s = l_2 \cdot \alpha_s \cdot t \]

3. **Condition of Equal Length Increase**:
According to the problem, the increase in length for both rods is the same:
\[ \Delta L_a = \Delta L_s \]
Therefore, we can set the equations equal to each other:
\[ l_1 \cdot \alpha_a \cdot t = l_2 \cdot \alpha_s \cdot t \]

4. **Simplifying the Equation**:
Since \( t \) is common on both sides, we can cancel it out (assuming \( t \neq 0 \)):
\[ l_1 \cdot \alpha_a = l_2 \cdot \alpha_s \]

5. **Finding the Ratio**:
Rearranging the equation gives us:
\[ \frac{l_1}{l_2} = \frac{\alpha_s}{\alpha_a} \]
Now, we can express \( l_2 \) in terms of \( l_1 \):
\[ l_2 = l_1 \cdot \frac{\alpha_a}{\alpha_s} \]

6. **Total Length**:
The total length of the combined rod is:
\[ L = l_1 + l_2 = l_1 + l_1 \cdot \frac{\alpha_a}{\alpha_s} = l_1 \left(1 + \frac{\alpha_a}{\alpha_s}\right) \]

7. **Finding the Desired Ratio**:
We need to find the ratio \( \frac{l_1}{l_1 + l_2} \):
\[ \frac{l_1}{l_1 + l_2} = \frac{l_1}{l_1 \left(1 + \frac{\alpha_a}{\alpha_s}\right)} = \frac{1}{1 + \frac{\alpha_a}{\alpha_s}} = \frac{\alpha_s}{\alpha_s + \alpha_a} \]

### Final Result:
Thus, the ratio \( \frac{l_1}{l_1 + l_2} \) is:
\[ \frac{l_1}{l_1 + l_2} = \frac{\alpha_s}{\alpha_s + \alpha_a} \]