Rope around the Earth
|Rope around the Earth|
Rope around the Earth
- This is a puzzle, about a rope tied taut around the equator. How much must it be lengthened so that, if made to levitate, there is a one foot gap at all points between the rope and the Earth? This is a puzzle in that the answer is surprising. 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.
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 extended rope were to be lifted up, the maximum clearance under it is quite large. For this specific case it would be 614.771 feet, enough room to fit two Statues of Liberty under it, base and all. This result is, however, 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:
- Lrope 2 is the length of the extended rope.
- Cearth is the Circumference of the Earth and the original length of the rope (Lrope 1).
- Rrope 2 is the radius of the circle made by the extended rope.
- Rearth is the radius of the Earth and the original radius of the rope (Rrope 1).
When the rope is taut around the globe, its length equals the circumference 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:
- Rrope 1= Rearth
- Rrope 2= Rearth+1 ft.
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 by definition be the same no matter the initial radius of the object being enclosed.
Maximum Height of Rope
- Note: A familiarity with trigonometry, series, and approximations is recommended for this section.
Why It's Interesting
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.
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- 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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