An isosceles triangle is formed with a thin rod of length l1 and coefficient of linear expansion α1, as the base and two thin rods each of length l2 and coefficient of linear expansion α2 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 (l1/l2) is

Question

An isosceles triangle is formed with a thin rod of length l1 and coefficient of linear expansion α1, as the base and two thin rods each of length l2 and coefficient of linear expansion α2 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 (l1/l2) is (A) 2 √(α1/α2) (B) √(α2/α1) (C) 2 √(α2/α1) (D) (1/2) √(α2/α1)

Tardigrade Admin · Accepted Answer

The distance between the apex and the midpoint of the base, using Pythagoras theorem

l = (l_{2})^{2} - (\frac{l _{1}}{2})^{2}

or

l^{2} = (l_{2})^{2} - (\frac{l _{1}}{2})^{2}

...(i)
Differentiating (i) w.r.t. temperature

0 = 2 l_{2} \times \frac{d l _{2}}{d T} - 2 (\frac{l _{1}}{2}) \cdot \frac{1}{2} \times \frac{d l _{1}}{d T}

\frac{l _{1}}{2} \times l_{1} α_{1} = 2 l_{2} \times l_{2} α_{2}

(\frac{l _{1}}{l _{2}})^{2} = 4 \frac{α _{2}}{α _{1}}

\Rightarrow \frac{l _{1}}{l _{2}} = 2 \frac{α _{2}}{α _{1}}

$2 \frac{α _{1}}{α _{2}}$

$\frac{α _{2}}{α _{1}}$

$2 \frac{α _{2}}{α _{1}}$

$\frac{1}{2} \frac{α _{2}}{α _{1}}$

2α2​α1​​​

α1​α2​​​

2α1​α2​​​

21​α1​α2​​​

$2 \frac{α _{1}}{α _{2}}$

$\frac{α _{2}}{α _{1}}$

$2 \frac{α _{2}}{α _{1}}$

$\frac{1}{2} \frac{α _{2}}{α _{1}}$