Question

One of the tallest buildings in the world is the Taipei 101 in Taiwan, at a height of 1671 feet. Assume that this height was measured on a cool spring day when the temperature was 15.5ºC. You could use the building as a sort of giant thermometer on a hot summer day by carefully measuring its height. Suppose you do this and discover that the Taipei 101 is 0.471 foot taller than its official height. What is the temperature, assuming that the building is in thermal equilibrium with the air and that its entire frame is made of steel?

Short answer

The temperature at which the length is increased by 0.471 foot is $39.0°C$.

Step-by-step solution

Concept of Linear Expansion in solids.

The linear change in length when the temperature change is given by the linear expansion expression,

$\mathbf{L}\mathbf{=}{\mathbf{L}}_{\mathbf{0}}\left(1+\alpha ∆T\right)$ ...(i)

Here, L is the expanded length, L₀ is the original length, α is the coefficient of expansion of the material and T is the absolute temperature.

 Determination of the change in length of the steel deck when temperature increases from -5.0ºC to 18.0ºC.

For steel, the coefficient of linear expansion,

$\alpha =1.2\times {10}^{-5}{\left(C°\right)}^{-1}$

Rearrange equation (i) to get the change in temperature that causes the length increase of the tower.

$$\begin{gathered} ∆L=L_0\alpha ∆T \\ ∆T=\frac{∆L}{L_0\alpha } \end{gathered}$$

Substitute all values in above equation,

$$\begin{gathered} ∆T=\frac{0.471\mathrm{ft}}{\left[1.2\times {10}^{-5}{\left(\mathrm{C}°\right)}^{-1}\left[1671\mathrm{ft}\right]\right]} \\ =23.5C° \end{gathered}$$

Thus, the temperature at which the length increases by 0.471 ft is,

$$\begin{gathered} {\mathrm{T}}_2=15.5°C+23.5°C \\ =39.0°C \end{gathered}$$