Question

How much taller does the Eiffel Tower become at the end of a day when the temperature has increased by 15ºC? Its original height is 321 m and you can assume it is made of steel.

Short answer

The Eiffel Tower gets taller by a length of 0.058 m by the end of the day.

Step-by-step solution

Introduction

We calculate the change in height of the Eiffel Tower by the formula for linear expansion in solids.

Given parameters and formula for linear expansion

Coefficient of linear expansion of marble = $1.2\times {10}^{-5}/°\mathrm{C}$

The formula for length expansion of solids is,

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

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

Increase in height

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

$\begin{array}{rcl}\mathbf{∆}\mathbf{L}& \mathbf{=}& \mathbf{\alpha L}\mathbf{∆}\mathbf{T}\\ & \mathbf{=}& \left(1.2\times {10}^{-5}/°C\right)\mathbf{\times }\mathbf{321}\mathbf{m}\mathbf{\times }\left(15°C\right)\\ & \mathbf{=}& \mathbf{0}\mathbf{.}\mathbf{058}\mathbf{}\mathbf{m}\end{array}$

Therefore, the Eiffel Tower gets taller by a length of 0.058 m by the end of the day.