Question

A steel tape is placed around the earth at the equator when the Temperature is ${{10}^{o}}C$. What will be the clearance between tape and the ground (assumed to be uniform) if the temperature of the tape rises to ${{40}^{o}}C$? Neglect expansion of the earth. Radius of the earth is $6400km$ & $\alpha =1.2\times {{10}^{-5}}\,{{K}^{-1}}$.
A. $2.3m$
B. $2.1m$
C. $2.3km$
D. $230m$

Answer 1

Hint: Expansion of solid will happen due to rise in temperature which will give rise to the clearance between the tape and ground.
Especially linear expansion of steel tape will happen as linear thermal expansion coefficient is given as $\alpha =1.2\times {{10}^{-5}}\,{{K}^{-1}}$
Thermal expansion coefficient explains how an object changes size when exposed to change in temperature.

 Complete step by step answer:
Due to increase in temperature the steel tape will linearly expand making a clearance or gap between it and the radius of earth at the equator.
Change in length due to rise in temperature can be expressed as
$\Delta L={{L}_{o}}\alpha \Delta T$ where ${{L}_{o}}$ is the original length of the object and $\Delta T$ is change in temperature.
This is a case of radial expansion as steel tape is tied around the earth
The initial length of the tape will be equal to radius of earth because it forms a steel ring around earth.
Initial length of steel ring
${{R}_{o}}=6400km$
Change in temperature:${{\left( 40-10 \right)}^{o}}C$
$\alpha =1.2\times {{10}^{-5}}\,{{K}^{-1}}$
$\Rightarrow \dfrac{\Delta R}{{{R}_{o}}}=\alpha \Delta T$
$\Rightarrow \Delta R=\left( 6.4\times {{10}^{6}}\times 1.2\times {{10}^{-5}}\times 30 \right)m$

$\Delta R=2.3\times {{10}^{3}}m$
$\Delta R=2.3Km$
So, on solving the correct option for this question is C.

Note:
There are 2 other types of expansion too.
Area expansion and volume expansion.
Change in area compared to initial area is called area expansion.
Change in volume compared to initial volume is called volume expansion.
Coefficient of area expansion $\beta $ coefficient of volume expansion $\gamma $ and coefficient of linear expansion $\alpha $ are inter related and this relation is used many a times
$\gamma =2\beta =3\alpha $