Question

The figure shows a frame $ABCD$ made of rods, initially stress free. The coefficient of linear expansion of rods $AB$ , $CD$ is $7\alpha$ and for rods $BC$ , $AD$ is $2\alpha$ . Coefficient of linear expansion of rods $AO$ , $BO$ , $CO$ and $DO$ is $\gamma$ . The value of $\gamma$ for which there should be no stress in any rod at any temperature will be $n\alpha$ . Find the value of $2n$ ( $\alpha ,\gamma$ are very small).

Answer 1

Answer: 12

$(BD{)}^2=(BC{)}^2+(DC{)}^2$
$z^2=y^2+x^2$
$2Z\frac{dZ}{dT}=2y\frac{dY}{dT}+2x\frac{dX}{dT}$
$2\sqrt{5}l\times \sqrt{5}lr=2l\left(2l\alpha \right)+2\left(2l\right)\left(14l\alpha \right)$
$10r=4\alpha +56\alpha$
$r=6\alpha$