Read before you practise · 6 min

Separate Free Expansion from Constrained Expansion

Heating makes a solid expand by a fixed fraction per degree, but what you calculate depends on whether the object is free to expand. Free expansion changes dimensions but creates no stress; blocked expansion creates stress and force without changing the length.

NSEJS · Thermal Physics · 6 linked questions · 100% coverage

Know this first

Thermal Expansion and Thermal Stress is the central idea in this question pattern. Identify the condition in the stem before applying the method below.

Study deeper: NCERT Science · Thermal Physics

Coefficients and the formulas

Linear expansion follows

\Delta l = l0 \alpha \Delta T, so \alpha = \Delta l / (l0 \Delta T)

To compare materials, compute Δl / (l0 ΔT) for each rod rather than comparing Δl alone: a rod with l0 = 1.5, ΔT = 50 and Δl = 3 has α = 3 / (1.5 x 50) = 0.04, twice the 0.02 of a rod with l0 = 1, ΔT = 100 and Δl = 2.

Areas expand with

β = 2α and volumes with γ = 3α

Every dimension of an isotropic solid grows by the same fraction, including the size of any hole, so a hole in a heated copper plate gets larger, exactly as if it were filled with the same copper.

Free expansion develops no strain

A rod resting on a smooth horizontal floor is free to lengthen, so nothing resists the expansion and no stress or longitudinal strain develops in it. The number α ΔT (here 5 x 10⁻⁵ x 20 = 10⁻³) is the free thermal strain, the fractional change in length, but the strain that is developed by constraint is zero. Reading the question for the words free, smooth floor or unclamped tells you which quantity is wanted.

Blocked expansion produces thermal stress

If the same rod is clamped so that it cannot lengthen, a compressive stress Y α ΔT appears and the force on the supports

F = Y A α ΔT

For an iron bar with

Y = 10¹¹ N/m², α = 10⁻⁶ per °C and A = 10⁻³ m² heated by 100 °C: stress = 10¹¹ x 10⁻⁶ x 100 = 10⁷ N/m² and force = 10⁴ N

The original length does not appear in the result. For a rod hung from a frame above a floor, the rod reaches the floor only if it expands more than the frame that holds it, which needs α(rod) > α(frame).

OlympIQ exam lens

✓α = Δl / (l0 ΔT); compare materials with this ratio, not with Δl alone
✓Area coefficient is about 2α and volume coefficient about 3α for isotropic solids
✓A hole in a heated plate expands along with the plate
✓Free rod: no stress and no developed strain; clamped rod: stress = Y α ΔT and force = Y A α ΔT
✓For a gap between a hanging rod and the floor, compare the expansion of the rod against the expansion of its frame

OlympIQ trap

Do not quote α ΔT as the strain developed in a rod that is free to expand. A rod on a smooth floor lengthens without any stress, so the developed strain is zero; α ΔT only turns into stress when something prevents the expansion.

Ready to try the pattern?

100% coverage across the 6 linked questions.

Practise 6 linked questions

← All NSEJS Prep Paths