Question

A rod of length $L$ and radius $r$ is held between two rigid walls so that it is not allowed to expand. If its temperature is increased then the stress force developed in it is proportional to

• A. $r^5$
• B. $r^2$
• C. $r^3$
• D. $r^4$

Answer 1

The correct option is B $r^2$

Due to thermal expansion,
$\Delta L=L\alpha \Delta T$, ..............$\left(1\right)$
where,
$L\to$ original length
$\alpha \to$ coefficient of linear expansion
$\Delta T\to$ change in temperature
Now, Young's modulus,
$Y=\frac{FL}{A\Delta L}$
where,
$F\to$ stress force
$A\to$ area of cross-section of rod
$\Rightarrow F=\frac{YA\Delta L}{L}$
$\Rightarrow F=\frac{YAL\alpha \Delta T}{L}$ $[$ from $\left(1\right)]$
$\Rightarrow F=YA\alpha \Delta T$
So, $F\propto A$
$\Rightarrow F\propto r^2$ $[A=\pi r^2]$