Mechanical Properties Of Solids Question 6
Question 6 - 31 January - Shift 1
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. $(.$ Take $.g=10 m s^{-2})$
Show Answer
Answer: (60)
Solution:
Formula: Young’s modulus
If $\Delta \ell$ is decease in length of rod due to decease in temperature
$ \begin{aligned} & \Delta \ell=\ell \alpha \Delta T \\ & \begin{aligned} \alpha=2 \times 10^{-5} K^{-1}, \Delta T & =(210-160) \\ & =50 K \end{aligned} \end{aligned} $
$\Delta \ell=1 \times 2 \times 10^{-5} \times 50=10^{-3} m$
Young Modulus $=Y=\frac{F / A}{\Delta \ell / \ell} \quad A=3 \times 10^{-6} m^{2}$
$ 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 $
Ans is 60.
