Difference between revisions of "Rope around the Earth"
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|AuthorName=Harrison Tasoff | |AuthorName=Harrison Tasoff | ||
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− | |WhyInteresting=Though it may seem that this is minuscule amount of extra rope needed to to produce such a considerable result, a look at the ratios will show otherwise. | + | |WhyInteresting= |
+ | <font color=red>(Maurer) At first this puzzle is interesting because the results are non-intuitive. But then it is doubly interesting if we can make them intuitive after all, and we can!</font> Though it may seem that this is minuscule amount of extra rope needed to to produce such a considerable result, a look at the ratios will show otherwise. | ||
The radius of the Earth is roughly 20,920,000 feet. There is 1 foot of difference between the radius of the circle made by the lengthened rope and the radius of the Earth. This foot of difference is a mere fraction of the radius of the Earth: about five one-hundred millionths, or .000000047, of the Earth's radius. A foot doesn't seem so large anymore. | The radius of the Earth is roughly 20,920,000 feet. There is 1 foot of difference between the radius of the circle made by the lengthened rope and the radius of the Earth. This foot of difference is a mere fraction of the radius of the Earth: about five one-hundred millionths, or .000000047, of the Earth's radius. A foot doesn't seem so large anymore. |
Revision as of 16:34, 24 May 2011
Rope around the Earth |
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Rope around the Earth
- This is a puzzle about by how much a rope tied taut around the equator must be lengthened so that, if made to hover, there is a one foot gap at all points between the rope and the Earth? Although finding the answer requires only basic geometry, even professional mathematicians find the answer strangely counter-intuitive. There is a related problem about stretching the rope taut again where the answer is even more surprising. A question similar to this appeared in William Whiston's The Elements of Euclid circa 1702.
Contents
Basic Description
Suppose a rope was tied taut around the Earth's equator. It would have the same circumference as the Earth (24,901.55 miles). The question is: by how much would the rope have to be lengthened such that, if made to hover, it would be one foot off the ground at all points around the Earth?
Despite the enormous size of the Earth, and the 1 foot gap around the entire circumference, the rope would have to be lengthened by a mere 2π feet, or roughly 6.28 feet.
In fact, this result is independent of the size of the ball around which the rope is wrapped.
Just as bizarre is that, if one point on the lengthened rope were to be lifted up so that the rope was taut again, the maximum clearance under it is quite large. For the specific case of a rope looped around the Earth, a 2π foot extension would provide 614.771 feet of clearance if the rope were lifted. This is enough room to fit two Statues of Liberty under it, base and all. Unlike the previous question, however, this result is dependent on the size of the ball.
A More Mathematical Explanation
- Note: understanding of this explanation requires: *High-school algebra and High-school geometry
The circumference of a circle is given by the equation: , where r is [...]
In the image to the right:
- L_{rope 2} is the length of the extended rope.
- C_{earth} is the Circumference of the Earth and the original length of the rope (L_{rope 1}).
- R_{rope 2} is the radius of the circle made by the extended rope.
- R_{earth} is the radius of the Earth and the original radius of the rope (R_{rope 1}).
When the rope is taut around the globe, its length equals the circumference of the Earth.
The puzzle states that we've lengthened the rope and made the rope hover 1 foot of the surface of the earth. Lengthening the rope so that it is 1 foot off the ground at all points simply means changing the radius of the circle it forms from:
- R_{rope 1}= R_{earth}
to
- R_{rope 2}= R_{earth}+1 ft.
So:
Distributing the 2 π yields:
The new length of the rope is merely 2 π feet longer than the original length. Indeed, one can see that the additional 2 π is a result of extending the radius of the rope circle by one foot, an extension that will be the same no matter the initial radius of the object being enclosed. Because the value of R_{earth} was not any number in particular throughout this proof, the answer didn't depend on the specific radius of the Earth in any way. Hence, this 2π extension would be the same for a ball, planet, or star of any size.
Maximum Height of Rope
- Note: A familiarity with trigonometry, series, and approximations is recommended for this section.
Why It's Interesting
(Maurer) At first this puzzle is interesting because the results are non-intuitive. But then it is doubly interesting if we can make them intuitive after all, and we can! Though it may seem that this is minuscule amount of extra rope needed to to produce such a considerable result, a look at the ratios will show otherwise.
The radius of the Earth is roughly 20,920,000 feet. There is 1 foot of difference between the radius of the circle made by the lengthened rope and the radius of the Earth. This foot of difference is a mere fraction of the radius of the Earth: about five one-hundred millionths, or .000000047, of the Earth's radius. A foot doesn't seem so large anymore.
Similarly, 2 π feet is 4.7 x 10^{-8} of the circumference of the Earth (which is about 131,000,000 feet). And, unsurprisingly, the ratio of 1 foot to the Earth's radius is the same as that of 2 π feet to the Earth's circumference.
So, in this perspective, a small change in the length of the rope yields a proportionally equivalent small change in the radius of the rope circle.
Teaching Materials
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References
- Pickover, C. A. (2009). The Math Book. New York: Sterling Publishing Co.
- (2009, March 3). Roping the Earth. Message posted to:
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