A metal bar of length L and area of cross-section A is clamped between two rigid supports. For the material of the rod, its Young’s modulus is Y and coefficient of linear expansion is α. If the temperature of the rod is increased by Δt°C, the force exerted by the rod on the supports is

The correct option is Y AL α Δt (which represents the physical formula $YA\alpha \Delta t$).

Step-by-step Derivation:

1. Thermal Expansion:
When the temperature of a metal rod of length $L$ and coefficient of linear expansion $\alpha$ is increased by $\Delta t$, its change in length, if it were free to expand, would be:
$\Delta L=L\alpha \Delta t$

2. Thermal Strain:
Since the rod is clamped between two rigid, unyielding supports, it is prevented from expanding. This restriction introduces a thermal strain in the rod:
$\text{Strain}=\frac{\Delta L}{L}=\frac{L\alpha \Delta t}{L}=\alpha \Delta t$

3. Thermal Stress:
Using Young's modulus ($Y$), which is the ratio of stress to strain:
$Y=\frac{\text{Stress}}{\text{Strain}}\Rightarrow \text{Stress}=Y\times \text{Strain}=Y\alpha \Delta t$

4. Force Exerted:
Stress is also defined as the restoring force ($F$) developed per unit cross-sectional area ($A$):
$\text{Stress}=\frac{F}{A}$
Equating the two expressions for stress gives:
$\frac{F}{A}=Y\alpha \Delta t$
Solving for the force ($F$) exerted on the supports:
$F=YA\alpha \Delta t$