A thin rod having a length of 1 m and area of cross-section is suspended vertically from one end. The rod is cooled from to . After cooling, a mass M is attached at the lower end of the rod such that the length of the rod again becomes 1 m. [2023]

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Solution

Q.
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 $210 ° C$ to $160 ° C$ . After cooling, a mass M is attached at the lower end of the rod such that the length of the 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 $g = 10 {ms}^{- 2}$ ) [2023]

Ans.
(60)

$If Δ l is decrease in length of rod due to decrease in temperature$

$Δ l = l α Δ T$

$α = 2 \times 10^{- 5} K^{- 1}$

$Δ T = (210 - 160) = 50 K$

$Δ l = 1 \times 2 \times 10^{- 5} \times 50 = 10^{- 3} m$

$Young's Modulus = Y = \frac{F / A}{Δ l / l}, A = 3 \times 10^{- 6} m^{2}$

$2 \times 10^{11} = \frac{M g / 3 \times 10^{- 6}}{10^{- 3} / 1}$

$M g = 2 \times 10^{11} \times 3 \times 10^{- 9} = 6 \times 10^{- 2}$

$\Rightarrow M = 60 kg$

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