A copper rod of length `l_(0)` at `0^(@)`C is placed on smooth surface. Now, the rod is heated upto `100^(@)`C. Find the longitudinal strain developed.
(`alpha`=coefficient of linear expansion)

Options

A. `(100 - alpha)/(l_0 + 100 l_0 alpha)`
B. `100 alpha`
C. Zero
D. None of these

Answer

To find the longitudinal strain developed in a copper rod when it is heated from \(0^\circ C\) to \(100^\circ C\), we can follow these steps:

### Step 1: Understand the concept of linear expansion
When a material is heated, it expands. The change in length (\(\Delta L\)) of a rod due to temperature change can be expressed using the formula:
\[ \Delta L = L_0 \cdot \alpha \cdot \Delta T \]
where:
- \(L_0\) = original length of the rod
- \(\alpha\) = coefficient of linear expansion
- \(\Delta T\) = change in temperature

### Step 2: Calculate the change in temperature
The change in temperature (\(\Delta T\)) when the rod is heated from \(0^\circ C\) to \(100^\circ C\) is:
\[ \Delta T = 100^\circ C - 0^\circ C = 100^\circ C \]

### Step 3: Substitute the values into the linear expansion formula
Now substituting the values into the linear expansion formula:
\[ \Delta L = L_0 \cdot \alpha \cdot 100 \]

### Step 4: Define longitudinal strain
Longitudinal strain (\(\epsilon\)) is defined as the ratio of the change in length to the original length:
\[ \epsilon = \frac{\Delta L}{L_0} \]

### Step 5: Substitute \(\Delta L\) into the strain formula
Now substituting \(\Delta L\) into the strain formula:
\[ \epsilon = \frac{L_0 \cdot \alpha \cdot 100}{L_0} \]

### Step 6: Simplify the expression
The \(L_0\) terms cancel out:
\[ \epsilon = 100 \cdot \alpha \]

### Final Result
Thus, the longitudinal strain developed in the copper rod when heated from \(0^\circ C\) to \(100^\circ C\) is:
\[ \epsilon = 100 \alpha \]

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