Chapter 13, Problem 7P

The Eiffel Tower (Fig. 13-31 [□) is built of wrought iron approximately 300 m tall. Estimate how much its height changes between January (average temperature of 2°C) and July (average temperature of 25°C). Ignore the angles of the iron beams and treat the tower as a vertical beam

Figure 13-31

To determine

The change in the height of the Eiffel Tower due to temperature variation.

Answer to Problem 7P

Solution: The correct answer is about $0.07797m$ or $7.797cm$ .

Explanation of Solution

Given:

The height of the Eiffel Tower is 300 m and temperature varies from $2℃$ to $25℃$ .

Formula used: New length after a temperature change, $l_2=l_1\left(1+\alpha \Delta t\right)$

Calculation: Let $l_1$ be the original height of the Eiffel Tower, i.e., in January (average temperature of $t_1=2℃$ ) and it becomes $l_2$ in July (average temperature of $t_2=25℃$ )when the temperature is changed by $\Delta t$ .

Here, $l_1=300m$

  ${\text{Δt=t}}_{\text{2}}{\text{-t}}_{\text{1}}\text{=25-2=23}℃$

For wrought iron, the value of the coefficient of thermal expansion is about,

  $\alpha =11.3\times {10}^{-6}per℃$

Therefore, from the formula $l_2=l_1\left(1+\alpha \Delta t\right)$

The height of the Eiffel Tower in July becomes,

  $l_2=300\left(1+11.3\times {10}^{-6}\times 23\right)$

  $=300.07797m$

Therefore, the change in height of the Eiffel Tower between January and July is

  $\Delta l=l_2-l_1$

  $\begin{array}{l}=300.07797–300.00000\hfill \\ =0.07797m\hfill \\ =7.797cm\hfill \end{array}$

Conclusion:

Therefore, the change in the height of the Eiffel Tower between January and July is about 7.797 cm.