The area of cross-section of a railway track is 0.01 m². The temperature variation is 10°C. Coefficient of liner expansion of material of track is 10⁻⁵/°C. The energy stored per meter in the track is ____________ J/m.
(Young's modulus of material of track is 10¹¹ Nm⁻²)

Solution

Physics sir · 4 steps

The track is not allowed to expand, so the blocked expansion becomes strain, and strain stores energy.

1. Thermal strain. A free rail would grow by Δ L = LαΔ T. The track is held fixed, so it is compressed by that amount instead. The strain is αΔ T = 10⁻⁵ × 10 = 10⁻⁴.
2. Thermal stress. Stress = Y × strain = 10¹¹ × 10⁻⁴ = 10⁷ N m⁻².

   U/V = 1/2×stress×strain = 1/2Y(αΔ T)²

3. Energy per volume. U/V = 1/2 × 10⁷ × 10⁻⁴ = 500 J m⁻³.
4. So. One metre of track has volume A × 1 m = 0.01 m³. The energy stored per metre is 500 × 0.01 = 5 J m⁻¹.

The common slip: 10. Leaving out the 1/2 in the energy density formula gives 10, twice the right value.

The idea to keep. A rod that is stopped from expanding behaves as if it were compressed by αΔ T. Use that as the strain, then energy per volume is 1/2Y(strain)².

Answer from the NTA final answer key. Solution by PrepWiser.