Question

The tallest building in the world, according to some architectural standards, 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 13.0 ∘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 0.481 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?

Answer 1

Answer:

35.14°C

Explanation:

The equation for linear thermal expansion is $\Delta L = \alpha L_0\Delta T$, which means that a bar of length $L_0$ with a thermal expansion coefficient $\alpha$ under a temperature variation $\Delta T$ will experiment a length variation $\Delta L$.

We have then $\Delta L$ = 0.481 foot, $L_0$ = 1671 feet and $\alpha$ = 0.000013 per centigrade degree (this is just the linear thermal expansion of steel that you must find in a table), which means from the equation for linear thermal expansion that we have a $\Delta T =\frac{\Delta L }{\alpha L_0}$ = 22.14°. As said before, these degrees are centigrades (Celsius or Kelvin, it does not matter since it is only a variation), and the foot units cancel on the equation, showing no further conversion was needed.

Since our temperature on a cool spring day was 13.0°C, our new temperature must be $T_f=T_0+\Delta T$ = 35.14°C