Let me start with an acknowledgement and a question.
First, the acknowledgement: This question came from the book, The Man Who Knew Infinity: A Life of the Genius Ramanujan, by Robert Kanigel. The book, of course, is about Srinivasa Ramanujan and he is the person who developed the basis for this question. So, the first part of my article may seem familiar to readers of the book (and if you already know the answer, don’t shout it out!) – but I have added wrinkles as you will see from my mention of a nanoparticle which Ramanujan probably didn’t deal with very often 100 years ago, and it is enjoyable nevertheless. The second part comes from me and that is: What makes this question so interesting?
I know the IIT mind, and several of you are already checking with Google if I’m right
Now let me set the background for the question: The circumference of the Earth at the equator is approximately 24,901.55 miles or 40,075.16 kilometers. I know the IIT mind, and several of you are already checking with Google if I’m right . Don’t worry about the precision of these numbers, because I’m about to make some simplifying assumptions anyway: Assume that the Earth is a perfect sphere (we know that it isn’t since it is more an ellipsoid shape with the circumference at equator being larger than the one touching the poles), and that we can approximate the circumference to 25,000 miles, which when converted into kilometers is 40,234 km.
So, if a tight metal belt is strapped around the circumference of the Earth at the equator, the metal belt is exactly 25,000 miles or 40,234 km long.
Did you think I made an error in my copy-paste action, and used the same answers by mistake?
Now, as I often do after a heavy meal, loosen the metal belt one “notch” of 50 ft. or 15.24 m. The tight belt was affixed exactly at the circumference of the equator, so now assume that the loosened belt stands uniformly above the surface of the Earth, as shown in the figure, obviously exaggerated for clarity.
Note that 50 ft. or 15.24 m is about the length of a large room, so it is very small increase in the length of a belt of 25,000 miles or 40,234 km. The loosened belt actually represents an increase of 0.0000004 – that’s six zeros after the decimal point (or, expressed as a percentage, 0.00004%) — so, that’s what I meant by the gap in the figure being obviously exaggerated for clarity.
Here’s Question 1: What will just fit in the gap that is created?
- A nanoparticle?
- A sheet of paper?
- A mouse?
- A cat?
- A crawling baby?
- Something smaller than (1)?
- Something larger than (6)?
Okay, I know all the readers of Fundamatics are really smart and can compute the answer, but don’t cheat – here, we are looking for your immediate first impression. Write it down.
Here’s Question 2: I now repeat this scenario on Jupiter which has 11.2 times Earth’s equatorial diameter, and yet keep the size of the loosened notch the same (50 ft. or 15.24 m). The picture shows Earth in the left front and Jupiter in the left rear row. In this case, the belt circumference increases by 0.00000003 (seven zeros after the decimal) or 0.000003%.
What will just fit in the gap that is created?
- A nanoparticle?
- A sheet of paper?
- A mouse?
- A cat?
- A crawling baby?
- Something smaller than A?
- Something larger than F?
Write down your answer.
Answer to Question 1: Even though the belt is loosened by only 50 ft. or 15.24 m, which is an increase of 0.0000004, the gap created is large enough for the tallest person you probably have ever seen to walk through. The gap created is 7.96 ft. or 2.43 m tall!
Answer to Question 2: Even though the belt is loosened by only 50 ft. or 15.24 m, which is an increase of 0.00000003, the gap created is large enough for the tallest person you probably have ever seen to walk through. The gap created is 7.96 ft. or 2.43 m tall!
Did you think I made an error in my copy-paste action, and used the same answers by mistake?
Here is the solution to the question:
Let C = Circumference
And r = Radius
Situation 1 is the tight belt: C1 = 2 π r1 So, r1 = C1 / 2 π
Situation 2 is the very slightly loosened belt: C2 = 2 π r2 So, r2 = C2 / 2 π
Therefore: r2 – r1 = (C2 – C1) / 2 π
The circumference changes by 15.24 m, so:
r2 – r1 = 15 / (2 * 3.14)
= 15 / 6.28
= 2.43 m (or 7.96 ft.)
Note: This answer is independent of the circumference, so the solution for Jupiter is the same!
Now, it gets more interesting.
Why does this seem so counterintuitive?
The belt size (circumference) is increased by such an incredibly small proportion: 0.0000004 (0.00004%), and even adding an additional zero for Jupiter, that we cannot conceive how that could make any difference at all. That’s why it’s so puzzling.
A different perspective broadens our understanding and lessens the limiting effect of our background
Although we increased the belt size (circumference) by such an incredibly small proportion: 0.0000004 (0.00004%), we cannot conceive how that would lead to such a large gap of 2.43 m or almost 8 feet. That’s why it’s so counterintuitive.
The solution to the quandary lies in our limited perspective.
Actually, while we are focused on the gap and the size of the object that can fit through the gap, we might have failed to realize that the relevant item is the radius of the loosened belt. The radius increased by the same incredibly small proportion: 0.0000004 (0.00004%)! It is a ~15 m increase on a radius of ~40+ million m.
It’s just that we humans are so used to standing on the surface of the Earth that everything is perceived relative to our ground level – we don’t think of things relative to the center of the Earth (which is the “anchor” for the mathematical solution). We just think of things relative to where we stand.
And, so it is for so many other things – we just think of things relative to where we stand. Such are our limitations – whether we are talking of our state, language, country, gender, organization, or background. We are not the anchor for everything. A different perspective broadens our understanding and lessens the limiting effect of our background.
