Question

A steel tape is placed around the earth at the equator when the temperature is $10{}^{\circ }\text{C}$. What will be the clearance between the tape and the ground (assumed to be uniform) if the temperature of the tape rises to $40{}^{\circ }\text{C}$ ?
Neglect expansion of the earth. The radius of earth at the equator is $6400\text{km}$ and ${\alpha }_{steel}=1.2\times {10}^{-5}{\text{K}}^{-1}.$

• A. $2.3\text{m}$
• B. $2.1\text{m}$
• C. $2.3\text{km}$
• D. $230\text{m}$

Answer 1

The correct option is C $2.3\text{km}$


The length of the steel tape required to measure the earth's equator is $2\pi R$.

The length of the tape after the temperature is increased from $10{}^{\circ }\text{C}$ to $40{}^{\circ }\text{C}.$

$L^{\prime }=L+\Delta L=2\pi R+2\pi R\alpha \Delta T$

$=2\pi R(1+\alpha \Delta T)$

Comparing with $L^{\prime }=2\pi R^{\prime }$, we get

$R^{\prime }=R(1+\alpha \Delta T)$

Now, the clearance between the earth's surface and tape

$=R^{\prime }-R=R+R\alpha \Delta T-R=R\alpha \Delta T$

$=6.4\times {10}^6\times 1.2\times {10}^{-5}\times 30$

$=2.3\times {10}^3\text{m}$

$=2.3\text{km}$

Hence, option $\left(\text{C}\right)$ is correct.