Question

A rod of length $1000 \, mm$ and coefficient of linear expansion $\alpha=10^{-4}$ per degree is placed symmetrically between fixed walls separated by $1001\, mm$. 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
$\Delta \ell=\ell \propto \Delta T=1000 \times 10^{-4} \times 20\, mm =2\, mm .$
But the rod can expand upto $1001\, mm$ only.
At that temperature its natural length $=1002\, mm$.
$\therefore$ Compression $=1 mm$
$\therefore$ Mechanical stress $=Y \frac{\Delta \ell}{\ell}=10^{11} \times \frac{1}{1000}=10^8\, N / m ^2$