Two Aluminium rods and a steel rod of equal cross-sectional area and equal length λ 0 are joined rigidly side by side as shown in figure. Initially the rods are at 0ºC. Find the length of the rod at the temperature θ if young's modulus of elasticity of the aluminium and steel are Y a and Y s respectively and coefficient of linear expansion of aluminium and steel are α a and α s respectively.
Text Solution
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at 0 º
at θ ºC
If rods are free to expand.
λ s = λ 0 (1 + α s θ )
λ a = λ 0 (1 + α a θ )
Y = 
x = 
if α s > α a
for steel (compression) x = 
= λ 0 (1 + α s Δθ ) – λ ......(i)
for aluminium (expansion) x = 
= λ – λ 0 (1 + α a Δθ ) ......(ii)
by solving (i) and (ii)
we get
then λ = λ 0 
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