Playground
Interactive metal bar that visually expands or contracts as you change the thermal expansion coefficient, original length, and temperature change.
Variables
| Symbol | Name | SI | Dimension | Range |
|---|---|---|---|---|
| Change in lengthoutput Increase (or decrease) in length due to temperature change | m | L | 0 – 0.1 | |
| Coefficient of linear expansion Fractional length change per degree temperature change | 1/K | Θ⁻¹ | 1e-7 – 0.0001 | |
| Original length Length of the object at the reference temperature | m | L | 0.01 – 1000 | |
| Temperature change Change in temperature from reference | K | Θ | -200 – 500 |
Deep dive
Derivation
At the atomic level, thermal expansion arises from the asymmetry (anharmonicity) of the interatomic potential energy curve. As temperature increases, atoms vibrate with larger amplitude, but because the potential well is steeper on the compression side, the average bond length increases. The linear coefficient α = (1/L₀)(dL/dT) is approximately constant over moderate temperature ranges.
Experimental verification
Harrison's bimetallic strip experiments (1730s). Modern measurement via dilatometry: a sample is heated in a furnace while a push-rod or laser interferometer tracks length changes to nanometer precision. NIST maintains reference data for α of metals, ceramics, and polymers.
Common misconceptions
- Holes in a material expand too — a heated ring gets larger on both the inside and outside, it does not close up
- Water is anomalous: it contracts when heated from 0°C to 4°C, then expands normally above 4°C
- α is not truly constant — it varies with temperature, especially near phase transitions and at cryogenic temperatures
Real-world applications
- Bridge expansion joints and railway rail gaps
- Bimetallic strip thermostats
- Shrink-fitting mechanical parts (heat the outer ring to slip it over the shaft)
- Dental fillings must match the expansion coefficient of tooth enamel to prevent cracking
Worked examples
Steel bridge expansion
Given:
- alpha:
- 0.000012
- L_0:
- 100
- Delta_T:
- 25
Find: Delta_L
Solution
ΔL = αL₀ΔT = 1.2×10⁻⁵ × 100 × 25 = 0.03 m = 3 cm
Aluminum power line sag
Given:
- alpha:
- 0.000023
- L_0:
- 200
- Delta_T:
- 40
Find: Delta_L
Solution
ΔL = αL₀ΔT = 2.3×10⁻⁵ × 200 × 40 = 0.184 m ≈ 18.4 cm
Scenarios
What if…
- scenario:
- What if you heat a metal ring — does the hole shrink or expand?
- answer:
- The hole expands. Every linear dimension scales by the same factor (1 + αΔT), including holes. Imagine drawing the ring on a rubber sheet and stretching it uniformly — all distances grow.
- scenario:
- What if you use Invar alloy (α ≈ 1.2×10⁻⁶ /K)?
- answer:
- Expansion drops by a factor of 10 compared to steel. Invar was invented specifically for precision instruments (clock pendulums, survey tapes) where thermal stability is critical.
Limiting cases
- condition:
- Delta_T → 0
- result:
- Delta_L → 0
- explanation:
- No temperature change, no expansion.
- condition:
- alpha → 0 (Invar alloy)
- result:
- Delta_L → 0
- explanation:
- Materials with near-zero expansion coefficient barely change length.
- condition:
- Large Delta_T
- result:
- Linear approximation breaks down
- explanation:
- Higher-order terms and phase changes become significant at extreme temperatures.
Context
John Harrison · 1730
Harrison exploited differential thermal expansion in his marine chronometers using bimetallic strips, enabling accurate longitude determination at sea.
Hook
Why do bridges have expansion joints with gaps in them?
A steel bridge beam (α = 12×10⁻⁶ /K) is 100 m long at 15°C. Find how much it lengthens on a 40°C summer day.
Dimensions: [ΔL] = [α][L₀][ΔT] → L = (Θ⁻¹)(L)(Θ) = L ✓
Validity: Valid for small temperature changes where expansion is approximately linear. Breaks down near phase transitions, at cryogenic temperatures, and for very large ΔT where higher-order terms matter.