First slide

Thermal expansion

Question

An isosceles triangle is formed with a thin rod of length $l_{1}$ and coefficient of linear expansion $α_{1}$ , as the base and two thin rods each of lenglh /, and coefficient of linear expansion $α_{2}$ , 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 $\frac{l_{1}}{l_{2}}$ is

Difficult

A

$2 \sqrt[]{\frac{α_{1}}{α_{2}}}$
B

$\sqrt{\frac{α_{2}}{α_{1}}}$
C

$2 \sqrt[]{\frac{α_{2}}{α_{1}}}$
D

$\frac{1}{2} \sqrt{\frac{α_{2}}{α_{1}}}$

Solution

The distance between the apex and the midpoint of the base, using Pythagoras theorem l = $\sqrt{{(l_{2})}^{2} - {(\frac{l_{1}}{2})}^{2}}$
or $l^{2} = {(l_{2})}^{2} - {(\frac{l_{1}}{2})}^{2}$ ------------(i)

Differentiating (i) w.r.t. temperature

$0 = 2 l_{2} \times \frac{{dl}_{2}}{dT} - 2 (\frac{l_{1}}{2}) . \frac{1}{2} \times \frac{{dl}_{1}}{dT}$
$\frac{l_{1}}{2} \times l_{1} α_{1} = 2 l_{2} \times l_{2} α_{2}$
${(\frac{l_{1}}{l_{2}})}^{2} = 4 \frac{α_{2}}{α_{1}} \Rightarrow \frac{l_{1}}{l_{2}} = 2 \sqrt{\frac{α_{2}}{α_{1}}}$

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