Question

Distance between two places is $200km$, $\alpha$ of metal is $2.5\times {10}^{-5}{/}^{o}C$. Total space that must be left between steel rails to allow for a change of temperature from ${36}^{o}F$ to ${117}^{o}F$ is :

• A. 2.25 km
• B. 0.23 km
• C. 22.5 km
• D. 0.0225 km

Answer 1

Answer: (B)

0.23 km

Given that the distance between the two places is $200\text{km}$.

Thus, the maximum length of rails is $200\text{km}$.

Let the initial length of rails be $l$.

Change in temperature is $\Delta T=(117-36{)}^{\circ }\text{F}=(117-36)\times \frac{5}{9}{}^{\circ }\text{C}={45}^{\circ }\text{C}$

Thus, the final length of rails is $l_f=l(1+\alpha \Delta T)=(1+2.5\times {10}^{-5}\times 45)l=1.001125l$

But, $l_f=200\text{km}$

$\therefore$ $200\text{km}=1.001125l$

$\Rightarrow l=\frac{200}{1.001125}=199.77\text{km}$

Thus, the amount of gap in rails is $200-199.77=0.23\text{km}$