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)

$(OR)^2 = (PR)^2 - (PO)^2 =l^2 - (\frac{l}{2})^2$
$[l ( 1 + \alpha_2t)]^2 - [\frac{1}{2} ( 1 + \alpha t)]^2$
$l^2 - \frac{l^2}{4} = l^2 ( 1 + a_2^2 t^2 + 2\alpha_2t ) - \frac{l^2}{4} ( 1 + a_1^2 t^2 + 2\alpha_1t)$
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 )$
$\Rightarrow 2\alpha_2 = \frac{2\alpha_1}{4}$
$\Rightarrow ; \alpha_1 = 4\alpha_2$