Question 20 of 126

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 $g=10m{s}^{-2}$)

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Answer: 60

Let $\Delta l$ be the decease in length of rod due to decease in temperature
Then, $\Delta l=l\alpha \Delta T$, here, coefficient of linear expansion is $\alpha =2\times {10}^{-5}{K}^{-1}$ and $\Delta T=(483-433)K$$=50K$
Thus, $\Delta l=1\times 2\times {10}^{-5}\times 50={10}^{-3}m$
Young's modulus is stated as $Y=\dfrac{\dfrac{F}{A}}{\dfrac{\Delta l}{l}}$, here, $F=Mg$.
So, $2\times {10}^{11}=\dfrac{Mg\times 1}{3\times {10}^{-6}\times {10}^{-3}}$
$\Rightarrow Mg=2\times {10}^{11}\times 3\times {10}^{-9}=6\times {10}^{-2}$
Thus, $M=60kg$