String Around the World

This is one of my favorite examples of counter-intuitive conundrum. I was fascinated by this problem (and its solution) for the first time when I was probably 15. I happened to come across this again sometime ago. It goes like this:

Assuming that the Earth is a perfect sphere, let's say that we tie a long string all around the equator so that it perfectly fits. Now, imagine if we cut the string at some point, and add an extra 2 meters of string, and tie it back again. The slightly longer string will be a bit loose now and can be raised by some height all around the Earth. How big will be that gap?

Intuition suggests that since the string is so long, adding an extra 2 meters would hardly make a difference. The height by which the string raises would be to the order of microns, and won't be even visible to the naked eye. But that is not the case.

The surprising answer is that the gap would be big enough for most people to crawl under! It will stand approximately 32 centimeters above the surface all the way around. Hard to believe, right? Let us look at the calculations:

Length of the string = Circumference of the earth (Pi x Diameter of the earth)

New length of the string = Old length + 2

Also, new length = Pi x (Diameter of the earth + 2 x Gap)

Therefore, Pi x Diameter of earth + 2 = Pi x (Diameter of earth + 2 x Gap)

Which means that the Gap = 1/Pi = 1/3.14 = 0.32 meters

The gap here is independent of the diameter of earth. So even if you do this with a tennis ball or a football, the answer would be the same (which is somewhat easier to visualize and more believable than in the earth's case).