Q.A straight rail track made of steel, of length $L = 10$ m, is clamped rigidly to a railway line at both of its ends so that it cannot expand freely. On a summer day the temperature rises by $20\,^\circ$C, so the track's natural expansion has nowhere to go and it buckles: it takes the shape of two equal straight segments meeting at the centre, which is pushed out by a distance $x$ from the original straight line. The distance between the two clamped ends stays fixed at $L$, while each of the two buckled segments has length $\tfrac{1}{2}(L + \Delta L)$, where $\Delta L$ is the total increase in the track's length caused by the heating. Taking the coefficient of linear expansion of steel as $\alpha_{steel} = 1.2 \times 10^{-5}\ /^\circ\mathrm{C}$, find $x$, the displacement of the centre.

Thermal Expansion: Why Things Grow When Heated

You already know that matter is made of atoms or molecules. In a solid, these particles are locked into a lattice, but they are not still — they vibrate around fixed positions. When you heat a substance, you give its particles more energy. They vibrate faster and with larger amplitude. That larger vibration pushes the particles slightly farther apart from each other. The result? The whole object gets bigger. That is thermal expansion.

The key intuition: heat increases atomic vibration, which increases the average distance between atoms, which makes the object expand in every direction.

The Precise Physics

For small temperature changes, the expansion is directly proportional to the temperature change. This is an excellent approximation for most engineering and exam problems.

Linear expansion (change in one dimension — length):

where $\alpha$ is the coefficient of linear expansion, $L_0$ is the original length, and $\Delta T$ is the temperature change.

Area expansion (change in two dimensions):

where $\beta$ is the coefficient of area expansion.

Volume expansion (change in three dimensions):

where $\gamma$ is the coefficient of volume expansion.

This ratio is not a coincidence — it follows directly from geometry. Imagine a cube of side $L$. Its area is $L^2$ and its volume is $L^3$. If each side expands by a factor $(1 + \alpha \Delta T)$, then:

• New area = $L^2 (1 + \alpha \Delta T)^2 \approx L^2 (1 + 2\alpha \Delta T)$ → so $\beta = 2\alpha$
• New volume = $L^3 (1 + \alpha \Delta T)^3 \approx L^3 (1 + 3\alpha \Delta T)$ → so $\gamma = 3\alpha$

The approximations hold because $\alpha \Delta T$ is tiny (typically $10^{-5}$ per °C), so squares and cubes of it are negligible.

Why Liquids Are Different

Liquids have no fixed shape, so they have no "linear" or "area" expansion — only volume expansion matters. A liquid expands to fill its container. The coefficient of volume expansion for a liquid is typically much larger than for a solid (about 10 times larger for water compared to steel).

A classic exam point: when a liquid is heated in a container, both the liquid and the container expand. The apparent expansion you see (liquid rising in a graduated cylinder) is the difference between the liquid's true expansion and the container's expansion.

Real-World Examples You Already Know …