hard
6. A circular ring of wire of radius R and cross-sectional area A has a coefficient of linear expansion α and Young's modulus Y. If the temperature of the ring is raised by ΔT, what is the tension developed in the ring if it is prevented from expanding radially by a rigid, non-conducting circular core?
- Y A α ΔT / 2
- 2π Y A α ΔT
- Y A α ΔT
- Zero
Show answer and explanation
Answer: (C) Y A α ΔT
The unrestrained expansion in circumference would be ΔC = 2π R α ΔT. Since it is prevented from expanding, the compressive circumferential strain is ΔC/C = (2π R α ΔT)/(2π R) = α ΔT. The restoring compressive/tensile stress developed along the ring circumference is σ = Y × strain = Y α ΔT. Therefore, the force or tension developed along the cross-section is T_force = σ A = Y A α ΔT.