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 $1m$. 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 $g =10 \ m s ^{-2}$)

Answer 1

If Δℓ is decease in length of rod due to decease in temperature
$Δℓ = ℓαΔT$
$α = 2×10^{−5} K^{−1}$
$ΔT = (210−160) = 50K$
$Δℓ =1×2×10^{−5}×50 = 10^{−3} m$
Young Modulus $Y = \frac {\frac FA}{\frac {​Δℓ}{ℓ}}$ $(A = 3×10^{−6} m^2)$

$2×10^{11}=\frac {\frac {mg}{3 \times 10^{-6}}}{\frac {10^{-3}}{1}}$
$Mg = 2×10^{11}×3×10^{−9}$
$Mg =6×10^{2}$
$M \times 10 = 600$
$M=60 \ kg$

So, The correct answer is 60.