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Ch 17: Temperature and Heat
Young & Freedman Calc - University Physics 14th Edition
Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Non è quello che usi tu?Cambia libro di testo
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Capitolo 17, Problema 12

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?

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1
Identify the concept: This problem involves thermal expansion, which is the tendency of matter to change in volume in response to a change in temperature.
Use the formula for linear thermal expansion: \( \Delta L = \alpha L_0 \Delta T \), where \( \Delta L \) is the change in length, \( \alpha \) is the coefficient of linear expansion for steel, \( L_0 \) is the original length, and \( \Delta T \) is the change in temperature.
Substitute the known values into the formula: \( \Delta L = 0.471 \text{ ft} \), \( L_0 = 1671 \text{ ft} \), and \( \alpha \approx 11 \times 10^{-6} \text{ °C}^{-1} \) for steel.
Rearrange the formula to solve for \( \Delta T \): \( \Delta T = \frac{\Delta L}{\alpha L_0} \).
Calculate the final temperature by adding the change in temperature \( \Delta T \) to the initial temperature of 15.5°C: \( T = 15.5 + \Delta T \).

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Thermal Expansion

Thermal expansion refers to the tendency of matter to change in volume in response to a change in temperature. For solids like steel, this expansion is typically linear, meaning the length increases proportionally with temperature. Understanding this concept is crucial for calculating how much the height of a structure changes with temperature variations.
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Volume Thermal Expansion

Coefficient of Linear Expansion

The coefficient of linear expansion is a material-specific constant that quantifies how much a material expands per degree change in temperature. For steel, this coefficient is approximately 12 x 10^-6 per degree Celsius. This value is essential for determining the change in dimensions of the building as the temperature changes.
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Linear Thermal Expansion

Thermal Equilibrium

Thermal equilibrium occurs when two objects or systems reach the same temperature and no heat flows between them. In this scenario, the building is assumed to be in thermal equilibrium with the surrounding air, meaning its temperature matches the air temperature. This assumption allows us to use the measured height change to infer the air temperature.
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