Linear Thermal Expansion
Change in length due to temperature change.
Formula
Variables
Example
Bridges need expansion joints for this
How it works
When a solid is heated, its atoms vibrate more strongly and tend to sit slightly farther apart on average, so the object grows in length; cooling does the opposite. For small temperature changes and uniform materials, that change is proportional to the original length and the temperature change, with the material’s coefficient of linear expansion, \alpha, setting how sensitive it is. This formula is a useful approximation for solids that expand evenly in one main direction.
When to use it
Use this when you need the length change of a rod, rail, wire, beam, or similar solid due to heating or cooling. It is especially common in engineering, construction, manufacturing tolerances, and physics problems involving dimensional changes.
Worked example, step by step
A 2.5 m steel rod is heated from 18°C to 68°C. If steel has \alpha = 12 \times 10^{-6} \; ^\circ\mathrm{C}^{-1}, find the change in length. Step 1: Write the temperature change. \Delta T = 68 - 18 = 50 ^\circ\mathrm{C} Step 2: Substitute into the formula. \Delta L = \alpha L_0 \Delta T \Delta L = (12 \times 10^{-6})(2.5)(50) Step 3: Calculate. \Delta L = 1500 \times 10^{-6} \text{ m} \Delta L = 0.0015 \text{ m} Step 4: Convert if helpful. 0.0015 \text{ m} = 1.5 \text{ mm} Final answer: the rod lengthens by 1.5 mm.
Common mistakes to avoid
- Using the final length instead of the original length L0
- Forgetting to convert temperature change to the correct units or sign
- Mixing up linear expansion with area or volume expansion formulas
Did You Know?
Expansion joints prevent buckling in bridges and rails.
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