Question

A rod of length $1000mm$ and coefficient of linear expansion $\alpha =10^{-4}$ per degree is placed symmetrically between fixed walls separated by $1001mm$. The Young's modulus of the rod is $10^{11}N/m^2$. If the temperature is increased by $20^{\circ }C$, then the stress developed in the rod is (in $N/m^2$ )

• A. $10$
• B. $10^8$
• C. $2\times 10^8$
• D. cannot be calculated

Answer 1

Answer: (B)

The change in length of rod due to increase in temperature in absence of walls is
$\Delta \ell =\ell \propto \Delta T=1000\times 10^{-4}\times 20mm=2mm.$
But the rod can expand upto $1001mm$ only.
At that temperature its natural length $=1002mm$.
$\therefore$ Compression $=1mm$
$\therefore$ Mechanical stress $=Y\frac{\Delta \ell }{\ell }=10^{11}\times \frac{1}{1000}=10^{8}N/m^2$