In the shown planar frame made of thin uniform rods, the length of section AB and EF is $l_{1}$ and its thermal linear coefficient of expansion is $α_{1}$ . The length of section CD is $l_{2}$ and its thermal linear coefficient of expansion is $α_{2}$ . CB and DE are of same length having thermal linear coefficient of expansion $α_{2}$ . Points A, B, E and F reside on same line, that is, sections AB and EF overlap. Then the ratio of $\frac{l_{2} α_{2}}{l_{1} α_{1}}$ for which the distance between end A and end F remains the same at all temperatures, is:

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answer is 2.

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Detailed Solution

For distance between A and F to remain constant, extension in CD= extension in AB + extension in EF
$∴ Δ l_{2} = 2 Δ l_{1} \Rightarrow l_{2} α_{2} Δ θ_{2} = 2 l_{1} α_{1} Δ θ$
$(3 l \infty Δ T - x) \frac{A Y}{l} = (2 l \infty Δ + x) \frac{2 Y A}{2 l}$
$l \infty Δ T = 2 x$
$x = \frac{l \infty Δ T}{2}$