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Question

Two steel rods and an aluminium rod of equal length l0 and equal cross-sectional areas are joined rigidly together as shown in the figure below. All the rods are in a state of zero tension at 0 C. Find the length of the system when the temperature is raised to θ. Coefficient of linear expansion of aluminium and steel are αa and αs respectively. Young's modulus of aluminium is Ya and of steel is Ys.


A
[1+2Ysαsθ+Yaαaθ2Ys+Ya]l0
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B
[12Ysαsθ+Yaαaθ2Ys+Ya]l0
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C
2[1+2Ysαsθ+Yaαaθ2Ys+Ya]l0
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D
2[12Ysαsθ+Yaαaθ2Ys+Ya]l0
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Solution

The correct option is A [1+2Ysαsθ+Yaαaθ2Ys+Ya]l0
Given:
Length of each rod =l0
Change in temperature, ΔT=θ0=θ
Coefficient of linear expansion of steel =αs
Young's modulus of steel =Ys
Coefficient of linear expansion of aluminium =αa
Young's modulus of aluminium =Ya
To find:
Length of the system, l=?


We know that,
Total strain = Total stress / Total Young's modulus
Δll0=2×Ys×Δlsl0+Ya×Δlal02Ys+Ya
[ from formula of Young's modulus ]

Δll0=2×Ys×l0αsθl0+Ya×l0αaθl02Ys+Ya
[ from formula of linear expansion ]

Δll0=2Ysαsθ+Yaαaθ2Ys+Ya
Δl=[2Ysαsθ+Yaαaθ2Ys+Ya]l0
l=l0+Δl=[1+2Ysαsθ+Yaαaθ2Ys+Ya]l0

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