Q9PE - College Physics (Urone)

Question

The height of the Washington Monument is measured to be 170 m on a day when the temperature is 35.0ºC. What will its height be on a day when the temperature falls to –10.0ºC? Although the monument is made of limestone, assume that its thermal coefficient of expansion is the same as marble’s.

Step-by-Step Solution

Answer

The new height of the Washington monument is 169.98 meters. 

Step 1: Introduction

We calculate the change in height of the monument and hence, by subtracting it from the original height we find the new height.

Step 2: Given parameters and formula for linear expansion

Coefficient of linear expansion of marble = $25\times {10}^{-6}/°C$

The formula for length expansion of solids is,

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

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

Step 3: decrease in height

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

$\begin{array}{rcl}\mathbf{∆}\mathbf{L}& \mathbf{=}& \mathbf{\alpha L}\mathbf{∆}\mathbf{T}\\ & \mathbf{=}& \mathbf{(}\mathbf{2}\mathbf{.}\mathbf{5}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{/}\mathbf{°}\mathbf{C}\mathbf{)}\mathbf{\times }\mathbf{170}\mathbf{m}\mathbf{\times }\mathbf{(}\mathbf{35}\mathbf{°}\mathbf{C}\mathbf{+}\mathbf{10}\mathbf{°}\mathbf{C}\mathbf{)}\\ & \mathbf{=}& \mathbf{1}\mathbf{.}\mathbf{92}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{2}}\mathbf{m}\end{array}$

Step 4: Calculate the new height

$\begin{array}{rcl}\mathbf{L}\mathbf{\text{'}}& \mathbf{=}& \mathbf{L}\mathbf{-}\mathbf{∆}\mathbf{L}\\ & \mathbf{=}& \mathbf{170}\mathbf{m}\mathbf{-}\mathbf{1}\mathbf{.}\mathbf{92}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{2}}\mathbf{m}\\ & \mathbf{=}& \mathbf{169}\mathbf{.}\mathbf{98}\mathbf{m}\end{array}$

Therefore, the new height is  169.98 m