Question

Question: Suppose that a steel hoop could be constructed to fit just around the earth’s equator at 20.0ºC. What would be the thickness of space between the hoop and the earth if the temperature of the hoop were increased by 0.500ºC?

Short answer

Answer

The thickness of space between the hoop and the earth is 38m .

Step-by-step solution

  Concept of thermal expansion.

Thermal expansion concept is manifested as the measurable changes in dimensions of an object when subjected to a change in temperature. The tendency of any matter is to expand whenever there is a rise in temperature.

Determination of the thickness of space between the hoop and the earth after  0.5000C change in temperature.

Initially, the radius of the circular hoop $R_0$ is equal to the radius of the earth,

$$\begin{gathered} R_0=R_E \\ =6.37\times {10}^6\text{m} \end{gathered}$$

The thickness is the increase in radius of the hoop due to expansion. Use the formula for radius expansion,

$R=R_0\alpha \Delta T$ ...(i)

Here, R is the expanded radius, $\alpha$ is the coefficient of thermal expansion of the material and is the change in temperature.

For steel, value of $\alpha$ is $1.2\times {10}^{-5}{K}^{-1}$. Substitute the values in equation (i)

$$\begin{gathered} R=\left(6.37\times {10}^6\text{m}\right)\times \left(1.2\times {10}^{-5}{\text{K}}^{\text{- 1}}\right)\times \left(0.5\text{K}\right) \\ =38\text{m} \end{gathered}$$

Thus, the value of thickness is 38m .