Question

If the coefficient of linear expansion of a metal is $0.00002 \,K^{-1}$ , then the necessary increase in temperature of the metal rod in order to increase its length by $2\%$ is

• A. $100 \,K$
• B. $373 \,K$
• C. $400 \,K$
• D. $1000 \,K$

Answer 1

Answer: (D)

$1000 \,K$

Answer 2

Let $L$ be the length of the metal rod. On increasing its length by $2\%$, its new length becomes
$L'=L+\frac{2}{100}$
$L=\frac{102}{100}$
$L=\frac{51}{50}L$
$\therefore$ The increase in the length of the metal rod is
$\Delta\,L=L'-L=\frac{51}{50}L-L=\frac{1}{50}L$
If $\Delta\, T$ is the necessary increase in temperature of metal rod, then
$\Delta\, L=L \, \alpha\,\Delta\,T$
(where $\alpha$ is the coefficient of linear expansion)
or $\Delta T=\frac{\Delta\,L}{L\,\alpha}=\left(\frac{1}{50}\right) \left(\frac{L}{0.00002\,K^{-1}}\right)$
$=\frac{10^{5}}{100} K=1000\,K$