Question

An isosceles triangle is formed with a thin rod of length $l_1$ and coefficient of linear expansion ${\alpha }_1$, as the base and two thin rods each of length $l_2$ and coefficient of linear expansion ${\alpha }_2$ as the two sides. The distance between the apex and the midpoint of the base remain unchanged as the temprature is varied. If The ratio $l_l:l_2$ is equal to $n\sqrt{\frac{{\alpha }_2}{{\alpha }_1}}$, then the value of $n$ is (answer upto two decimal places)

Answer 1

The distance between the apex and the midpoint of the base, using Pythagoras theorem
$l^2={\left(l_2\right)}^2-{\left(\frac{l_1}{2}\right)}^2\text{Differentiating with respect to temperature}$

$0=2l_2\times \frac{d{l}_2}{dT}-2\left(\frac{l_1}{2}\right)\frac{1}{2}\times \frac{d{l}_1}{dT}\frac{l_1}{2}\times l_1{\alpha }_1=2l_2\times l_2{\alpha }_2{\left(\frac{l_1}{l_2}\right)}^2=4\frac{{\alpha }_2}{{\alpha }_1}\Rightarrow \frac{l_1}{l_2}=2\sqrt{\frac{{\alpha }_2}{{\alpha }_1}}$