A steel wire of length ' $L$ ' at $40^{\circ}\,C$ is suspended from the ceiling and then a mass ' $m$ ' is hung from its free end. The wire is cooled down from $40^{\circ}\,C$ to $30^{\circ}\,C$ to regain its original length ' $L$ '. The coefficient of linear thermal expansion of the steel is $10^{-5} { }^{\circ}\,C$, Young's modulus of steel is $10^{11}\, N /$ $m ^2$ and radius of the wire is $1\, mm$. Assume that $L \gg $ diameter of the wire. Then the value of ' $m$ ' in $kg$ is nearly
Medium
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(c)
$Y=\frac{m g / A}{\Delta \ell / \ell}=\frac{m g \ell}{A \Delta \ell}$
Also $\Delta \ell=\ell \alpha \Delta T$
From (1) and (2)
$Y=\frac{m g \ell}{A \ell \alpha \Delta T}=\frac{m g}{A \alpha \Delta T}$
$\therefore m=\frac{Y A \alpha \Delta T}{g}$
$=\frac{10^{11} \times \pi\left(10^{-3}\right)^2 \times 10^{-5} \times 10}{10}=\pi \approx 3$
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