Question

Steel wire of length ${}^{\prime }{L}^{\prime }$ at ${40}^{\circ }C$ is suspended from the ceiling and then a mass ${}^{\prime }{m}^{\prime }$ is hung from its free end. The wire is cooled down from ${40}^{\circ }C$ to ${30}^{\circ }C$ to regain its original length ${}^{\prime }{L}^{\prime }$. 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 $1mm$. Assume that $L>>$ diameter of the wire. Then the value of ${}^{\prime }{m}^{\prime }$ in $kg$ is nearly.

Answer 1

$Y=\frac{stress}{strain}=\frac{mg/\Delta }{\Delta l/l}$

$Y=\frac{mgl}{A\Delta l}$

$Y=\frac{mgl}{Al\propto \Delta T}=\frac{mg}{A\propto \Delta T}$

$m=\frac{YA\propto \Delta T}{g}$

$=\frac{{10}^4\times \pi ({10}^{-3}{)}^2\times {10}^{-3}\times 10}{10}$

$n\approx 3$

$m=3$