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Question

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

A
2:3
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B
1:1
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C
3:2
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D
4:9
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Solution

The correct option is C 3:2
Data given:
For rod 1,
Coefficient of thermal expansion =α1
Young's modulus =Y1
For rod 2,
Coefficient of thermal expansion =α2
Young's modulus =Y2
Ratio of coefficients of thermal expansion, α1α2=23
To find:
Ratio of Young modulii, Y1Y2=?

Let us suppose, L is the initial length of the rods and ΔT is the equal change in temperature.
By definition of linear expansion,
For rod 1,
ΔL1=α1LΔT ...........(1)
For rod 2,
ΔL2=α2LΔT ...........(2)
From (1) and (2)
ΔL1ΔL2=α1α2=23 [given]
ΔL1/LΔL2/L=23
ϵ1ϵ2=23 ............(3)
[ϵ Thermal Strain ]

Now, by definition of Young's modulus,
Y1Y2=σ1ϵ1σ2ϵ2=σ1ϵ2σ2ϵ1
[σ Thermal Stress ]
Y1Y2=ϵ2ϵ1
[ given thermal stresses are equal ]
Y1Y2=32=3:2
[from (3)]

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