Difference between revisions of "Rope around the Earth"
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If the puzzle were different, having a different amount of slack added to the rope, we could still find ''h'' if given a value for ''l''. We can create a general formula that will find ''h'' for any value of ''l'', in addition to when ''l'' = 2π. As mentioned above, due to the cos<sup>-1</sup> (''R'' / (''R'' + ''h'')) there is no explicit formula to find the height. Nevertheless, using series, on can obtain very good approximations. Using [[series approximations]] one can derive the following formulas for height achieved by lengthening the rope by length ''l''; the first is a 1<sup>st</sup> order approximation, and the second, a more accurate 2<sup>nd</sup> order approximation. The approximations are more accurate the smaller the added slack is compared to the original radius of the circle. | If the puzzle were different, having a different amount of slack added to the rope, we could still find ''h'' if given a value for ''l''. We can create a general formula that will find ''h'' for any value of ''l'', in addition to when ''l'' = 2π. As mentioned above, due to the cos<sup>-1</sup> (''R'' / (''R'' + ''h'')) there is no explicit formula to find the height. Nevertheless, using series, on can obtain very good approximations. Using [[series approximations]] one can derive the following formulas for height achieved by lengthening the rope by length ''l''; the first is a 1<sup>st</sup> order approximation, and the second, a more accurate 2<sup>nd</sup> order approximation. The approximations are more accurate the smaller the added slack is compared to the original radius of the circle. | ||
− | <div id="Image4"></div>[[Image:Series approximations | + | <div id="Image4"></div>[[Image:Series approximations after anna.jpg|Image 4. The series approximations for a large value of ''R'' relative to ''l''. Image created by Harrison Tasoff|thumb|400px|left]] |
Revision as of 14:27, 22 June 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 there is a one foot gap at all points between the rope and the Earth if the rope is made to hover. 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
In this puzzle, we treat the Earth as though it were a prefect sphere, even though it actually bulges toward the equator. 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 so 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, 2π feet the answer regardless of the size of the ball around which the rope is wrapped.
Just as bizarre is what happens when one point on the lengthened rope is lifted up so that the rope is taut again, as in Image 1. The maximum clearance under the rope proves to be 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 Image 2:
- 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 have to lengthen 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.
Thus:
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 2π is a result of distributing the 2π to the 1 in Eq. 2, which yields an increase of 2π no matter what the radius of the ball is. Also, 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
At first this puzzle is interesting because of its non-intuitive results. It becomes even more interesting if we can then make them intuitive after all, and indeed and we can. Though it may seem that 2π, or roughly 6.28, feet is a 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, though this varies because the Earth is not a perfect sphere. 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 .000000047 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
- ↑ Modified from original content at http://rst.gsfc.nasa.gov/Sect16/Sect16_1.html
- ↑ Modified from original content at http://extellireader.wordpress.com/2009/03/03/rope_around_earth/
- 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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