The radius of the Earth is approximately $6670 \mathrm{~km}$. Suppose a wire is adjusted to fit exactly over the Equator, which is a circle with a radius approximately equal to $6670 \mathrm{~km}$.

![](https://cdn.mathpix.com/cropped/2024_05_01_a444074f1f8ae50444d5g-06.jpg?height=236&width=228&top_left_y=1633&top_left_x=1466)

Next, suppose the length of the wire is increased by $1 \mathrm{~m}$, so that the wire and the Equator form concentric circles around the Earth. Can a man standing, an ant, or an elephant pass under this wire?

## List 6
Given that the radius of the Earth is very large, and only an additional $1 \mathrm{~m}$ was added to the length of the wire, it seems that the clearance between the wire and the Equator is very small. Moreover, if we replace the Earth with Jupiter or a marble and perform the same experiment, it seems that the height of the clearance between the extended wire and the equator of the sphere also changes, such that the larger the sphere considered, the smaller the clearance between the wire and the equator of the sphere.

Let's see that this intuitive idea is false and that the height of the clearance between the wire and the Equator is approximately $16 \mathrm{~cm}$, regardless of the radius of the sphere on which the experiment is performed.

Consider a circle of radius $R$. Its circumference is equal to $2 \pi R$. Let's also consider a circle with the same center, but with a circumference equal to $2 \pi R + 1$.

![](https://cdn.mathpix.com/cropped/2024_05_01_a444074f1f8ae50444d5g-25.jpg?height=329&width=331&top_left_y=2153&top_left_x=1407)

This circle has a radius equal to $R + h$, where $h$ is the height of the clearance between the two circles. Since a circle with radius $R + h$ has a circumference of $2 \pi (R + h)$, we obtain the equation $2 \pi (R + h) = 2 \pi R + 1$. Simplifying this expression, we get $h = \frac{1}{2 \pi} \approx \frac{1}{6.28} \approx 0.16$. Therefore, for any value of $R$, the height of the clearance is approximately $16 \mathrm{~cm}$.

Thus, only the ant is able to pass under the wire.

## List 6