Animated Solution for Physics - Properties of Solids and Liquids: A steel rod with $Y=2.0\times 10^{11}{\text{ Nm}}^{-2}$ and $\alpha =10^{-5}{}^{\circ }{\text{C}}^{-1}$ of length $4\text{ m}$ and area of cross-section $10{\text{ cm}}^2$ is heated from $0^{\circ }\text{C}$ to $400^{\circ }\text{C}$ without being allowed to extend. The tension produced in the rod is $x\times 10^5\text{ N}$, where the value of $x$ is ............. .

A Trapped Rod

Imagine you are an engineer designing a massive steel structure. You place a solid steel rod, exactly $4\text{ m}$ long, perfectly fitted between two immovable, rigid concrete walls. It is completely trapped. There is absolutely no room for it to stretch or shrink. This is our physical setup, and it sets the stage for a classic battle between heat and mechanics.

Thermal Strain

Now, what happens when we turn up the heat? We are raising the temperature of this rod from $0^{\circ }\text{C}$ to $400^{\circ }\text{C}$. Naturally, the steel rod wants to expand. The formula for this natural, unrestricted expansion is $\Delta l=l\alpha \Delta T$.

But wait! The rigid walls completely block this expansion. Because the rod is forced to stay at its original length, it experiences a massive internal frustration. It wants to be longer, but it can't be. In physics, we quantify this frustration as thermal strain. Since the prevented change in length is $\Delta l$, the strain developed is simply the ratio of this prevented length to the original length:

The Pushback

Because of this strain, an immense force builds up inside the rod. To calculate this force, we bring in Hooke's Law. Young's modulus, denoted by $Y$, is the ratio of stress to strain. Stress is defined as the internal restoring force per unit area ($F/A$).

Rearranging this beautifully simple relationship, we get our master equation for thermal force:

Notice something fascinating here? The original length of the rod ($l=4\text{ m}$) doesn't even appear in the final equation! It completely cancels out. A $4\text{ m}$ rod and a $100\text{ m}$ rod of the same material and thickness will exert the exact same force on the walls for a given temperature change.

Let's carefully plug in the numbers. This is where many students make a silly mistake, so pay close attention to the units. Young's modulus $Y$ is $2.0\times 10^{11}{\text{ Nm}}^{-2}$. The area $A$ is given as $10{\text{ cm}}^2$. We must convert this to square meters by multiplying by $10^{-4}$, giving us $10^{-3}{\text{ m}}^2$. The coefficient of linear expansion $\alpha$ is $10^{-5}{}^{\circ }{\text{C}}^{-1}$, and our temperature change $\Delta T$ is $400^{\circ }\text{C}$.

Let's group the powers of ten first. We have $10^{11}$, minus $3$ from the area, and minus $5$ from $\alpha$. $11-3-5=3$. So we are left with $10^3$. Now multiply the remaining numbers: $2\times 400=800$.

We can rewrite this in standard scientific notation as:

The problem asks us to find the value of $x$, where the force is given as $x\times 10^5\text{ N}$. Comparing our calculated result with the given format, it is perfectly clear that:

This is the immense force the rod exerts on the walls, and by Newton's third law, the force the walls exert back on the rod to keep it in place.