A steel rod of length `L_0` . has a cross sectional area A. The force required to stretch this rod by the same amount as the expansion produced by heating it through `DeltaT` is (coefficient of linear expansion of steel is `alpha` and young's modulus for steel is Y ).

Options

A. `YA alphaDeltaT`
B. `1//2YAalphaDeltaT`
C. `2YAalphaDeltaT`
D. `3YAalphaDeltaT`

Answer

To solve the problem of finding the force required to stretch a steel rod by the same amount as the expansion produced by heating it through a temperature change \(\Delta T\), we can follow these steps:

### Step-by-Step Solution

1. **Understand the Expansion Due to Heating**:
The change in length (\(\Delta L\)) of the steel rod when heated can be expressed using the coefficient of linear expansion (\(\alpha\)):
\[ \Delta L = L_0 \cdot \alpha \cdot \Delta T \]
where \(L_0\) is the original length of the rod.

2. **Define Strain**:
Strain (\( \epsilon \)) is defined as the change in length divided by the original length:
\[ \epsilon = \frac{\Delta L}{L_0} = \frac{L_0 \cdot \alpha \cdot \Delta T}{L_0} = \alpha \cdot \Delta T \]

3. **Define Stress**:
Stress (\( \sigma \)) is related to strain through Young's modulus (\( Y \)):
\[ \sigma = Y \cdot \epsilon \]
Substituting the expression for strain:
\[ \sigma = Y \cdot (\alpha \cdot \Delta T) \]

4. **Relate Stress to Force**:
Stress is also defined as force (\( F \)) per unit area (\( A \)):
\[ \sigma = \frac{F}{A} \]
Therefore, we can express force in terms of stress:
\[ F = \sigma \cdot A \]

5. **Combine the Equations**:
Substitute the expression for stress into the force equation:
\[ F = (Y \cdot (\alpha \cdot \Delta T)) \cdot A \]
This simplifies to:
\[ F = Y \cdot \alpha \cdot \Delta T \cdot A \]

### Final Expression
Thus, the force required to stretch the steel rod by the same amount as the expansion produced by heating it through \(\Delta T\) is given by:
\[ F = Y \cdot \alpha \cdot \Delta T \cdot A \]