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Three metal rods of the same length and area of cross section form an equilateral triangle as shown in Fig. D is the mid-point of side BC. If AD is independent for small change in temperature, then: ( $α_{1}$ is the coefficient of linear expansion for rod BC and $α_{2}$ for rods AB and AC).

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$α_{1} = 2 α_{2}$

$α_{1} = \frac{α_{2}}{2}$

$α_{1} = 4 α_{2}$

$α_{2} = α_{1}$

answer is D.

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Detailed Solution

${AD}^{2} = l^{2} - \frac{l^{2}}{4} = {[l (1 + α_{2} t)]}^{2} - {[\frac{l}{2} (1 + α_{1} t)]}^{2},$
expanding binomially and neglecting $α^{2}$ and higher order terms, keeping AD² independent of t.

We get $α_{1} = 4 α_{2}$ .

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