The radius of the Earth measures approximately $6378 \mathrm{~km}$ at the Equator. Suppose a wire is exactly adjusted over the Equator.

Then, 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. Would a man standing, an ant, or an elephant be able to pass under this wire?
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 along the Equator, it seems that the gap 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 gap between the extended wire and the "equator" of this sphere also changes, such that the larger the sphere considered, the smaller the gap between the wire and the "equator" of the sphere.

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

Consider the circumference of length $2 \pi R$ of a circle with radius $R$ and also the circumference of length equal to $2 \pi R+1$ of another circle with the same center, with radius equal to $R+h$. Thus, $h$ is the height of the "gap" between the two circumferences. Since the circumference of a circle with radius $R+h$ has a length equal to $2 \pi(R+h)$, we obtain the equality

$$
2 \pi R+1=2 \pi(R+h)=2 \pi R+2 \pi h
$$

which, simplified, gives $1=2 \pi h$, or

$$
h=\frac{1}{2 \pi} \approx \frac{1}{6.28} \approx 0.16
$$

![](https://cdn.mathpix.com/cropped/2024_05_01_30c9a294a58a6e4b8190g-112.jpg?height=308&width=306&top_left_y=840&top_left_x=1523)

Therefore, regardless of the value of $R$, the height of the gap obtained with an additional $1 \mathrm{~m}$ of wire is always approximately $16 \mathrm{~cm}$. In particular, only an ant is capable of passing underneath this wire.