A rod of length 2m rests on smooth horizontal floor. If the rod is heated from `0^(@)C` to `20^(@)C`. Find the longitudinal strain developed ? `(alpha=5xx10^(-5)//.^(@)C)`

Options

A. `10^(-3)`
B. `2xx10^(-3)`
C. zero
D. `10^(-4)`

Answer

To solve the problem of finding the longitudinal strain developed in a rod when it is heated, we can follow these steps:

### Step 1: Understand the Concept of Longitudinal Strain
Longitudinal strain is defined as the change in length per unit original length of the rod. It can be mathematically expressed as:
\[ \text{Longitudinal Strain} = \frac{\Delta L}{L_0} \]
where \( \Delta L \) is the change in length and \( L_0 \) is the original length of the rod.

### Step 2: Calculate the Change in Length (\(\Delta L\))
The change in length due to thermal expansion can be calculated using the formula:
\[ \Delta L = \alpha \cdot L_0 \cdot \Delta T \]
where:
- \( \alpha \) is the coefficient of linear expansion,
- \( L_0 \) is the original length of the rod,
- \( \Delta T \) is the change in temperature.

Given:
- \( \alpha = 5 \times 10^{-5} \, \text{°C}^{-1} \)
- \( L_0 = 2 \, \text{m} \)
- \( \Delta T = 20 \, \text{°C} - 0 \, \text{°C} = 20 \, \text{°C} \)

Substituting the values:
\[ \Delta L = (5 \times 10^{-5}) \cdot (2) \cdot (20) = 2 \times 10^{-3} \, \text{m} = 0.002 \, \text{m} \]

### Step 3: Calculate the Longitudinal Strain
Now, we can substitute the values into the longitudinal strain formula:
\[ \text{Longitudinal Strain} = \frac{\Delta L}{L_0} = \frac{0.002 \, \text{m}}{2 \, \text{m}} = 0.001 \]

### Step 4: Conclusion
Since the rod is resting on a smooth horizontal floor, there is no external force acting on it to develop strain. Therefore, while the rod will expand, the longitudinal strain developed will effectively be zero.

Thus, the final answer is:
\[ \text{Longitudinal Strain} = 0 \]