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 DeltaT`
B. `A Y alpha DeltaT`
C. `(AY)/(alphaDeltaT)`
D. `I^(2) Y alpha DeltaT`

Answer

To solve the problem, we will analyze the effects of the compressive force and the thermal expansion of the steel rod. The goal is to find the expression for the force \( F \) when the net change in length is zero.

### Step-by-step Solution:

1. **Understanding Thermal Expansion**:
When the temperature of the rod increases by \( \Delta T \), the change in length \( \Delta L \) due to thermal expansion can be expressed as:
\[ \Delta L = L \alpha \Delta T \]
where:
- \( L \) is the original length of the rod,
- \( \alpha \) is the coefficient of linear expansion,
- \( \Delta T \) is the change in temperature.

2. **Understanding Young's Modulus**:
Young's modulus \( Y \) is defined as the ratio of stress to strain. Mathematically, it can be expressed as:
\[ Y = \frac{\text{Stress}}{\text{Strain}} = \frac{F/A}{\Delta L/L} \]
Rearranging this gives us the change in length \( \Delta L \) in terms of the applied force \( F \):
\[ \Delta L = \frac{F}{A} \cdot \frac{L}{Y} \]

3. **Setting Up the Equation**:
Since the net change in length is zero, the thermal expansion \( \Delta L \) due to heating must equal the contraction \( \Delta L \) due to the compressive force:
\[ L \alpha \Delta T = \frac{F}{A} \cdot \frac{L}{Y} \]

4. **Canceling Common Terms**:
We can cancel \( L \) from both sides of the equation (assuming \( L \neq 0 \)):
\[ \alpha \Delta T = \frac{F}{A} \cdot \frac{1}{Y} \]

5. **Solving for the Force \( F \)**:
Rearranging the equation to solve for \( F \):
\[ F = A Y \alpha \Delta T \]

Thus, the expression for the compressive force \( F \) is:
\[ F = A Y \alpha \Delta T \]