If two rods of length L and 2L having coefficients of linear expansion `alpha` and `2alpha` respectively are connected so that total length becomes 3L, the average coefficient of linear expansion of the composite rod equals

Options

A. `(3)/(2)alpha`
B. `(5)/(2)alpha`
C. `(5)/(3)alpha`
D. None of these

Answer

To find the average coefficient of linear expansion of the composite rod made of two rods with different lengths and coefficients of linear expansion, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the lengths and coefficients of the rods:**
- Let the length of the first rod (Rod 1) be \( L \) with a coefficient of linear expansion \( \alpha \).
- Let the length of the second rod (Rod 2) be \( 2L \) with a coefficient of linear expansion \( 2\alpha \).

2. **Determine the total length of the composite rod:**
- The total length of the composite rod when both rods are connected is:
\[ L + 2L = 3L \]

3. **Use the formula for linear expansion:**
- The change in length due to thermal expansion for each rod can be expressed as:
\[ \Delta L_1 = L \cdot \alpha \cdot \Delta T \quad \text{(for Rod 1)} \]
\[ \Delta L_2 = 2L \cdot (2\alpha) \cdot \Delta T = 4L \cdot \alpha \cdot \Delta T \quad \text{(for Rod 2)} \]

4. **Calculate the total change in length:**
- The total change in length of the composite rod is:
\[ \Delta L_{total} = \Delta L_1 + \Delta L_2 = (L \cdot \alpha \cdot \Delta T) + (4L \cdot \alpha \cdot \Delta T) = 5L \cdot \alpha \cdot \Delta T \]

5. **Relate the total change in length to the average coefficient of linear expansion:**
- For the entire composite rod of length \( 3L \), the change in length can also be expressed as:
\[ \Delta L_{total} = 3L \cdot \alpha_{effective} \cdot \Delta T \]

6. **Set the two expressions for total change in length equal to each other:**
\[ 5L \cdot \alpha \cdot \Delta T = 3L \cdot \alpha_{effective} \cdot \Delta T \]

7. **Cancel out common terms:**
- Since \( L \) and \( \Delta T \) are common in both sides, we can cancel them out:
\[ 5\alpha = 3\alpha_{effective} \]

8. **Solve for the average coefficient of linear expansion:**
\[ \alpha_{effective} = \frac{5\alpha}{3} \]

### Final Answer:
The average coefficient of linear expansion of the composite rod is:
\[ \alpha_{effective} = \frac{5\alpha}{3} \]