9.2.2. Coefficient of Linear Expansion

The coefficient of linear expansion $\alpha$ is a property of the material and represents the fractional change in length per unit temperature change. Materials with high $\alpha$ expand more with a temperature increase.

The coefficient of linear expansion is a measure of how much a material expands per degree of temperature change. When heated, the particles in a material vibrate more and tend to move further apart, which results in an increase in length. A higher value of $\alpha$ means the material will expand more when its temperature increases.

Think of a balloon. When you warm it up, the air inside expands and the balloon gets larger. Just like how different materials react to heat, some balloons can stretch more than others depending on their material type.

The change in length of a solid when its temperature changes is termed linear expansion. It is given by the formula: $\Delta L = \alpha L_0 \Delta T$ Where:

• $\Delta L$ = Change in length (in meters)
• $\alpha$ = Coefficient of linear expansion (in per °C)
• $L_0$ = Original length of the solid (in meters)
• $\Delta T$ = Change in temperature (in °C)

This formula allows us to calculate the change in length of a material when its temperature changes. Here, $\Delta L$ is how much longer the material gets when heated or cooler, $L_0$ is its original size, and $\Delta T$ is the difference between the new temperature and the original temperature. The coefficient $\alpha$ helps give context to how much that material expands based on its properties.

Consider a metal rod. If you know its original length, the temperature increase, and the rod's expansion rate, you can easily find out how much longer it will be after being heated. It's like baking a loaf of bread; knowing the initial dough size and the temperature of the oven can tell you how much the bread will rise!

A metal rod of length 3 m is heated from 20°C to 100°C. If the coefficient of linear expansion is $1.5 \times 10^{-5}$ °C⁻¹, the change in length is: $\Delta L = (1.5 \times 10^{-5}) \times 3 \times (100 - 20) = 0.0036 \text{ m}$ Hence, the length increases by 3.6 mm.

In this example, we start with a metal rod that is 3 meters long. As the temperature increases from 20°C to 100°C, we calculate the change in length using the formula provided. By substituting the values into the formula, we find that the rod expands by 0.0036 meters, which is equivalent to 3.6 millimeters. This illustrates how significant temperature changes can affect the size of materials.

Imagine a metal ruler that you left in the sun. When you take it inside, it is slightly longer than it was when you measured it cold. This effect, while minor, demonstrates the principle of linear expansion and shows how materials can change size due to temperature.

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