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The Earth's Waist-Line

Imagine that the earth is a perfect sphere with a metal wire wrapped around its equator, fitting snugly against the earth's surface. Now, imagine that we cut this wire at some convenient spot, and splice in an additional 15 meters length of wire.

Of course, this causes some slack in the wire to occur, however slight it may be. We equalize this slack all around the world by placing the wire on some posts or sticks of an appropriate height, thus creating a larger circle than we had before the cut.

Would you be able to:

  • Walk under the wire normally,
  • Walk under it but by bending your body down a lot,
  • Just barely squeeze under it (if at all), or
  • None of the above?
Finally, how would your answer change if you did this with a wire around the equator of a planet with a different radius?

 


Hint: Visualize the problem...do you see two concentric circles? What are the radii and circumferences of the two circles?

 


Solution Commentary: Reread the hint. Suppose the earth's radius at the equator is R meters (you can look up the "exact" value if you want) and that the radius of the wire-formed circle is R+r. Then, the circumferences for the two circles are 2(PI)R and 2(PI)(R+r) respectively, yet differ by what length? With the aid of some algebra, take it from there.......to find a real surprise!

And what happens if you change the planet's radius? Notice that in the preceding solution commentary, the problem could be solved using the variable radius R...which means that the radius is irrelevant. That is, you could actually peform the experiment using a basketball...all you need is about 16 meters of wire and some posts.