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Three rods of equal lengths are joined to form an equilateral triangle ABC.D is the mid-Point of AB. The Coefficient of linear expansion is $α_{1}$ for material of rod AB and BC, $α_{2}$ for material of rod AC. If the distance DC remains constant for small changes in temperature, then

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$α_{1} = 2 α_{2}$

$α_{1} = 8 α_{2}$

$α_{1} = α_{2}$

$α_{1} = 4 α_{2}$

answer is B.

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Detailed Solution

$C D^{2} = A C^{2} - A D^{2}$

$l^{2} - \frac{l^{2}}{4} = {[l (1 + α_{2} θ)]}^{2}$

$l^{2} - \frac{l^{2}}{4} = {[l (1 + α_{2} θ)]}^{2} - {[\frac{l}{2} [1 + α_{1} θ]]}^{2}$

Neglecting $α_{1}^{2} θ^{2} & α_{2}^{2} θ^{2}$

By simplifying $α_{1} = 4 α_{2}$

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