## Roping the Earth

### 2009/03/03

I have utterly enjoyed the very stimulating online physics book by Christoph Schiller Motion Mountain. It includes some 1700 challenges for the reader ranging from fun little puzzles to standing research problems. This is a great example of how textbooks should be written to engage the student.

Here is Challenge 70 from the book.

A rope is put around the Earth, on the Equator, as tightly as possible. The rope is then lengthened by 1 m. Can a mouse slip under the rope?

I have seen this problem previously and in both cases was dissatisfied with how understated the solution was. Schiller’s answer is:

Yes, it can. In fact, many cats can slip through as well.

The author’s reasoning must be as follows. Since the circumference of a circle is $P = 2\pi r$, then adding one meter to P adds 16 cm to the radius. Therefore, a cat could crawl under the rope. The result may be counterintuitive because of the huge proportions involved: 1 meter seems so insignificant to Earth’s circumference that it shouldn’t change anything. Yet mathematics must trump intuition’s whims.

However, this calculation assumes that the rope retains its circular shape and remains centered on Earth’s center (as in Figure a below). If one pulls the rope up as shown in Fig. b, the 1-meter slack will allow 121 meters of clearing under the rope. That’s enough to slip the Titanic under it, let alone `many cats.’

A rope around the earth

Here is the calculation as shown in the figure below. Imagine a tower h = 121.37 m high. Then the distance from the top of the tower to the horizon is $L_1 = \sqrt{(R+h)^2-R^2}=39,282.48 \text{ m}$, where $R = 6,357,000 \text{ m}$ is the Earth radius. Meanwhile, the ground distance from the bottom of the tower to the same point on the horizon is $L_0 = R\cos^{-1}\frac R {R+h} = 39,281.98 \text{ m}$, a difference of only 0.5 m. Therefore, the straight path from the horizon to the top of the tower and again to the horizon on the opposite side is 1 m longer than the ground path between the two points on the horizon $2L_1-2L_0=1 \text{ m}$.

Calculation of the height of the object that could be slipped under the rope lengthened by 1 meter.

Amazingly, adding just 1 mm of slack to the rope will make a clearing high enough for a child to walk under the rope: 1.25 m.

### 3 Responses to “Roping the Earth”

1. Christoph Schiller Says:

Thank you for this! It is the first time I heard about it, and I will add the result in the book.

Regards

Christoph Schiller

2. Todd Brown Says:

Dimitri,

Hi, this is Todd Brown from GE. I stumbled on your blog and read the ‘rope around the earth’ problem. Very interesting as the answer is not intuitive (as you pointed out). Thinking in the limit, it would be more intuitive to think that there is a point where the initial radius is so large that a 1m increase would have no discernable impact to the radius, but as you pointed out, the math does not lie. A little off the subject, but I have often thought before that math is the one great truth in the world. If a mathematical model can be created for something or some process, and holds up under the scrutiny of testing/observation, it represents truth. Few things in life seem to be so deterministic.

3. Dimitri Says:

Hi Todd,

Good to hear from you again. I just shot you an email.

Quite the opposite happens in the limit. As the globe increases in radius, the clearance get larger and much faster. This one simple problem, for some reason, challenges intuitions more than a lot of other problems. A good case study for cognitive neuroscientists more than for mathematicians: the naive intuition is the real subject of study here.

I have the following view of truth in mathematics. Mathematics does not contain any truths. It’s a language of consistent manipulations that can help reformulate one expression into another under well defined rules (eg linearity). For example, the Fourier transform takes a time series and re-expresses it in the frequency space. The amazing thing is that many natural phenomena obey many simple rules and submit themselves to mathematical manipulation. Math does not add any truth about the world, only tautology. However, by allowing us to reformulate observations into other equivalent forms, mathematics helps us detect patterns, gain insight, and make new predictions, which can be tested empirically and lead to truth.

Perhaps the most eloquent and vivid recapitulation of the relationship between truth and mathematics was given by Murray Gell-Mann at the TED conference: http://www.ted.com/index.php/talks/murray_gell_mann_on_beauty_and_truth_in_physics.html .