Two metallic rods of length l and 3l have coefficient of linear expansion `alpha` and `3alpha` respectively. The coefficient of linear expansion ofr their series combinations, is

Options

A. `alpha`
B. `2.5alpha`
C. `4alpha`
D. `2alpha`

Answer

To find the coefficient of linear expansion for the series combination of two metallic rods, we can follow these steps:

### Step 1: Understand the Problem
We have two metallic rods:
- Rod 1: Length = \( l \), Coefficient of linear expansion = \( \alpha \)
- Rod 2: Length = \( 3l \), Coefficient of linear expansion = \( 3\alpha \)

When these rods are connected in series, we need to find the equivalent coefficient of linear expansion for the combined length.

### Step 2: Write the Formula for Change in Length
The change in length (\( \Delta L \)) of a rod due to temperature change can be expressed as:
\[ \Delta L = L \cdot \alpha \cdot \Delta \theta \]
where \( L \) is the original length of the rod, \( \alpha \) is the coefficient of linear expansion, and \( \Delta \theta \) is the change in temperature.

### Step 3: Calculate the Change in Length for Each Rod
For Rod 1:
\[ \Delta L_1 = l \cdot \alpha \cdot \Delta \theta \]

For Rod 2:
\[ \Delta L_2 = 3l \cdot (3\alpha) \cdot \Delta \theta = 9l \cdot \alpha \cdot \Delta \theta \]

### Step 4: Calculate the Total Change in Length
The total change in length (\( \Delta L_{\text{total}} \)) for the series combination of the two rods is the sum of the changes in length of each rod:
\[ \Delta L_{\text{total}} = \Delta L_1 + \Delta L_2 = (l \cdot \alpha \cdot \Delta \theta) + (9l \cdot \alpha \cdot \Delta \theta) \]
\[ \Delta L_{\text{total}} = (l \cdot \alpha \cdot \Delta \theta + 9l \cdot \alpha \cdot \Delta \theta) = 10l \cdot \alpha \cdot \Delta \theta \]

### Step 5: Relate the Total Change in Length to the Equivalent Rod
Let the equivalent length of the combined rods be \( L_{\text{eq}} = l + 3l = 4l \) and the equivalent coefficient of linear expansion be \( \alpha_{\text{eq}} \). The change in length for the equivalent rod can be expressed as:
\[ \Delta L_{\text{eq}} = L_{\text{eq}} \cdot \alpha_{\text{eq}} \cdot \Delta \theta = 4l \cdot \alpha_{\text{eq}} \cdot \Delta \theta \]

### Step 6: Set the Two Expressions for Change in Length Equal
Since both expressions represent the same total change in length, we can set them equal:
\[ 10l \cdot \alpha \cdot \Delta \theta = 4l \cdot \alpha_{\text{eq}} \cdot \Delta \theta \]

### Step 7: Simplify the Equation
We can cancel \( l \) and \( \Delta \theta \) from both sides (assuming they are not zero):
\[ 10\alpha = 4\alpha_{\text{eq}} \]

### Step 8: Solve for the Equivalent Coefficient of Linear Expansion
Now, divide both sides by 4:
\[ \alpha_{\text{eq}} = \frac{10\alpha}{4} = 2.5\alpha \]

### Final Answer
The coefficient of linear expansion for the series combination of the two rods is:
\[ \alpha_{\text{eq}} = 2.5\alpha \]

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