Animated Solution for Physics - Properties of Solids and Liquids: The pressure that has to be applied to the ends of a steel wire of length $10\text{ cm}$ to keep its length constant when its temperature is raised by $100^{\circ }\text{C}$ is (For steel, Young's modulus is $2\times 10^{11}{\text{ Nm}}^{-2}$ and coefficient of thermal expansion is $1.1\times 10^{-5}{\text{ K}}^{-1}$)

When you first read this problem, your eyes probably darted straight to the length of the wire: $10\text{ cm}$. It feels like a crucial piece of the puzzle. But in the beautiful world of thermal stress, this is a classic distractor!

Let's break down why. When a rod is heated, its natural tendency is to expand. The change in length is given by the formula $\Delta L=L\alpha \Delta T$. However, because the rod is clamped between rigid supports, it is not allowed to expand. The walls push back with a force just strong enough to compress the rod by that exact same amount, $\Delta L$.

To find the pressure (which is simply the compressive stress applied by the walls), we turn to Hooke's Law. Hooke's Law tells us that:

The strain here is the fractional deformation the rod would have undergone if it were free.

Notice what just happened? The original length $L$ completely canceled out! The thermal strain depends only on the material's properties ($\alpha$) and the temperature change ($\Delta T$). Therefore, the thermal stress (or pressure) is:

Now, it is just a matter of plugging in the given values. We have the Young's modulus $Y=2\times 10^{11}{\text{ Nm}}^{-2}$, the coefficient of thermal expansion $\alpha =1.1\times 10^{-5}{\text{ K}}^{-1}$, and the temperature rise $\Delta T=100^{\circ }\text{C}$.

Let's group the numbers and the powers of $10$ to avoid any silly mistakes:

This is an immense amount of pressure—equivalent to thousands of atmospheres—generated simply by heating a constrained piece of steel by $100^{\circ }\text{C}$. This is exactly why engineers must leave expansion gaps in railway tracks and bridges!