A composite bar oflength L = L1 + L2 is made from a bar of material 1 and length

Question

A composite bar oflength L = L1 + L2 is made from a bar of material 1 and length L1 attached to a bar of material 2 and length L2, as shown in the figure. Show that the coefficient of linear expansion ? for this composite bar is given by ? = (?1L1 + ?2L2)/L

Explanation / Answer

Let equivalent coefficient of linear expansion be alphanet Lnew = L(1+alphanet*change in temp) Lnew = L1(1+alpha1*change in temp) + L2(1+alpha2*change in temp) Both values wl be equal => L1 + L1*alpha1*change in temp + L2 + L2*alpha2*change in temp = L + L*alphanet*change in temp Since L = L1 + L2 =>L1*alpha1*change in temp + L2*alpha2*change in temp = L*alphanet*change in temp =>change in temp*(L1*alpha1 + L2*alpha2) = L*alphanet*change in temp => alphanet = (L1*alpha1 + L2*alpha2) /L

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