Question

(I) 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 Problem 9.The Eiffel Tower in Paris.

Short answer

The change in the height of the Eiffel tower is \(8.28\;{\rm{cm}}\).

Step-by-step solution

Identification of the given data

• The coefficient of thermal expansion of the wrought iron is\(\alpha = 12 \times {10^{ - 6}}\;{\rm{/}}^\circ {\rm{C}}\).
• The height of the Eiffel tower is\(h = 300{\rm{ m}}\).
• The average temperature in July is\({T_7} = 25^\circ {\rm{C}}\).
• The average temperature in January is \({T_1} = 2^\circ {\rm{C}}\).

Understanding the change in the height of the Eiffel tower with the variation in temperature

The change in the height of the Eiffel tower depends on the initial height of the Eiffel tower, the coefficient of thermal expansion, and the change in temperature.

The greater is the temperature, the greater will be the change in the height of the tower.

Determination of the change in the height of the Eiffel tower

The change in the height of the Eiffel tower can be expressed as

\(\begin{aligned}{c}\Delta h &= \alpha h\Delta T\\ &= \alpha h\left( {{T_7} - {T_1}} \right).\end{aligned}\)

Substitute the values in the above equation.

\(\begin{aligned}{c}\Delta h &= 12 \times {10^{ - 6}}\;{\rm{/^\circ C}} \times 300{\rm{ m}} \times \left( {25^\circ {\rm{C}} - 2^\circ {\rm{C}}} \right)\\ &= 3.6 \times {10^{ - 3}}{\rm{ m/^\circ C}} \times 23^\circ {\rm{C}}\\ &= 8.28 \times {10^{ - 2}}{\rm{ m}}\\ = 8.28\;{\rm{cm}}\end{aligned}\)

Thus, the change in the height of the Eiffel tower is\(8.28\;{\rm{cm}}\).