A steel rod with y = 2.0 × 10¹¹ Nm⁻² and α = 10⁻⁵ °C⁻¹ of length 4 m and area of cross-section 10 cm² is heated from 0°C to 400°C without being allowed to extend. The tension produced in the rod is x × 10⁵ N where the value of x is ____________.

Solution

Physics sir · 4 steps

The rod wants to grow by LαΔ T but cannot, so the walls squeeze out exactly that much strain.

1. Blocked expansion. Heated freely, the rod would lengthen by Δ L = LαΔ T. The supports push it back to its original length, so the rod carries a strain (Δ L)/L = αΔ T. The length L cancels.
2. Stress to force. Stress is Y times strain, and force is stress times area. This gives the thermal force formula.

   F = YAαΔ T

3. Put in numbers. A = 10 cm² = 10 × 10⁻⁴ m² = 10⁻³ m² and Δ T = 400 °C. So F = 2.0 × 10¹¹ × 10⁻³ × 10⁻⁵ × 400 = 8 × 10⁵ N.
4. So. The force in the rod is 8 × 10⁵ N, so x = 8.

The common slip: 800. Converting 10 cm² with 10⁻² instead of 10⁻⁴ makes the area 0.1 m² and the force 8 × 10⁷ N, so x comes out as 800.

The idea to keep. When expansion is fully blocked, the thermal strain is αΔ T and the force is YAαΔ T. The rod's length never matters. Always convert the area to m² first.

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