Question

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

• A. $\frac{\alpha_{2}}{2 \alpha_{1}}$
• B. $\frac{2 \alpha_{2}}{\alpha_{1}}$
• C. $\frac{2 \alpha_{1}}{\alpha_{2}}$
• D. $\frac{\alpha_{1}}{2 \alpha_{2}}$

Answer 1

Answer: (A)

For distance between $A$ and $F$ to remain constant, extension in $C D=$ extension in $A B+$ extension in $E F$
$\therefore \Delta l_{2}=2 \Delta l_{1}$
$\Rightarrow l_{2} \alpha_{2} \Delta \theta_{2}=2 l_{1} \alpha_{1} \Delta \theta$
or $\frac{l_{1}}{l_{2}}=\frac{\alpha_{2}}{2 \alpha_{1}}$