String theory

My brother posed an interesting puzzle earlier today, one which I knew about and which caused me to search this site to confirm I’d not talked about it before. I hadn’t, so here it is.

Get a piece of string and wrap it tightly around a football. Your string will be about 70cm long. Untie it and add a metre to this string, and lay it in a circle on the ground around the football. The string will be about 16 centimetres from the edge of the football.

Now get a longer piece of string and wrap it tightly around the earth. You may need more than one ball of string for this—tie the ends together. Let me know when you’re done.

Now take this string and add a metre to it, and make it hover a constant distance above the earth. You may not believe it, but this string will also be about 16 centimetres away from the earth’s surface.

The extra metre gives it a full 16cm clearance. The human mind makes you think that an extra metre won’t make a jot of difference when you’re talking about something as big as the earth; but it does.

The mathematical rationale for this is as follows. c is the smaller circumference, r/R is the smaller/larger radius.

c = 2πr, so r = c/2π

Add 1 to the circumference, so c + 1 = 2πR.

Substitute c with 2πr and you have 2πr + 1 = 2πR

So 2π(R – r) = 1

So R – r = 1/(2π), or else R = r + 1/(2π)

So whatever the radius of your original circle, the new radius will be 16cm longer for that metre of string added.

I suppose the best way to describe the logic behind this is to take evenly-spaced concentric circles, each 16cm apart. The first circle has a radius of 16cm and a circumference of 1m. The next circle has a radius of 32cm and a circumference of 2m. The next, 48cm and 3m […] a 637,810,000cm radius gives a circumference of 40,074,784.21m, and a radius of 637,810,016cm gives a circumference of 40,074,785.21m.

The last two in that series were the earth and "the earth + 16cm", but you already knew this right, given that you’ve just tied a piece of string around it?

So there you have it.

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