Calculate the compressional force required to prevent the metallic rod of length l cm and cross sectional area `Acm^2` when heated through `t^@C` from expanding lengthwise. Young's modulus of elasticity of the metal is E and mean coefficient of linear expansion is `alpha` per degree celsius.

Options

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

Explanation

To solve the problem of calculating 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 (F) required to prevent a metallic rod of length \( l \) cm and cross-sectional area \( A \) cm² from expanding when heated through \( t \) °C. The Young's modulus of the material is \( E \) and the mean coefficient of linear expansion is \( \alpha \).

### Step 2: Determine the change in length due to thermal expansion
The change in length (\( \Delta L \)) of the rod when heated can be calculated using the formula for linear expansion:
\[ \Delta L = l \cdot \alpha \cdot t \]
where:
- \( l \) is the original length of the rod,
- \( \alpha \) is the coefficient of linear expansion,
- \( t \) is the change in temperature.

### Step 3: Apply Young's modulus
Young's modulus (\( E \)) relates stress and strain in a material. The formula for Young's modulus is given by:
\[ E = \frac{F/A}{\Delta L / L} \]
Rearranging this gives:
\[ F = E \cdot A \cdot \frac{\Delta L}{L} \]

### Step 4: Substitute the change in length
Substituting \( \Delta L = l \cdot \alpha \cdot t \) into the equation for force, we get:
\[ F = E \cdot A \cdot \frac{l \cdot \alpha \cdot t}{l} \]
This simplifies to:
\[ F = E \cdot A \cdot \alpha \cdot t \]

### Step 5: Final expression
Thus, the compressional force required to prevent the rod from expanding is:
\[ F = E \cdot A \cdot \alpha \cdot t \]

### Summary
The required compressional force \( F \) to prevent the metallic rod from expanding when heated is given by:
\[ F = E \cdot A \cdot \alpha \cdot t \]