Question

A thin rod having a length of $1 m$ 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 $1 m$. 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____ $kg$. $\left(\right.$ Take $g =10 m s ^{-2}$ )

Answer 1

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) $
$ =50 K $
$ \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=60\, kg $