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

Answer

To find the average coefficient of linear expansion of the composite rod formed by two rods of lengths \(L\) and \(2L\) with coefficients of linear expansion \(\alpha\) and \(2\alpha\) respectively, we can follow these steps:

### Step 1: Understand the lengths and coefficients
We have two rods:
- Rod 1: Length = \(L\), Coefficient of linear expansion = \(\alpha\)
- Rod 2: Length = \(2L\), Coefficient of linear expansion = \(2\alpha\)

### Step 2: Calculate the total initial length
The total initial length \(L_{\text{total}}\) of the composite rod is the sum of the lengths of the two rods:
\[ L_{\text{total}} = L + 2L = 3L \]

### Step 3: Calculate the change in length for each rod
The change in length \(\Delta L\) for each rod due to a temperature change \(\Delta T\) can be calculated using the formula:
\[ \Delta L = L \cdot \alpha \cdot \Delta T \]

For Rod 1:
\[ \Delta L_1 = L \cdot \alpha \cdot \Delta T \]

For Rod 2:
\[ \Delta L_2 = 2L \cdot (2\alpha) \cdot \Delta T = 4L \cdot \alpha \cdot \Delta T \]

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

### Step 5: Calculate the average coefficient of linear expansion
The average coefficient of linear expansion \(\bar{\alpha}\) of the composite rod can be defined as:
\[ \bar{\alpha} = \frac{\Delta L_{\text{total}}}{L_{\text{total}} \cdot \Delta T} \]

Substituting the values we found:
\[ \bar{\alpha} = \frac{5L \cdot \alpha \cdot \Delta T}{3L \cdot \Delta T} \]

### Step 6: Simplify the expression
Cancel \(L\) and \(\Delta T\) from the numerator and denominator:
\[ \bar{\alpha} = \frac{5\alpha}{3} \]

### Final Answer
Thus, the average coefficient of linear expansion of the composite rod is:
\[ \bar{\alpha} = \frac{5\alpha}{3} \]