Steel expands 11 parts in a million for each 1 degree C change. Consider a 40,000 km steel pipe that forms a ring to fit snugly all around the circumference of the Earth. Suppose people along its length breathe on it so as to raise its temperature 0.80 degree C. The pipe gets longer. It also is no longer snug. How high does it stand above ground level? (To simplify, consider only the expansion of its radial distance from the center of Earth, and apply the geometry formula that relates circumference C and radius r, C = 2 \pi r. The result is surprising!)

Question

Steel expands 11 parts in a million for each 1 degree C change. Consider a 40,000 km steel pipe that forms a ring to fit snugly all around the circumference of the Earth. Suppose people along its length breathe on it so as to raise its temperature 0.80 degree C. The pipe gets longer. It also is no longer snug. How high does it stand above ground level? (To simplify, consider only the expansion of its radial distance from the center of Earth, and apply the geometry formula that relates circumference C and radius r, C = 2 \pi r. The result is surprising!)

Answer 1

To find out how high the steel pipe stands above ground level, we need to calculate the change in its radius due to the increase in temperature and then use the formula that relates the circumference to the radius.

First, let's determine the change in the radius. We know that steel expands 11 parts in a million for each 1 degree Celsius change. Since the temperature increased by 0.80 degrees Celsius, the change in the radius can be calculated as follows:

Change in radius = (0.80 degrees Celsius) * (11 parts per million) * (40,000 km)

Converting the length of the pipe to meters (1 km = 1000 m), we have:

Change in radius = (0.80) * (11/1,000,000) * (40,000,000 m)

Simplifying the calculation further:

Change in radius = 35.2 m

Now, let's use the formula for the circumference of a circle, C = 2πr, to determine the height above ground level. Since the pipe forms a ring around the Earth's circumference, the circumference of the pipe after expansion would be:

C' = C + 2π(change in radius)

Substituting the known values:

C' = 2πr + 2π(35.2 m)

Since the original circumference of the pipe is equal to the circumference of the Earth (40,000 km), which is 2π times the radius of the Earth, we have:

C' = (2π * radius of Earth) + 2π(35.2 m)

Simplifying further:

C' = 2π(r + 35.2 m)

We can equate this circumference to the formula for the circumference of a circle:

2πr' = 2π(r + 35.2 m)

This allows us to solve for r', the new radius after expansion:

r' = r + 35.2 m

Here, r is the radius of the Earth, which is approximately 6,371 km (or 6,371,000 m). Substituting this value, we can determine the new radius:

r' = 6,371,000 m + 35.2 m

r' ≈ 6,371,035.2 m

Finally, to find out how high the steel pipe stands above ground level, we subtract the radius of the Earth from the new radius:

Height above ground level = r' - r

Height above ground level ≈ 6,371,035.2 m - 6,371,000 m

Height above ground level ≈ 35.2 m

Therefore, the steel pipe stands approximately 35.2 meters above ground level.