Question

A rod has a variable coefficient of linear expansion $\alpha =\frac{x}{5000},$ where $x$ is the distance from one end of the rod. If the length of the rod is $1m$ then find the increase in the length of the rod (in $cm$ ) on increasing the temperature of the rod by $100^\circ C.$

Answer 1

Increase in length $\left(d x\right)=\alpha \left(d x\right)\Delta T$
$\left(d l\right)=\frac{x}{5000}\times \left(d x\right)\times 100=\frac{x}{50}dx$
$\therefore $ total expansion, $\Delta l=\displaystyle \int _{0}^{1}\frac{x}{50}dx=\frac{1}{100}m=1cm$