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The concept of the Coefficient of Linear Expansion is crucial in understanding how materials expand in response to temperature changes. This coefficient, denoted as \( \alpha \), measures how much a material's length will change per degree change in temperature. For steel, which is the material in question for the steel hoop, the coefficient \( \alpha \) is \( 11 \times 10^{-6} \ \/^{\circ}C \). This means for every degree Celsius increase in temperature, the steel will expand by a fraction of its original length, proportional to this coefficient. Knowing the coefficient allows engineers and scientists to predict how much a material will expand, which is essential for designing structures that must withstand temperature variations. The formula used to calculate the change in length due to thermal expansion is \( \Delta L = \alpha L_0 \Delta T \), where \( L_0 \) is the original length and \( \Delta T \) is the change in temperature.

Circumference Calculation is a fundamental step when dealing with circular objects such as the steel hoop in this exercise. The circumference \( C \) of a circle is determined by the formula \( C = 2\pi R \), where \( R \) is the radius of the circle. In the problem, the initial circumference of the hoop equals Earth's circumference, because it fits snugly around the equator.

Calculating Earth's circumference involves using its average radius, approximately \( 6,371 \ km \) or \( 6,371,000 \ m \). Thus, the initial circumference of the hoop is \( 40,030,173 \ m \). Accurate calculation of circumference is key, as it forms the baseline for understanding how much the hoop expands after a temperature increase.

The Temperature Change Effect explains how changes in temperature impact the dimensions of materials. When the temperature of a material such as steel increases, the atoms in the steel move more vigorously, causing the material to expand. This is why, when the temperature of our hoop increases by \( 0.500 \ ^{\circ}C \), its circumference also increases.

The change in circumference \( \Delta L \) can be calculated using the formula \( \Delta L = \alpha L_0 \Delta T \), where \( \alpha \) is the coefficient of linear expansion, \( L_0 \) is the original circumference, and \( \Delta T \) is the temperature change.
The temperature change in this exercise leads to a \( 220.165 \ m \) increase in the hoop's circumference, illustrating how even a small increase in temperature can result in a significant dimensional change when dealing with large objects like the hoop.

Radius Calculation becomes necessary after determining how much the circumference changes due to thermal expansion. Once the new circumference of the hoop is known, calculating the new radius follows from the circumference formula \( C = 2\pi R_{new} \). The formula can be rearranged to solve for the new radius \( R_{new} = \frac{C}{2\pi} \).

In the exercise, the expanded circumference becomes \( 40,030,173 + 220.165 \ m \). Calculating the new radius inserted into this formula yields an increase in the radius. Finally, the gap between the Earth and the hoop, or the expansion space, is calculated as the difference between the new and original radii, resulting in approximately \( 35 \ m \). This gives us a full understanding of how the increase in the hoop's size translates into an increased radius and ultimately a space between the hoop and Earth.