Question

A thin rod having a length of $1m$ and area of cross-section $3\times 10^{-6}{m}^2$ is suspended vertically from one end. The rod is cooled from $210^{\circ }C$ to $160^{\circ }C$. After cooling, a mass $M$ is attached at the lower end of the rod such that the length of rod again becomes $1m$. Young's modulus and coefficient of linear expansion of the rod are $2\times 10^{11}N{m}^{-2}$ and $2\times 10^{-5}{K}^{-1}$, respectively. The value of $M$ is____ $kg$. $($ Take $g=10m{s}^{-2}$ )

Answer 1

Answer: 60

If $\Delta \ell$ is decease in length of rod due to decease in temperature

$\Delta \ell =\ell \alpha \Delta T$
$\alpha =2\times 10^{-5}{K}^{-1},\Delta T=(210-160)$
$=50K$
$\Delta \ell =1\times 2\times 10^{-5}\times 50=10^{-3}m$
$\text{ Young Modulus }=Y=\frac{F/A}{\Delta \ell /\ell }A=3\times 10^{-6}{m}^2$
$2\times 10^{11}=\frac{Mg/3\times 10^{-6}}{10^{-3}/1}$
$Mg=2\times 10^{11}\times 3\times 10^{-9}=6\times 10^{-2}$
$M=60kg$