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

Explanation

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

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

### Step 2: Calculate the change in length of each rod due to temperature change
- The change in length \( \Delta L \) for a rod due to a temperature change \( \Delta T \) is given by the formula:
\[ \Delta L = \alpha \cdot L \cdot \Delta T \]
- For Rod 1:
\[ \Delta L_1 = \alpha \cdot L \cdot \Delta T \]
- For Rod 2:
\[ \Delta L_2 = 2\alpha \cdot (2L) \cdot \Delta T = 4\alpha \cdot L \cdot \Delta T \]

### Step 3: Find the total change in length of the composite rod
- The total change in length \( \Delta L_{total} \) of the composite rod is the sum of the changes in length of both rods:
\[ \Delta L_{total} = \Delta L_1 + \Delta L_2 = \alpha \cdot L \cdot \Delta T + 4\alpha \cdot L \cdot \Delta T \]
- Simplifying this gives:
\[ \Delta L_{total} = 5\alpha \cdot L \cdot \Delta T \]

### Step 4: Relate the total change in length to the average coefficient of linear expansion
- The total length of the composite rod is \( 3L \). If we denote the average coefficient of linear expansion as \( \alpha_{avg} \), then the change in length can also be expressed as:
\[ \Delta L_{total} = \alpha_{avg} \cdot (3L) \cdot \Delta T \]

### Step 5: Set the two expressions for total change in length equal to each other
- Equating the two expressions for \( \Delta L_{total} \):
\[ 5\alpha \cdot L \cdot \Delta T = \alpha_{avg} \cdot (3L) \cdot \Delta T \]

### Step 6: Cancel out common terms
- Since \( L \) and \( \Delta T \) are common in both sides, we can cancel them out:
\[ 5\alpha = \alpha_{avg} \cdot 3 \]

### Step 7: Solve for the average coefficient of linear expansion
- Rearranging gives:
\[ \alpha_{avg} = \frac{5\alpha}{3} \]

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

### Final Answer
Thus, the correct option is option 3: \( \frac{5}{3} \alpha \).

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