Question

Railway lines are laid with gaps to allow for expansion. The gap between steel rails $66m$ long is $3.63cm$ at $10^\circ C$. At what temperature will the rails just touch? (Coefficient of linear expansion of steel $= \left( {11 \times {{10}^{ - 6}}} \right)^\circ {C^{ - 1}}$)
A. $40^\circ C$
B. $50^\circ C$
C. $60^\circ C$
D. $70^\circ C$

Answer 1

We know that expansion is nothing but increase in length. If the increase in length is along one dimension over the volume, then it is known as linear expansion. Here, the expansion has caused the change in temperature. Here, we will use the relation between change in length, coefficient of linear expansion and change in temperature to find the answer.

Formula used:
$\Delta l = {\alpha _L}\Delta Tl$,
where, $\Delta l$ is the change in length due to linear expansion, ${\alpha _L}$ is the coefficient of linear expansion, $\Delta T$ is the change in the temperature and $l$ is the original length.

Complete step by step answer:
Here, we are given the data about railway lines. We are given the following data in the question: $l = 66m$, $\Delta l = 3.63cm = 0.0363m$, ${T_1} = 10^\circ C$, ${\alpha _L} = \left( {11 \times {{10}^{ - 6}}} \right)^\circ {C^{ - 1}}$
We need to find the temperature ${T_2}$ at which the rails will just touch. We know that;
$\Delta l = {\alpha _L}\Delta Tl \\\ \Rightarrow \Delta l = {\alpha _L}\left( {{T_2} - {T_1}} \right)l \\\ \Rightarrow \left( {{T_2} - {T_1}} \right) = \dfrac{{\Delta l}}{{{\alpha _L}l}} \\\ \Rightarrow \left( {{T_2} - {T_1}} \right) = \dfrac{{0.0363}}{{11 \times {{10}^{ - 6}} \times 66}} \\\ \Rightarrow \left( {{T_2} - {T_1}} \right) = 50 \\\ \Rightarrow {T_2} = 50 + {T_1} \\\ \Rightarrow {T_2} = 50 + 10 \\\ \therefore {T_2} = 60^\circ C$
Thus, the temperature at which the rails will just touch is $60^\circ C$.

Hence, option C is the right answer.

Note: We have solved this question by using the concept of linear expansion of the material. By this concept, we can understand how much material can withstand its original shape and size under the influence of heat radiation. The coefficient of linear expansion considered in this problem can be defined as the rate of change of unit length per unit degree change in temperature.