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 temperature is varied. The ratio of lengths $l_1/l_2$ is $\sqrt{\frac{a{\alpha }_2}{{\alpha }_1}}$. Find the value of $a$.(Answer upto two digit after the decimal point)

Answer 1

The distance between the apex and the midpoint of the base, using Pythagoras theorem
$l=\sqrt{(l_2{)}^2-{\left(\frac{l_1}{2}\right)}^2}$
or, $\begin{array}{}{l}^2=(l_2{)}^2-{\left(\frac{l_1}{2}\right)}^2& \text{(i)}\end{array}$
Differentaiting (i) w.r.t. temperature
$0=2l_2\times \frac{d{l}_2}{dT}-2\left(\frac{l_1}{2}\right).\frac{l}{2}\times \frac{d{l}_1}{dT}$
$\frac{l_1}{2}\times l_1{\alpha }_1=2l_2\times l_2{\alpha }_2$
${\left(\frac{l_1}{l_2}\right)}^2=4\frac{{\alpha }_2}{{\alpha }_1}\Rightarrow \frac{l_1}{l_2}=2\sqrt{\frac{{\alpha }_2}{{\alpha }_1}}$