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A metal $rod A$ of $25 cm$ length expands by $0.050 cm$ when its temperature is raised from $0^{°} C$ to $100^{°} C.$ Another rod B of a different metal of length $40 cm$ expands by $0.040 cm$ for the same rise in temperature. A third rod C of $50 cm$ length is made up of pieces of rods A and B placed end to end expands by $0.03 cm$ on heating from $0^{°} C$ to $50^{°} C.$ Find the difference of the lengths of each portion of the composite rod (in cm)

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Detailed Solution

From $Δ l = l α Δ θ$ we have $0.05 = 25 α_{A} (100)$

$α_{A} = 0.00002 {per}^{°} C$

$0.04 = 40 α_{B} (100)$

$α_{B} = 0.00001 {per}^{°} C$

In third case let $l$ is the length of $rod A .$ Then length of $rod B$ will be $(50 - l) cm$

$Δ l = Δ l_{1} + Δ l_{2}$ or $0.03 = l (0.00002) (50) + (50 - l) (0.00001) (50)$

Solving we get $l = 10 cm$ and $50 - l = 40 cm$

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