Question

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 $1mm$. Assume that $L\gg$ diameter of the wire. Then the value of ' $m$ ' in $kg$ in nearly.

• A. 1
• B. 2
• C. 3
• D. 5

Answer 1

Answer: (C)

Youngs modulus
We know that
$E=\frac{F/A}{\Delta L/L}$
$\Rightarrow \frac{\Delta L}{L}=\frac{F}{AE}\dots$ (i)
Also
$\frac{\Delta L}{L}=\alpha \Delta T\dots$ (ii)
$\frac{F}{AE}=\alpha \Delta T$
$\Rightarrow mg=(\alpha \Delta T)AE$
$\Rightarrow m=\frac{\alpha \Delta T}{g}AE$
$=\frac{10^{-5}\times 10\times \pi \times 10^{-6}\times 10^{11}}{10}$
$=\pi \approx 3$