Question

A $5m$ long cylindrical wire with radius $2*{{10}^{-3}}m$ is suspended vertically from a rigid support and carries a bob mass $100kg$ at the other end. If the bob gets snapped, calculate the change in temperature of the wire ignoring radiation losses. ( take $g=10m/{{s}^{2}}, Y=2.1 \times {{10}^{11}}N/{{m}^{2}}, density=7860kg/{{m}^{3}}, S = 420J/kgC$)

Answer 1

The heat generated by the wire suspended is equal to the elastic potential energy gained by the wire. Therefore, find out the two terms and solve them, you will get the change in temperature which is included in the heat generated formula.

Formula used:
$\begin{aligned} & Q=ms\Delta t \\\ & U=\dfrac{1}{2}\dfrac{F}{A}\dfrac{\Delta l}{l} \\\ & m=dv \\\ \end{aligned}$

Complete answer:
Given the length of the steel wire $l=5m$, radius of the wire $r=2*{{10}^{-3}}m$, mass of the bob $m=100kg$, $Y=2.1 \times {{10}^{11}}N/{{m}^{2}}$, density $d=7860kg/{{m}^{3}}$ and specific heat of the steel as $S=420J/kgC$
Now, let’s find out the elastic potential energy of the wire, which is,
$\begin{aligned} & U=\dfrac{1}{2}\dfrac{F}{A}\dfrac{\Delta l}{l} \\\ & U=\dfrac{1}{2}\dfrac{mg}{\pi {{r}^{2}}}\dfrac{F}{AY} \\\ & U=\dfrac{1}{2}\dfrac{{{m}^{2}}{{g}^{2}}}{\begin{aligned} & {{\pi }^{2}}{{r}^{4}}Y \\\ & \\\ \end{aligned}} \\\ \end{aligned}$
Next, let's find out the heat change occurred in the process, which is,
$\begin{aligned} & Q=mS\Delta t \\\ & Q=dvS\Delta t \\\ \end{aligned}$
Now, as the heat change occurred must be equal to the elastic potential energy of the wire,
$\begin{aligned} & U=Q \\\ & \dfrac{1}{2}\dfrac{{{m}^{2}}{{g}^{2}}}{{{\pi }^{2}}{{r}^{4}}Y}=dvS\Delta t \\\ & \Delta t=\dfrac{{{m}^{2}}{{g}^{2}}}{2{{\pi }^{2}}{{r}^{4}}dYS} \\\ & \Delta t=4.54 \times {{10}^{-3}}^{{}} \\\ & \\\ \end{aligned}$

Additional Information:
Young’s modulus describes the elastic properties of a solid undergoing tension or compression in only one direction. It is a measure of the ability of the material to withstand change in the length when any tension or compression is applied. Mathematically, the Young’s modulus is equal to the longitudinal stress divided by the strain. Units of Young’s modulus is pound per square inch or newton per square meter.

Note:
Young’s modulus is a specific form of Hooke’s law of elasticity. As young’s modulus changes with temperature, the major drawback is that the value changes as temperature changes. It doesn’t tell if there’s any change in shape of the body.