Question

Two rods of different materials having coefficients of linear expansion α₁ ; α₂ and Young’s moduli Y₁ and Y₁ respectively are fixed between two rigid massive walls. The rods are heated such that they undergo the same increase in temperature. There in no bending of the rods. If α₁ : α₂ = 2 : 3 , the thermal stresses developed in the two rods are equal provided Y₁ : Y₂ is equal to 

(1) 2 : 3 

(2) 1 : 1 

(3) 3 : 2 

(4) 4 : 9

Answer 1

(3) 3 : 2 

For an increase in temperature Δθ; increase in length Δl = αlΔθ . Longitudinal strain due to rise in temperature = Δl/l = αΔθ . We know

Strees/Strain = Y; therefore

Thermal stress = Y × strain = Ya Δθ

Given thermal stress in the two wires is same

∴ Y₁a₁Δθ = Y₂a₂Δθ