# A railway track(made of iron) is laid in winter when temp. is 18 degree C. The track consists of...

## Question:

A railway track(made of iron) is laid in winter when temp. is {eq}18^{\circ}C {/eq}. The track consists of sections of 12m placed one after the other. How much gap shud be kept b/w two such sections so that there is no compression in summer at temp. {eq}48^{\circ}C {/eq}? Coeff. of linear expansion of iron = {eq}1.1 \times 10^{-6} {/eq}.

## Thermal Expansion:

Thermal expansion is the phenomenon wherein the physical dimensions of an object will increase as its temperature rises. The expansion itself originates from the fact that the molecules that make up the object will have more kinetic energy, allowing them to increase their average separation distance. The increse in separation distance between molecules extends all the way up to the macroscopic level. The change in length, macroscopically, can be written as:

{eq}\displaystyle \rm \Delta L =\alpha L \Delta T {/eq}

where:

• {eq}\displaystyle \Delta L {/eq} is the change in length
• {eq}\displaystyle \alpha {/eq} is the linear expansion coefficient
• {eq}\displaystyle \Delta T {/eq} is the change in temperature
• L is the original length of the object

Given:

• {eq}\displaystyle \rm T_0 = 18^\circ C {/eq} is the initial temperature
• {eq}\displaystyle \rm L_0 = 12\ m {/eq} is the initial length of a single track
• {eq}\displaystyle \rm T = 48^\circ C {/eq} is the elevated temperature
• {eq}\displaystyle \rm \alpha = 1.1\ \times\ 10^{-6}\ ^\circ C^{-1} {/eq} is the coefficient of thermal expansion for the track

Let us first determine the change in the length of a single track when the temperature has increased. We can do so by using the concept of thermal expansion. The equation for thermal expansion can be written as:

{eq}\displaystyle \rm \Delta L =\alpha L (T - T_0) {/eq}

We substitute:

{eq}\displaystyle \rm \Delta L = (1.1\ \times\ 10^{-6}\ ^\circ C^{-1})(12\ m)(48^\circ C - 18^\circ C) {/eq}

We will get:

{eq}\displaystyle \rm \Delta L = 0.000396\ m\ \approx\ 0.396\ mm {/eq}

Note that since the tracks are adjacent to each other, the gap has to be enough to accomodate the thermal expansion of two adjacent railway tracks. To get the gap distance, we just simply multiply our answer by two:

{eq}\displaystyle \rm D = (2)(0.396\ mm) {/eq}

We will thus get:

{eq}\displaystyle \rm \boxed{\rm D = 0.792\ mm} {/eq}

This is just the minimum needed gap. Of course, designers may opt for a little wider gap, just in case if the temperature exceeds {eq}\displaystyle \rm 48^\circ C {/eq}. 