Question

Distance between two places is 200 km, \[~\alpha ~\] of metal is \[2.5\times {{10}^{^{-5}}}{{/}^{\circ }}C\]. Total space that must be left between steel rails to allow for a change of temperature from \[{{36}^{\circ }}F~\]to ${{117}^{\circ }}F$ is:
(A) 2.25km
(B) 0.23 km
(C) 22.5 km
(D) 0.0225 km

Answer 1

Hint:Most of the materials have a tendency to expand themselves when they are heated. As long as the temperature change isn’t very large, thermal expansion occurs at a definite level. This causes a variation in the area as well as the volume of the material, say metals.

Formulas used:
Each dimension of the object undergoes thermal expansion as per the given equation:
$\begin{align}
& \Delta L={{L}_{0}}\alpha \Delta T \\
&\Rightarrow L={{L}_{0}}(1+\alpha \Delta T) \\
\end{align}$
Where ${{L}_{0}}$ is the initial length, L is the final length and $\Delta L$ is the difference between them. Also, $\Delta T$ is the difference in temperature and \[~\alpha ~\] is the coefficient of linear expansion of the given material.

Complete step by step answer:
We are given that the maximum distance between the two places is 200 km.
Hence, the maximum length of the rails can be taken as 200km itself.
We consider the initial length of the rails to be l.
Change in temperature $\begin{align}
& \Delta T={{(117-36)}^{\circ }}F={{(81\times \dfrac{5}{9})}^{\circ }}C={{45}^{\circ }}C \\
& \\
\end{align}$
Thus the final length of the rails can be calculated from the formula
\[\begin{align}
& L={{L}_{0}}(1+\alpha \Delta T) \\
&\Rightarrow L={{L}_{0}}(1+2.5\times {{10}^{^{-5}}}\times 45) \\
&\Rightarrow L=1.001125{{L}_{0}} \\
\end{align}\]
L=200km (given),Therefore,
$\begin{align}
& 200=1.001125{{L}_{0}} \\
& \Rightarrow {{L}_{0}}=\dfrac{200}{1.001125}=199.77km \\
\end{align}$

Thus, the amount of gap that must be left between the rails is 200-199.77=0.23km.Hence, option B is the correct answer among the given options.

Note:A bimetallic strip consists of two different materials with unique thermal expansion coefficients. If we take the strips from each material and if they have the same length at room temperature, the material with larger value of coefficient of linear expansion expands more on heating and the strip curves with that material outside the curve. The other material forms the outer curve upon cooling. They are used as switches in thermostats.