Add expansion over the temperature interval

For a freely heated 0.50 m rod, α(θ) = (5.0 + 0.020θ) × 10⁻⁶ K⁻¹, where θ is the numerical rise in kelvins from the initial temperature. The rod warms uniformly by 100 K.

What is the first-order extension?

Organize the givens & assumptions

L0 = 0.50 m
α changes linearly from 5 to 7 × 10⁻⁶ K⁻¹
Rise = 100 K

Uniform temperatures and the stated expansion coefficients; small thermal strains and first-order expansion unless a retained ratio is explicitly requested. Free expansion except for the named axial constraints. Constrained rods remain straight and linearly elastic; no buckling, yielding or moving supports.

A $0.60\ \mathrm{mm}$
B $0.30\ \mathrm{mm}$
C $0.35\ \mathrm{mm}$
D $0.25\ \mathrm{mm}$

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