Thermal Expansion Simulator

What Is the Thermal Expansion Simulator?

How One Number, α, Sets Each Rod's Stretch in the Thermal Expansion Sim

Heat a rod of aluminium and a rod of Invar by the same amount, and one lengthens while the other hardly stirs. That difference lives in a single material constant, α, the coefficient of linear expansion. This simulation lets you pick that constant from six buttons and watch it steer everything else. Choose Aluminium (α = 23), Brass (19), Copper (17), Steel (12), Glass (9), or Invar (1.2, in units of 10^-6 per °C) and the rod's response resets instantly — aluminium expanding roughly 19 times as much as Invar for identical settings.

Two sliders supply the rest of the recipe. Temperature change ΔT can run positive for heating or negative for cooling, and Original length L₀ sets the starting metre-scale. The readouts return the change in length ΔL, the original L₀, and the final length L = L0 + ΔL. Everything traces to ΔL = α·L0·ΔT, a product of three factors: double either ΔT or L₀ and ΔL doubles, while a negative ΔT contracts the rod.

Because α itself is only about 10^-5 per °C, a real ΔL is tiny — a fraction of a millimetre up to a few millimetres per metre — so the drawing magnifies the motion ×200 while the numbers stay honest. Since ΔL grows with L₀, long rails and bridge spans, not short offcuts, are what earn expansion gaps. Run the numbers in the thermal expansion calculator, then open the next experiment in our physics lab collection.

Where the ideal thermal expansion law breaks down

The first assumption to fail is that α is a constant. The sim holds it fixed from ΔT = -50 °C to +200 °C, but a tabulated α is an average taken near room temperature. Cool a solid towards absolute zero and α falls away towards zero; push it towards its melting point and α creeps upward. Over the range these sliders cover the error is a few per cent, which is why the linear law is taught first; a cryogenic pipe run cannot use one number.

Invar shows the same lesson more sharply. Its α = 1.2 is not stiffness but a magnetic accident: heating makes the alloy's magnetic ordering contract it by almost as much as ordinary thermal motion expands it, and the two nearly cancel. Heat it far enough, the magnetism goes, the cancellation stops and α climbs towards that of an unremarkable nickel steel — so the flat 1.2 the sim returns at every setting holds only inside a limited temperature window.

The larger idealisation is that the rod is free. Every readout here answers "how much longer would it get" — the real answer only if nothing is in the way. Clamp both ends and the length change is forbidden, so the same physics gives stress instead of motion: roughly σ = E·α·ΔT. Steel at L₀ = 2.00 m and ΔT = +100 °C reads ΔL = 2.40 mm; block that 2.40 mm and, with a Young's modulus near 200 GPa, you get of order 240 MPa of compression, comparable to the yield strength of structural steel. Notice what vanished: length. A 2 m bar and a 200 m rail build the same stress, which is how track buckles into a sun kink.

The rod is also assumed uniformly hot, but real heat arrives from outside: while it does, the surface is hotter than the core and expands more, so the object strains against itself. Metals conduct fast enough that this passes in seconds; glass does not, which is why the glass setting — α = 9, below steel's 12 — is the one that cracks when boiling water hits a cold tumbler.

Where this is actually used

Expansion gaps in railway track. Select Steel, set L₀ = 3.00 m and ΔT = +50 °C, and the sim reports ΔL = 1.80 mm. A 50 °C swing between a dark rail on a summer afternoon and the same rail on a winter night is ordinary, and since ΔL is strictly proportional to L₀, a 25 m jointed rail moves 25/3 as far — about 15 mm. That is the gap you see at every joint of older track, and why continuously welded rail is laid under tension instead.

The bimetallic strip in a thermostat. The useful quantity here is the difference between two ΔL values. Set L₀ = 2.00 m and ΔT = +100 °C and press Brass: the readout gives 3.80 mm. Press Steel without touching either slider: 2.40 mm. Bond those two strips face to face and the 1.40 mm mismatch cannot appear as a length difference, so the pair curls towards the steel side instead. Wound into a spiral, that curl becomes a usable rotation — enough to move a switch contact in a room thermostat or trip a thermal circuit breaker.

Invar in precision measurement. Set L₀ = 1.00 m and ΔT = +100 °C. Steel gives ΔL = 1.20 mm; Invar, with everything else identical, gives 0.120 mm — ten times less drift for the same swing. When the metre bar is the measurement, that factor decides the instrument: surveying tapes, length standards and pendulum rods all moved to Invar for exactly this reason.