May

11

That’s impossible!

By Antonio Cangiano. Filed under Essential Math, Suggested reading 


Imagine a rope that was just long enough to wrap tightly around the equator of a perfectly spherical earth. Now imagine that the length of that rope is increased by one meter and again wrapped around the earth, supported in a regular way, so as to form an annulus. Doing things in this way will form a certain gap between the earth and the extended rope. Now imagine that you repeat the process with a golf ball. How do the two gaps compare?

Most people who understand the problem correctly, will immediately assume that adding a single meter to the rope surrounding the huge spherical earth won’t create much of a gap, while adding a single meter to the rope around the golf ball, will create a large gap. In reality, the two gaps are identical. It’s counterintuitive, but it can be proven easily with elementary mathematics.

We know that:

C = 2 \pi r
C + 1 = 2 \pi R

Where C is the length of the rope around a given spherical object, C+1 is obviously the length of the longer (by one meter) rope, r is the radius of the object (e.g. the golf ball) and R the radius of the annulus. We can express the two equations above as such:

\displaystyle r = \frac{C}{2\pi}
\displaystyle R = \frac{C+1}{2\pi}

The width of the gap between the longer rope and the object that’s taken into consideration is R-r.

Annulus’ gap

Therefore:

\displaystyle R - r = \frac{C+1}{2\pi} - \frac{C}{2\pi} = \frac{1}{2\pi} \approx 0.159

The gap, as you can see, is constant at about 16 cm, and it doesn’t depend in any way on the size of the radius (r) of the object at hand. That means that the gap between the extended rope and the golf ball is the same as in the case of the spherical earth.

impossible.jpgQuite a surprising result, isn’t it? This exact example opens up the book Impossible?: Surprising Solutions to Counterintuitive Conundrums which I received as a media copy in the mail. So far it’s been a very enjoyable and easy read, chocked full of surprising paradoxes and results which common sense would have you deem (practically) impossible or counterintuitive. To make things even nicer, the math involved is not overly advanced, and anyone who grasped high school level math, should have no problem following this engaging book.


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Comments

4 Responses to “That’s impossible!”

  1. Enginerd on May 11th, 2008 6:46 pm

    The width of the gap might be constant, but the area wouldn’t be. It would be
    pi*( (r+1)^2 - r^2) = pi*(2*r + 1), which clearly depends on r. Simply asking how the two gaps compare is a bit vague.

    I think the real issue is that when you picture the problem, the gaps look different. A 16 cm wide gap doesn’t seem like much next to the radius of the Earth, but it seems like a lot next to the radius of a golf ball.

  2. Antonio Cangiano on May 11th, 2008 7:02 pm

    Yes Enginerd, the areas are obviously very different. The pardox is limited to the width of the gaps. :)

  3. bob saget on May 12th, 2008 2:55 am

    wait. this does sound impossible!

    So, if the world had a 32 cm bigger diameter, you’d only need one more meter of rope to wrap it…

    *thinks* actually that kind makes sense… because adding 1 meter to the circumference would increase the size by that ammount.

    The thinking relies on making the 1 meter sound like the small measurement adjustment causing a huge 16cm change.

  4. links for 2008-05-13 « Tom Altman’s Wedia Conversation on May 13th, 2008 7:33 am

    [...] That’s impossible! | Math-Blog (tags: math paradox) [...]

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