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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 $2 \times 10^{11} Nm ^{-2}$ and $2 \times 10^{-5} K ^{-1}$, respectively. The value of $M$ is $.......kg .\left(\right.$ Take $\left.g=10\,m s ^{-2}\right)$
$60$
$59$
$58$
$57$
Solution
If $\Delta l$ is decease in length of rod due to decease in temperature
$\Delta l=l \alpha \Delta T$
$\alpha=2 \times 10^{-5}\,K ^{-1}, \Delta T =(210-160)=50\,K$
$\Delta l=1 \times 2 \times 10^{-5} \times 50=10^{-3}\,m$
$\text { Young Modulus }= Y =\frac{ F / A }{\Delta l / l} \quad A =3 \times 10^{-6}\,m ^2$
$2 \times 10^{11}=\frac{ Mg / 3 \times 10^{-6}}{10^{-3} / 1}$
$Mg =2 \times 10^{11} \times 3 \times 10^{-9}=6 \times 10^{-2}$
$M =60\,kg$
