Question

# Two rods of different materials, having coefficients of linear expansion α1 and α2 and Young's moduli Y1 and Y2 respectively, are fixed between two rigid walls. The rods are heated to the same temperature. There is no bending of the rods. If α1:α2=2:3, the thermal stress developed in the two rods will be equal provided Y1:Y2 is equal to

A

2:3

B

1:1

C

3:2

D

4:9

Solution

## The correct option is B 3:2 Young's modulus Y=StressStrain=σϵ, where ϵ=lL. Now, if the temperature of a rod is increased by θ, the increase in its length due to thermal expansion is l=αLθ ∴ Strain, ϵ=lL=αθ. and stress, σ=Yϵ=Yαθ For the two rods, the stress is given by σ1=Y1α1θ and σ2=Y2α2θ But given that, σ1=σ2 ⇒Y1α1θ=Y2α2θ ⇒Y1Y2=α2α1=32. Hence, the correct choice is (c).

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