A metal bar of length `L` and area of cross-section A is clamped between two rigid supports. For the material of the rod. It Young's modulus is T and Coefficient if linear expansion is `alpha`. If the temperature of the rod is increased by `Deltat^(@)C`, the force exerted by the rod on the supports is

Options

A. `Y A L Delta t`
B. `Y A alpha Delta t`
C. `(Y L alpha Delta t)/(A)`
D. `Y alpha A L Delta t`

Answer

To solve the problem, we need to find the force exerted by the metal bar on the supports when the temperature of the rod is increased by \( \Delta t \) °C. We will use the concepts of linear expansion and Young's modulus.

### Step-by-Step Solution:

1. **Understanding Linear Expansion**:
When the temperature of a metal bar increases, it expands. The change in length \( \Delta L \) of the bar due to temperature change can be expressed as:
\[ \Delta L = L \cdot \alpha \cdot \Delta t \]
where:
- \( L \) = original length of the bar
- \( \alpha \) = coefficient of linear expansion
- \( \Delta t \) = change in temperature

2. **Finding the Total Length After Expansion**:
The new length \( L' \) of the bar after the temperature increase is given by:
\[ L' = L + \Delta L = L + L \cdot \alpha \cdot \Delta t = L(1 + \alpha \Delta t) \]

3. **Understanding the Constraints**:
Since the bar is clamped between two rigid supports, it cannot expand freely. Therefore, the increase in length \( \Delta L \) will create a stress in the bar, which leads to a force exerted on the supports.

4. **Calculating Strain**:
Strain \( \epsilon \) is defined as the change in length divided by the original length:
\[ \epsilon = \frac{\Delta L}{L} = \frac{L \cdot \alpha \cdot \Delta t}{L} = \alpha \Delta t \]

5. **Calculating Stress**:
Stress \( \sigma \) is related to strain by Young's modulus \( Y \):
\[ \sigma = Y \cdot \epsilon = Y \cdot (\alpha \Delta t) \]

6. **Calculating the Force**:
The force \( F \) exerted by the bar on the supports can be calculated using the formula for stress:
\[ F = \sigma \cdot A = (Y \cdot \alpha \Delta t) \cdot A \]
Therefore, the final expression for the force exerted by the rod on the supports is:
\[ F = Y \cdot A \cdot \alpha \cdot \Delta t \]

### Final Answer:
The force exerted by the rod on the supports when the temperature is increased by \( \Delta t \) °C is:
\[ F = Y \cdot A \cdot \alpha \cdot \Delta t \]