Linear Thermal Expansion
Length change of a solid when heated.
Formula
Variables
Example
A 10 m steel rail (α≈12×10⁻⁶) warmed 30 °C grows about 3.6 mm.
How it works
Most solids expand when their temperature rises because their atoms vibrate more and their average spacing increases. For small temperature changes, the increase in length is approximately proportional to three things: the original length, the temperature change, and a material constant called the linear expansion coefficient. The formula works well when the object is uniform and the temperature change is not so large that the material’s properties change significantly.
When to use it
Use this when you need to estimate how much a solid bar, rod, wire, bridge segment, or other long object changes length due to heating or cooling. It is especially useful in engineering and physics problems involving gaps, fits, or thermal stress.
Worked example, step by step
A brass rod is 2.5 m long at 20 °C. If its linear expansion coefficient is α = 19 × 10⁻⁶ °C⁻¹ and its temperature increases by 40 °C, how much does it expand? ΔL = αL₀ΔT ΔL = (19 × 10⁻⁶)(2.5)(40) ΔL = 1900 × 10⁻⁶ m ΔL = 1.9 × 10⁻³ m ΔL = 0.0019 m ΔL = 1.9 mm Final answer: the rod lengthens by 1.9 mm.
Common mistakes to avoid
- Using the final length instead of the original length L₀ in the formula
- Forgetting that ΔT is the temperature change, not the final temperature
- Mixing up units, such as using α in °C⁻¹ but writing the length change in cm or mm without converting consistently
Did You Know?
Bridges have expansion joints because steel grows measurably longer on a hot day.
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