Two elastic rods are joined between fixed supports, as shown. What is the condition for no change in the length of either rod with increasing temperature? $α_{1}, α_{2}$ are expansion coefficients; $A_{1}, A_{2}$ are cross-sectional areas; $Y_{1}, Y_{2}$ are Young moduli.

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Source diagram 102

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For the first rod to keep its length, F = Y 1 ​ A 1 ​ α 1 ​ Δ T With fixed total length, the second must also have zero extension: F = Y 2 ​ A 2 ​ α 2 ​ Δ T A 1 ​ Y 1 ​ α 1 ​ = A 2 ​ Y 2 ​ α 2 ​

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