A metallic rod l cm long, A square cm in cross-section is heated through `t^(@)"C"`. If Young’s modulus of elasticity of the metal is E and the mean coefficient of linear expansion is `alpha` per degree celsius, then the compressional force required to prevent the rod from expanding along its length is

Options

A. `Y Aalphat`
B. `(Yaalphat)/(1-alphat)`
C. `(Yaalphat)/(1+alphat)`
D. `(YA(1+alphat))/(alpha)`

Answer

To solve the problem step by step, we will follow the principles of thermal expansion and Young's modulus.

### Step 1: Understand the Problem
We have a metallic rod of length \( L \) cm and cross-sectional area \( A \) cm². The rod is heated to \( t \) °C. We need to find the compressional force required to prevent the rod from expanding due to heating.

### Step 2: Calculate the Change in Length
When the rod is heated, it will expand. The change in length (\( \Delta L \)) due to thermal expansion can be calculated using the formula:
\[ \Delta L = \alpha \cdot L \cdot \Delta T \]
where \( \Delta T = t \) °C (the change in temperature).

Thus, we can express the new length \( L' \) as:
\[ L' = L + \Delta L = L + \alpha \cdot L \cdot t \]

### Step 3: Determine the Strain
Strain (\( \epsilon \)) is defined as the change in length divided by the original length:
\[ \epsilon = \frac{\Delta L}{L} = \frac{\alpha \cdot L \cdot t}{L} = \alpha \cdot t \]

### Step 4: Calculate the Stress
Stress (\( \sigma \)) is defined as the force (\( F \)) applied per unit area (\( A \)):
\[ \sigma = \frac{F}{A} \]

### Step 5: Relate Stress and Strain using Young's Modulus
Young's modulus (\( E \)) relates stress and strain:
\[ E = \frac{\sigma}{\epsilon} \]
Substituting the expressions for stress and strain, we have:
\[ E = \frac{F/A}{\alpha \cdot t} \]

### Step 6: Solve for the Force
Rearranging the equation to solve for the force \( F \):
\[ F = E \cdot A \cdot \alpha \cdot t \]

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