Two metal rods of lengths `L_(1)` and `L_(2)` and coefficients of linear expansion `alpha_(1)` and `alpha_(2)` respectively are welded together to make a composite rod of length `(L_(1)+L_(2))` at `0^(@)C.` Find the effective coefficient of linear expansion of the composite rod.

Options

A. `(L_(1)alpha_(1)^(2)-L_(2)alpha_(2)^(2))/(L_(1)^(2)+L_(2)^(2))`
B. `(L_(1)^(2)alpha_(1)-L_(2)^(2)alpha_(2))/(L_(1)^(2)+L_(2)^(2))`
C. `(L_(1)alpha_(1)+L_(2)alpha_(2))/(L_(1)-L_(2))`
D. `(L_(1)alpha_(1)+L_(2)alpha_(2))/(L_(1)+L_(2))`

Explanation

To find the effective coefficient of linear expansion of the composite rod made by welding two metal rods together, we can follow these steps:

### Step 1: Understand Linear Expansion
The linear expansion of a rod is given by the formula:
\[ \Delta L = \alpha \cdot L \cdot \Delta T \]
where:
- \(\Delta L\) is the change in length,
- \(\alpha\) is the coefficient of linear expansion,
- \(L\) is the original length of the rod,
- \(\Delta T\) is the change in temperature.

### Step 2: Determine the Change in Length for Each Rod
For the first rod with length \(L_1\) and coefficient of linear expansion \(\alpha_1\):
\[ \Delta L_1 = \alpha_1 \cdot L_1 \cdot \Delta T \]

For the second rod with length \(L_2\) and coefficient of linear expansion \(\alpha_2\):
\[ \Delta L_2 = \alpha_2 \cdot L_2 \cdot \Delta T \]

### Step 3: Total Change in Length of the Composite Rod
The total change in length of the composite rod when both rods are welded together is the sum of the changes in length of each rod:
\[ \Delta L_{\text{total}} = \Delta L_1 + \Delta L_2 = \alpha_1 \cdot L_1 \cdot \Delta T + \alpha_2 \cdot L_2 \cdot \Delta T \]

### Step 4: Factor Out \(\Delta T\)
We can factor out \(\Delta T\) from the equation:
\[ \Delta L_{\text{total}} = \Delta T \cdot (\alpha_1 \cdot L_1 + \alpha_2 \cdot L_2) \]

### Step 5: Define the Effective Coefficient of Linear Expansion
Let \(\alpha\) be the effective coefficient of linear expansion for the composite rod. The total change in length can also be expressed as:
\[ \Delta L_{\text{total}} = \alpha \cdot (L_1 + L_2) \cdot \Delta T \]

### Step 6: Set the Two Expressions for \(\Delta L_{\text{total}}\) Equal
Now, we can set the two expressions for \(\Delta L_{\text{total}}\) equal to each other:
\[ \alpha \cdot (L_1 + L_2) \cdot \Delta T = \Delta T \cdot (\alpha_1 \cdot L_1 + \alpha_2 \cdot L_2) \]

### Step 7: Cancel \(\Delta T\) (Assuming \(\Delta T \neq 0\))
Assuming \(\Delta T \neq 0\), we can cancel \(\Delta T\) from both sides:
\[ \alpha \cdot (L_1 + L_2) = \alpha_1 \cdot L_1 + \alpha_2 \cdot L_2 \]

### Step 8: Solve for the Effective Coefficient of Linear Expansion
Now, we can solve for \(\alpha\):
\[ \alpha = \frac{\alpha_1 \cdot L_1 + \alpha_2 \cdot L_2}{L_1 + L_2} \]

### Final Result
The effective coefficient of linear expansion of the composite rod is given by:
\[ \alpha = \frac{\alpha_1 \cdot L_1 + \alpha_2 \cdot L_2}{L_1 + L_2} \]