Question

196.Two uniform bars of length L1 and L2 with coefficient of linear expansion alpha1 and alpha2 are joined to form a longer length L1+L2.The effective coefficient of linear expansion of the composite bar is

Answer 1

As given L₁ and L₂ be the length of two uniform bars and α₁ and α₂ be the coefficient of linear expansion. Then, change in lengths will be
ΔL₁ = L₁α₁ΔT
​ΔL₂ = L₂α₂ ΔT​
Let L be the equivalent length of two bars and α be their equivalent coefficient of linear expansion.​Then
ΔL = L₁α₁ΔT +​ L₂α₂ ΔT​​
Lα ΔT ​​= L₁α₁ΔT +​ L₂α₂ ΔT​​​
$$\begin{gathered} \alpha L={\alpha }_1{L}_1+{\alpha }_2{L}_2 \\ \Rightarrow a=\frac{L_1{\alpha }_1+L_2{a}_2}{L} \end{gathered}$$