Rope around the Earth

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Rope around the Earth
Rope around the Earth.jpg
Field: Geometry
Image Created By: Harrison Tasoff

Rope around the Earth

The puzzle of lengthening a rope tied taut around the equator so that, if made to levitate, there is a one foot gap at all points between the rope and the Earth.


Basic Description

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.

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: C=2\pi\,\!r Where r is t [...]

The Circumference of a circle is given by the equation: C=2\pi\,\!r Where r is the radius.
Radius circumference.jpg


In the image to the right:

Lrope is the length of the rope.
Cearth is the Circumference of the Earth.
Rrope is the radius of the circle made by the rope.
Rearth is the radius of the Earth.


When the rope is taut around the globe, its length equals the circumference of the Earth.

L_{rope}=C_{earth}=2\pi\,\!R_{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

to

Rrope 2= Rearth+1 ft.


So: L_{rope 2}=2\pi\,\!(R_{earth}+1)


Distributing the 2 π yields:

L_{rope 2}=2\pi\,\!R_{earth}+2\pi\,\!


Now it is clear that new length of the rope is merely 2 π feet longer than he 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

Were the lengthened rope again to be held taught, by raising it at an arbitrary point (as shown in the picture to the right), what would the distance form this point to the surface of the earth be?

In the diagram to the right:

x1 is the distance from the horizon to the highest point on the taut rope.
xo is the ground distance from the point where the rope leaves the globe, to the point below the apex of the rope.
R is the radius of the globe.
h is the height of the apex of the rope above the ground.

The Pythagorean Theorem: C^2=A^2+B^2 where A and B are the legs of a right triangle, and C is the hypotenuse.

Using this theorem, we know that: (R+h)^2=R^2+x_1^2, which is equivalent to x_1=\sqrt{(R+h)^2-R^2}.

To find the length of xo, we must remember what the length of an arc is:

L_{arc}=r\theta\,\!

Where θ is the angle formed between two radii from the center of the circle to the endpoints of the arc.

Thus, x_o=R cos^{-1}(\frac{R}{R+h}).

Where cos-1 represents the angle whose cosine is\tfrac{R}{R+h}.


Since we know that we lengthened the rope by 2 π feet, we know that 2x1= 2xo + 2 π, because 2x1 is the extra slack put in to the rope.

Thus: x_1=x_o+\pi\,\!

Running the three equations above through a numerical calculator resulted in h = 614.771 ft as the height that a 2π extension would yield.


As a result of the cos-1 (R / (R + h)), there is no explicit formula to find the height. Nevertheless, very accurate results can be achieved using Taylor approximations. In this way, we find the following formulas for height achieved by lengthening the rope by length l; the first is a first order approximation, and the second, a slightly more accurate second order approximation. The approximations are more accurate the smaller the added slack is compared to the original radius of the circle.

h \left (\mathit{l} \right )= \frac{1}{2} \left ( \frac{3}{2} \right )^{\left ( \frac{2}{3} \right )} \mathit{l}^{ \left ( \frac{2}{3} \right )} R^{ \left ( \frac{1}{3} \right )}


h \left (\mathit{l} \right )= \frac{1}{2} \left ( \frac{3}{2} \right )^{\left ( \frac{2}{3} \right )} \mathit{l}^{ \left ( \frac{2}{3} \right )} R^{ \left ( \frac{1}{3} \right )} + \cfrac {9 \left ( \frac{3}{2} \right )^{\left ( \frac{1}{3} \right )}}{80 R^{  \left (\frac{1}{3} \right )}}\  \mathit{l}^{\left ( \frac{4}{3} \right )}


Where l is the added slack, h (l) is the height as a function of l, and R is the initial radius of the rope circle.


Taking l = 2π feet of slack and R = R earth = 20925524.9 feet from the initial problem the first approximation yields 614.766, demonstrating the accuracy and precision of these approximations. The second approximation yields 614.771, which is more precise than the first, but really only a negligible amount, and simply highlights the accuracy of the first. On the scale of the Earth, there is a decent margin of error in measurements: the earth is not a perfect sphere, ropes stretch with strain, objects expand and contract with heat. The result is that these approximations are far more accurate and precise than our measurements can ever be.


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.


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:
http://extellireader.wordpress.com/2009/03/03/rope_around_earth/





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