Information

The earth and the orange

The earth and the orange

Imagine that the earth is a perfect sphere and that we have an orange that is too. Suppose we surround the globe with a rope that crosses the entire equator and is perfectly adjusted to the earth, that is, that touches the earth in its entire path.

Now we do the same with the orange and surround it with a string that wraps it around its larger circle. Suppose now that we lengthen both strings by 1 meter and separate them from the earth and the orange respectively so that the distance of the rope to the object is always constant.

In which of the two cases will this space between the string and the sphere that surrounds be greater?

Solution

Common sense leads us to think that the distance from the rope to the orange will be much greater than the distance from the rope to the ground since we only extend it one meter that is somewhat insignificant for a sphere of 40,000 km perimeter as is our planet.

In contrast, in orange, compared to its size, a meter is a huge amount and if it is added to the perimeter of the orange, the resulting clearance will be very noticeable.

We will perform the calculations. If we call C to the perimeter of the earth and c that of the orange we have that the radius of the Earth will be R = C / 2π and that of the orange, r = c / 2π.

After adding the piece of rope, the perimeter of the ring that girdles the Earth will be C + 1 and the one with the orange ring c + 1, their radii will be respectively (C + 1) / 2π and (c + 1) / 2π.

To calculate the distance that separates each string, we will subtract the initial radius of the sphere from that obtained after adding a meter of string.

In the case of land we will have C + 1 / 2π - C / 2π = 1 / 2π

And in the case of the orange c + 1 / 2pi - c / 2pi = 1 / 2π

Surprise! In both cases, The clearance resulting from lengthening the rope by one meter, regardless of the size of the Earth or the orange, will be the same: 1 / 2π meters, that is, approximately 16 cm.