Question

Three rods of equal length $l$ are joined to form an equilateral triangle $PQR.O$ is the mid point of $PQ$. Distance $OR$ remains same for small change in temperature. Coefficient of linear expansion for $PR$ and $RQ$ is same i.e. ${\alpha }_2$ but that for $PQ$ is ${\alpha }_1$. Then

• A. ${\alpha }_2=3{\alpha }_1$
• B. ${\alpha }_2=4{\alpha }_1$
• C. ${\alpha }_1=3{\alpha }_2$
• D. ${\alpha }_1=4{\alpha }_2$

Answer 1

Answer: (D)

$(OR{)}^2=(PR{)}^2-(PO{)}^2=l^2-(\frac{l}{2}{)}^2$
$[l(1+{\alpha }_{2}t){]}^2-[\frac{1}{2}(1+\alpha t){]}^2$
$l^2-\frac{l^2}{4}=l^2(1+a_2^2{t}^2+2{\alpha }_{2}t)-\frac{l^2}{4}(1+a_1^2{t}^2+2{\alpha }_{1}t)$
Neglecting ${\alpha }_2^2{t}^2$ and ${\alpha }_1^2{t}^2$
$0=l^2(2{\alpha }_{2}t)-\frac{l^2}{4}(2{\alpha }_{1}t)$
$\Rightarrow 2{\alpha }_2=\frac{2{\alpha }_1}{4}$
$\Rightarrow ;{\alpha }_1=4{\alpha }_2$