A compressive force, F is applied at the two ends of a long thin steel rod. It is heated, simultaneously, such that its temeprature increass by `DeltaT`. The net change in its length is zero. Let l be the length of the rod, A its area of cross-section, Y its Young's modulus, and `alpha` its coefficient of linear expansion. Then, F is equal to

Options

A. `IA Y alpha Delta T`
B. `A Y alpha Delta T`
C. `(AY)/(alpha DeltaT)`
D. `I^(2) Yalpha Delta T`

Answer

To solve the problem, we need to analyze the effects of both the compressive force and the temperature change on the length of the steel rod.

### Step-by-Step Solution:

1. **Understand the Effects of Temperature Change**:
- When the rod is heated, it tends to expand. The change in length (\( \Delta L \)) due to the temperature increase (\( \Delta T \)) can be expressed using the formula:
\[ \Delta L_{\text{thermal}} = \alpha L \Delta T \]
- Here, \( \alpha \) is the coefficient of linear expansion, \( L \) is the original length of the rod, and \( \Delta T \) is the change in temperature.

2. **Understand the Effects of Compressive Force**:
- The compressive force \( F \) causes a change in length in the opposite direction. The relationship between the force, area of cross-section \( A \), Young's modulus \( Y \), and change in length due to the force can be expressed as:
\[ \Delta L_{\text{compression}} = -\frac{F}{A} \cdot \frac{L}{Y} \]
- Here, the negative sign indicates that this change in length is a contraction.

3. **Set Up the Equation for Net Change in Length**:
- Since the net change in length is zero, we can set the two changes in length equal to each other:
\[ \Delta L_{\text{thermal}} + \Delta L_{\text{compression}} = 0 \]
- Substituting the expressions we derived:
\[ \alpha L \Delta T - \frac{F}{A} \cdot \frac{L}{Y} = 0 \]

4. **Solve for the Compressive Force \( F \)**:
- Rearranging the equation gives:
\[ \alpha L \Delta T = \frac{F}{A} \cdot \frac{L}{Y} \]
- Cancel \( L \) from both sides (assuming \( L \neq 0 \)):
\[ \alpha \Delta T = \frac{F}{A Y} \]
- Now, solving for \( F \):
\[ F = A Y \alpha \Delta T \]

5. **Final Result**:
- Therefore, the expression for the compressive force \( F \) is:
\[ F = A Y \alpha \Delta T \]