[net.puzzle] Band around the Earth - with solution

jeff@heurikon.UUCP (02/17/84)

Have you seen this one?  Assume the Earth is a perfect sphere.
Wrap a band of wire around it at the equator.  Pull it *tight*
and connect the ends of the band together.

Now, snip the band and add exactly ten feet of extra wire.  Go
around the Earth and adjust the band so that it is the same height
off the ground all the way around.  (It won't be tight any more, right?)

Okay, here's the question:  After you've done all that, would
you be able to:
    a) Just slide a piece of paper between the band and the ground?
    b) Fit your head between?
    c) Crawl under the band?
    d) Walk under it?
More, specifically, just how high will the band be above the ground?

Now, before you start calculating, which answer *sounds* most reasonable?
The fun part of this problem is that the answer will probably surprise
you.  So think about it a little first.

Beware, solution coming below...
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You can figure out the solution in a number of ways.  Here's the
mathematical solution:
	C = 2*pi*r	where r = radius of earth
	r = C/(2*pi)	
	delta r = (delta C)/(2*pi) = (10 feet)/(2*pi)
	delta r = 1.59 feet
So, you could crawl under it.

You don't even have to know what the radius of the Earth is.  In fact,
the problem could be restated using an *orange* or a point of zero radius
instead of the Earth and, of course, the answer would be the same.

The intuitive solution is to take the extra wire and add 1/4 of it at
each of four spots, 90 degrees apart.  Each new piece will be 2.5 feet
long and, in conjunction with the extra piece on the opposite side of
the globe, will tend to push the original band away from the Earth one
half that amount, or 1.25 feet.	  The difference between 1.25 and 1.59
is due to the slight discontinuity in the curvature thus obtained.

	  Before 	After adding two pieces 
				 2.5'
	    			 ---		 movement
	   (o)			( o )		<--    --> 
	    			 ---		1.25  1.25
 				 2.5'

When I first saw this I was amazed that a mere 10 feet could make such
a big difference in height, considering the size of the Earth.
This is a good one to ask over the lunch table.
-- 
/"""\	Jeffrey Mattox, Heurikon Corp, Madison, WI
|O.O|	{harpo, hao, philabs}!seismo!uwvax!heurikon!jeff  (news & mail)
\_=_/				     ihnp4!heurikon!jeff  (mail - fast)

jjb@pyuxnn.UUCP (J Bernardis) (02/22/84)

I think the interesting thing about this problem is that the
delta-R you are solving for is independent of the original radius.
You can take an infinitesimally (sp?) small circle, and if you
increase its circumference by 10 ft, the difference between the
two radii will still be 1.59 feet.  It's just one of those things
that's obvious when you see the algebra, but it's still very
difficult to believe.

		Jeff Bernardis, AT&T Technologies @ Piscataway NJ