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 15.5°C. 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?

Verified step by step guidance

1. Identify the coefficient of linear expansion for steel, which is typically around $\alpha = 12 \times 10^{-6} \, \text{°C}^{-1}$.
2. Set up the formula for linear expansion, $\Delta L = L_0 \alpha \Delta T$, where $\Delta L$ is the change in length, $L_0$ is the original length, $\alpha$ is the coefficient of linear expansion, and $\Delta T$ is the change in temperature.
3. Substitute the known values into the equation. Here, $\Delta L = 0.471 \, \text{foot}$ and $L_0 = 1671 \, \text{feet}$.
4. Rearrange the formula to solve for $\Delta T$: $\Delta T = \frac{\Delta L}{L_0 \alpha}$.
5. Calculate $\Delta T$ and then add this to the original temperature of 15.5°C to find the new temperature.

Key Concepts

Thermal Expansion

Thermal expansion refers to the tendency of materials to change their dimensions in response to temperature changes. In solids, such as steel, this expansion occurs uniformly, meaning that as temperature increases, the material expands in length, area, and volume. This principle is crucial for understanding how the height of the Taipei 101 changes with temperature, as the increase in height can be directly correlated to the increase in temperature.

Coefficient of Linear Expansion

The coefficient of linear expansion is a material-specific constant that quantifies how much a material expands per degree of temperature change. For steel, this coefficient is approximately 11 x 10^-6 /°C. By knowing this value, one can calculate the change in height of the building due to temperature variations, which is essential for determining the new temperature based on the observed height increase.

Thermal Equilibrium

Thermal equilibrium occurs when two objects in contact reach the same temperature, resulting in no net heat transfer between them. In this scenario, the Taipei 101 is assumed to be in thermal equilibrium with the surrounding air, meaning that its temperature will match that of the air on a hot summer day. This concept is vital for accurately measuring the temperature based on the building's height, as it ensures that the expansion observed is solely due to the air temperature.