Calculate the compressional force required to prevent the metallic rod length `l cm` and cross-sectional area `A cm^(2)` when heated through `t^(@)C`, from expanding along length wise. The Young's modulus of elasticity of the metal is `E` and mean coefficient of linear expansion is `alpha` per degree Celsius

Options

A. `E A alpha t`
B. `(EA alphat)/((1+alphat))`
C. `(EA alphat)/((1-alphat))`
D. `El alphat`

Answer

To calculate the compressional force required to prevent a metallic rod from expanding when heated, we can follow these steps:

### Step 1: Understand the Problem
We need to find the compressional force that will counteract the expansion of a metallic rod when it is heated. The rod has a length \( l \) cm, a cross-sectional area \( A \) cm², and it is heated through \( t \) degrees Celsius. The Young's modulus of the metal is \( E \) and the mean coefficient of linear expansion is \( \alpha \).

### Step 2: Calculate the Change in Length
The change in length (\( \Delta L_t \)) of the rod due to thermal expansion can be calculated using the formula:
\[ \Delta L_t = l \cdot \alpha \cdot t \]

### Step 3: Set Up the Equation for Constant Length
Since we want to prevent the rod from expanding, the change in length due to the applied compressional force (\( \Delta L \)) must equal the change in length due to thermal expansion:
\[ \Delta L_t - \Delta L = 0 \implies \Delta L = \Delta L_t \]
Thus, we have:
\[ \Delta L = l \cdot \alpha \cdot t \]

### Step 4: Relate Stress and Strain
Young's modulus \( E \) is defined as the ratio of stress to strain:
\[ E = \frac{\text{Stress}}{\text{Strain}} \]
Where:
- Stress = \( \frac{F}{A} \) (Force per unit area)
- Strain = \( \frac{\Delta L}{L} \)

### Step 5: Substitute Stress and Strain
Substituting the expressions for stress and strain into the Young's modulus equation gives:
\[ E = \frac{F/A}{\Delta L/l} \]
Rearranging this, we find:
\[ F = E \cdot A \cdot \frac{\Delta L}{l} \]

### Step 6: Substitute for \( \Delta L \)
Now, substituting \( \Delta L = l \cdot \alpha \cdot t \) into the equation for force:
\[ F = E \cdot A \cdot \frac{l \cdot \alpha \cdot t}{l} \]
This simplifies to:
\[ F = E \cdot A \cdot \alpha \cdot t \]

### Final Result
Thus, the compressional force required to prevent the rod from expanding is:
\[ F = E \cdot A \cdot \alpha \cdot t \]