A thin rod having a length of $1\ \text{m}$ and area of cross-section $3\times10^{-6}\ \text{m}^2$ is suspended vertically from one end. The rod is cooled from $210^\circ\text{C}$ to $160^\circ\text{C}$. After cooling, a mass $M$ is attached at the lower end of the rod such that the length of rod again becomes $1\ \text{m}$. Young's modulus and coefficient of linear expansion of the rod are $2\times10^{11}\ \text{N m}^{-2}$ and $2\times10^{-5}\ \text{K}^{-1}$, respectively. The value of $M$ is ______ kg. (Take $g=10\ \text{m s}^{-2}$)
Numerical value type. Enter your answer.
Answer: 60
Contraction on cooling: $\Delta L=L\alpha\Delta T=1\times2\times10^{-5}\times50=10^{-3}\ \text{m}$.
The load must stretch it back by the same amount:
$$Mg=\frac{YA\,\Delta L}{L}=2\times10^{11}\times3\times10^{-6}\times10^{-3}=600\ \text{N}\ \Rightarrow\ M=60\ \text{kg}$$
Solution by Sreeraj P, M.Sc Physics