Thursday, September 20, 2012

The "String Around the World" Problem

Now, imagine this - and I use the word "imagine" early on, because this problem really does require a certain feat of the imagination. One day, when you are emptying out your pockets, you find a really long length of string; a length of string so long, that it seems that it might be long enough to stretch all the way around the circumference of the earth. Now, this certainly defies logic, but it does not defy the imagination, so let's play pretend for a while.

As it so happens, you're standing right here, in Malaysia, which is located conveniently closely enough to the equator of the earth. A perfect place to start your experiment. The Greenwich would probably be an equally suitable spot, but from here you'll be able to lay your length of string all the way around the world, without having to pass through, say, the North Pole. So you jog on over to Indonesia with your string in your pocket, and begin laying it down on the ground all the way around the earth. Of course, the Ocean presents a bit of a problem, so what should we do? Let's build a World Bridge that joins all the countries on the Equator. Again, a bit of imagination may be required here. From Indonesia you travel due East, with one end of the string in your hand, and the other end waiting back in Indonesia. You pass through all the countries on the equator, until finally, you arrive back at Indonesia. You fuse both ends of the string together, using... fire, or super glue, or magic, or something... and, well what do you know, the string is just long enough! An inch less and you would not have been able to make the ends meet.

Quite satisified with how everything turned out, you bend down, and try to lift the string off the ground. But it won't budge a bit. It's as tightly bound around the earth as the belt around your waist. You ring up your friend in Uganda and ask them to try lifting the string off the ground. They reply that the string won't move. You call all your other friends in Brazil, Kenya, and Ecuador to try the same thing, and they all give the same results. You station some more friends all the way around the World Bridge, and they can't live the string off the ground. The string is a snug fit all the way around the earth at every point on the world. To put this in perspective, you can't even slide a sheet of paper between the string and the earth.

Being quite pleased, you stick your hands into your pockets, and discover, lo and behold! You have another piece of string in your pocket. This piece of string, however, is exactly 1 meter long. You decide that it wouldn't hurt to add this 1 meter of string to the one on the ground, so you cut the string at your feet, add the 1 meter of string to it, and fuse them together with... whatever method it was you imagined up just now. Then you ring up all your friends in Kenya, Brazil, Uganda, Ecuador, and the whole World Bridge, and tell them all to pull up on the string all at once. Of course, you in Indonesia will try to lift the string too.

The main question, after an unnecessarily long backstory, is this: On the count of three, how high off the ground will you and your friends be able to lift the string?

If the string was a perfect fit around the globe just now, it would surely be able to come off the ground now, after you've added 1 meter to it, right?

When I asked this question to my friends at school, most of them said that the string would come 1 meter off the ground. This answer certainly defies Logic, and Mathematics; but this is just the Georgian way of thinking, these days. If the string come 0 meters off the ground, and you add 1 meter to the string, then it would come 1 meter off the ground. In other words, when you are at loss for an answer, just take any information given in the question, and write it in the answer space. Maybe you'll get some points for that. As a friend of mine always says, "Only in SGI."

The answer that people who ask this question generally expect to get from other people is, the string would hardly come off the ground at all. After all, the circumference of the earth is 40, 074 kilometers long. Your string is exactly that length, and it would not have come off the ground at all. After you've added 1 meter to it... your string becomes 40, 074 kilometers and 1 meter long. Imagine that. Such a small addition to the string around the world, would hardly make a difference at all. So, the string probably still wouldn't move at all, and even if it did, the difference would hardly be discernible. You might be able to slide that piece of paper under the string now, but only just. This is what most people would answer - if you ask the right people, anyway - and on the whole, it appears to be a very logical answer. However, it is actually mathematically inaccurate. In other words, it is dead wrong.

The correct answer, that is to say, the answer that is mathematically accurate, still seems to defy Logic. If you tie a piece of string all the way around the earth, and then add 1 meter to it, and ask everyone to lift the string off the ground, everyone would be able to life the string to a height of 15.9 centimeters off the ground.

The explanation for it - what can I say? It all comes down to the mathematics. The equation for finding out how high off the ground the string will rise with an extra meter added to it, is configured in such a way that the diameter of the sphere becomes obsolete; meaning to say that, if you repeated this experiment with the Moon, the globe in your study, or the gumball in your pocket, you would find that the string came off the surface of the sphere to a height of 15.9 centimeters at all points when it was pulled away from the object evenly all the way around. When you consider how this would work in comparison of a basketball to a golfball, you may start to see how it works.

If you have an interest in mathematics and want to find out how it works, exactly, or if you think I'm trying to pull the fleece over your eyes here and want some real evidence, or if you got bored reading my long post just now and jumped to the end here, you can find a much shortened version of the above post together with all the calculations involved at Worsley School, i.e. by clicking anywhere here.

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