Physics - Elasticity Question with Solution | TestHub

Step 1: Understand thermal expansion.

When a material is heated, its length tends to increase. The change in length $\Delta L$ for an original length $L$ due to a temperature change $\Delta T$ is given by:

$$\Delta L=L\alpha \Delta T$$

Here, the temperature is raised by $t^{\circ }\mathrm{C}$, so $\Delta T=t$.

The fractional change in length (thermal strain) would be:

$$\frac{\Delta L}{L}=\alpha t$$

Step 2: Understand Young's Modulus and stress.

Young's modulus $Y$ is defined as the ratio of stress to strain:

$$Y=\frac{Stress}{Strain}$$

Stress is defined as force per unit area ($F/A$), and strain is the fractional change in length ($\Delta L/L$).

So,

$$Y=\frac{F/A}{\Delta L/L}$$

The problem states that the rod is held at its ends such that its length remains invariant. This means that even though the rod tries to expand due to the temperature increase, external forces prevent this expansion. These external forces create a stress within the rod, known as thermal stress.

Step 3: Calculate the thermal stress.

Since the length is held invariant, the actual strain is zero. However, the rod *attempts* to expand, and this attempted expansion is what causes the stress. We can think of the thermal strain ($\alpha t$) as the strain that *would* occur if the rod were free to expand. The stress developed is the stress required to counteract this thermal expansion.

Therefore, the thermal stress is:

$$Stress=Y\times ThermalStrain$$

$$Stress=Y\alpha t$$

The linear stress developed in the rod is $Y\alpha t$.

The final answer is $\boxed{Y\alpha t}$.