Two rods of lengths L₁ and L₂ are welded together to make a composite rod of length $(L_{1} + L_{2})$ . If the coefficients of linear expansion of the rods an $α_{1} a n d α_{2}$ respectively, the effective coefficient of linear expansion of the composite rod will be

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$\frac{\sqrt{L_{1} L_{2} α_{1} α_{2}}}{L_{1} + L_{2}}$

$\frac{1}{2} (α_{1} + α_{2})$

$\sqrt{α_{1} α_{2}}$

$\frac{L_{1} α_{1} + L_{2} α_{2}}{L_{1} + L_{2}}$

answer is C.

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Detailed Solution

Total length of the composite rod at $0 ° C$ is $L_{0} = L_{1} + L_{2} .$ When the composite rod is heated to $t ° C$ , its length at $t ° C$ will be $L = L_{1} (1 + α_{1} t) + L_{2} (1 + α_{2} t) = (L_{1} + L_{2}) + L_{1} α_{1} t + L_{2} α_{2} t$ , or
$L = L_{0} + (L_{1} α_{1} + L_{2} α_{2}) t$
$∴$ Effective coefficient of expansion of the composite rod is
$α = \frac{L - L_{0}}{L_{0} t} = \frac{L_{1} α_{1} + L_{2} α_{2}}{L_{1} + L_{2}}$ .