An iron rod of length 50 cm is joined at an end to an aluminum rod of length 100 cm. All measurements refer to `20^(@)C` . The coefficients of linear expansion of iron and aluminum are `12xx10^(-6)//^(@)C and 24xx10^(-6)//^(@)C`, respectively. The average coefficient of expansion of composite system is :

Options

A. `36xx10^(-6)//^(@)C`
B. `12xx10^(-6)//^(@)C`
C. `20xx10^(-6)//^(@)C`
D. `48xx10^(-6)//^(@)C`

Explanation

To find the average coefficient of linear expansion of the composite system made of an iron rod and an aluminum rod, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the lengths and coefficients of expansion:**
- Length of iron rod, \( L_{iron} = 50 \, \text{cm} \)
- Length of aluminum rod, \( L_{aluminum} = 100 \, \text{cm} \)
- Coefficient of linear expansion of iron, \( \alpha_{iron} = 12 \times 10^{-6} \, \text{°C}^{-1} \)
- Coefficient of linear expansion of aluminum, \( \alpha_{aluminum} = 24 \times 10^{-6} \, \text{°C}^{-1} \)

2. **Calculate the total length of the composite system:**
\[ L_{total} = L_{iron} + L_{aluminum} = 50 \, \text{cm} + 100 \, \text{cm} = 150 \, \text{cm} \]

3. **Write the formula for change in length (\( \Delta L \)) for each rod:**
- For the iron rod:
\[ \Delta L_{iron} = L_{iron} \cdot \alpha_{iron} \cdot \Delta T \]
- For the aluminum rod:
\[ \Delta L_{aluminum} = L_{aluminum} \cdot \alpha_{aluminum} \cdot \Delta T \]

4. **Express the total change in length (\( \Delta L \)) of the composite system:**
\[ \Delta L = \Delta L_{iron} + \Delta L_{aluminum} \]
\[ \Delta L = (L_{iron} \cdot \alpha_{iron} + L_{aluminum} \cdot \alpha_{aluminum}) \cdot \Delta T \]

5. **Substitute the values into the equation:**
\[ \Delta L = \left(50 \cdot 12 \times 10^{-6} + 100 \cdot 24 \times 10^{-6}\right) \cdot \Delta T \]

6. **Calculate the individual contributions:**
- For the iron rod:
\[ 50 \cdot 12 \times 10^{-6} = 600 \times 10^{-6} \]
- For the aluminum rod:
\[ 100 \cdot 24 \times 10^{-6} = 2400 \times 10^{-6} \]

7. **Combine the contributions:**
\[ \Delta L = (600 \times 10^{-6} + 2400 \times 10^{-6}) \cdot \Delta T = 3000 \times 10^{-6} \cdot \Delta T \]

8. **Find the average coefficient of linear expansion (\( \alpha_{average} \)):**
\[ \alpha_{average} = \frac{\Delta L}{L_{total} \cdot \Delta T} \]
\[ \alpha_{average} = \frac{3000 \times 10^{-6}}{150 \, \text{cm}} = 20 \times 10^{-6} \, \text{°C}^{-1} \]

### Final Answer:
The average coefficient of expansion of the composite system is:
\[ \alpha_{average} = 20 \times 10^{-6} \, \text{°C}^{-1} \]