Question

A metal rod is fixed rigidly at two ends so as to prevent its thermal expansion. If $L,\alpha$ and $Y$ respectively denote the length of the rod, coefficient of linear thermal expansion and Young's modulus of its material, then for an increase in temperature of the rod by $\Delta T$, the longitudinal stress developed in the rod is:

• A. inversely proportional to $\alpha$
• B. inversely proportional to $Y$
• C. directly proportional to $\Delta T/Y$
• D. independent of $L$

Answer 1

The correct option is D independent of $L$

Length of the rod as a function of temperature $L^{\prime }=L(1+\alpha \Delta T)$

where $\alpha$ is the coefficient of thermal expansion.

$\Delta T$ is the change in temperature

$\therefore$ $\Delta L=L^{\prime }-L=L\alpha \Delta T$

Strain $\frac{\Delta L}{L}=\alpha \Delta T$ ............(1)

From Hooke's law, $Y=\frac{Stress}{Strain}$ $⟹Stress=Y\times Strain$

$\therefore$ Stress $=Y\times \alpha \Delta T$

$⟹$ Stress is independent of $L$