A uniform rod of length l, cross-sectional area A, density ρ, Young's modulus Y and linear expansion coefficient α rests freely on a smooth horizontal floor. It is uniformly heated by ΔT. If elastic strain energy is (n/4)YAl(αΔT)², find n.
Physics · Mixed Chapter
JEE Mains

Question

A uniform rod of length l, cross-sectional area A, density ρ, Young's modulus Y and linear expansion coefficient α rests freely on a smooth horizontal floor. It is uniformly heated by ΔT. If elastic strain energy is (n/4)YAl(αΔT)², find n.

Physics · Mixed Chapter

G1 Physics · Accepted Answer

Step 1 — Distinguish free expansion from constrained heating: On a smooth floor the rod is not held between rigid stops, so it can increase its length freely. Step 2 — Thermal strain and mechanical strain: Natural thermal strain = αΔT. Because free expansion is allowed, there is no additional elastic (stress-producing) strain. Step 3 — Stress and elastic energy: σ = 0, so U = (1/2)∫σ εelastic dV = 0. Step 4 — Compare with nYAl(αΔT)²/4: n=0. Answer: 0.