Q12PE - College Physics (Urone)

Question

How large an expansion gap should be left between steel railroad rails if they may reach a maximum temperature 35.0ºC greater than when they were laid? Their original length is 10.0 m.

Step-by-Step Solution

Answer

The expansion gap that should be left should be of $4.2\times {10}^{-3} \mathrm{m}$ between the steel railroad rails.

Step 1: Introduction

We calculate the change in length of the steel railroads rails by the formula for linear expansion in solids.

Step 2: Given parameters and formula for linear expansion

Coefficient of linear expansion of marble =

The formula for length expansion of solids is,

$\mathbf{∆}\mathbf{L}\mathbf{=}\mathbf{\alpha L}\mathbf{∆}\mathbf{T}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{\left(}\mathbf{1}\mathbf{\right)}$

Here, is the coefficient of linear expansion, L is the original length, and $∆\mathrm{T}$ is the temperature change. 

Step 3: Expansion

Substitute the given values in equation (1), and we get,

$$\begin{gathered} \mathbf{∆}\mathbf{L}\mathbf{=}\mathbf{\alpha L}\mathbf{∆}\mathbf{T} \\ \mathbf{=}\mathbf{(}\mathbf{1}\mathbf{.}\mathbf{2}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{5}}\mathbf{/}\mathbf{°}\mathbf{C}\mathbf{)}\mathbf{\times }\mathbf{10}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{m}\mathbf{ }\mathbf{\times }\mathbf{(}\mathbf{35}\mathbf{°}\mathbf{C}\mathbf{)} \\ \mathbf{=}\mathbf{4}\mathbf{.}\mathbf{2}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{3}}\mathbf{ }\mathbf{m} \end{gathered}$$

Therefore, the expansion gap that should be left should be of $\mathbf{4}\mathbf{.}\mathbf{2}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{3}}\mathbf{ }\mathbf{m}$ between the steel railroad rails.