Friday, August 3, 2007

Can you answer this one?

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Suppose we take a long ribbon and wrap it around the earth, around the equator, so tight that not even a piece of paper can go between the ribbon and the earth. Now we increase the length of the ribbon by just 1 metre so that it becomes slack. This makes it possible to raise the ribbon from the surface of the earth because we have loosened it by introducing another 1 metre to it's length. You have to tell me by how much the ribbon can be raised from the earth's surface. Remember, the ribbon is raised not at one point on the earth but all around the earth, equally. The following diagram will make things clear.



By how many millimeter/cm/meter the ribbon can be raised? You will be surprised if I tell you the answer. The answer is 16 cm. Sounds unbelievable, isn't it? The circumference of the earth is approximately 40,075,160 metres, and so is the length of the ribbon. How can by increasing the ribbon length by just a tiny 1m enable it to be raised by 16cm all around the earth?

Here is the math:

Radius of earth = r (meters)
Circumference of earth = 2πr (meters)
Initial Length of ribbon = 2πr (meters)
New length of ribbon = 2πr + 1 (meters)

Height by which the ribbon can be raised = (Radius of the circle of ribbon) - (Radius of earth)

H = (2πr + 1)/2π - r (meters)
= r + 1/2π - r (meters)
= 1/2π (meters)
= .159 (meters)
= 16 cm

Notice that H doesn't depend on the radius of the earth, which means that whether you wrap the ribbon around a football, or around earth, or around the sun, if you increase the lenght by 1m it can always be raised by 16 cm!!

Amazing, isn't it?

25 comments:

  1. Lucky you don't work for NASA.

    0.0159m != 16cm

    0.0159m = 16mm or 1.6cm.

    It's still more than i'd have thought, but perhaps the rest of your math is incorrect too??

    Good luck.

    ReplyDelete
  2. Oops!! I made a mistake.
    It's not 0.0159 metres but 0.159 metres.

    I inserted an extra zero by mistake. The answer is still 16cm.

    The math is correct, I can assure you that. Why don't you do it yourself?

    ReplyDelete
  3. "approximately 40,07516 metre"

    Now what kind of number is that?
    It's ~40,075,160 meters anyway.

    ReplyDelete
  4. Dammit!! Another typo!!
    It's 40,075,160 meters.
    Thanks for correcting.

    ReplyDelete
  5. You're surprised by this?

    Circumference equals 2pi times radius. If you add (anything) to the radius of a circle, it's circumference increases by 2pi(anything). If you add (anything) to the circumference of a circle, it's radius increases by 1/2pi(anything).

    Incidentally, if you think about it proportionally, it makes a lot of sense. 1 meter doesn't seem like much compared to the circumference of the earth, and .159 meters isn't much compared to 6,378,100 meters (the radius of the earth).

    ReplyDelete
  6. The math is solid but the layout could be better (for me anyway). I prefer :

    let d be the diameter
    let r be the initial radius
    let R be the final radius

    So
    2*pi*r = d
    2*pi*R = d + 1

    Therefore

    2*pi*R = 2*pi*r + 1
    2*pi*R - 2*pi*r = 1
    2*pi(R - r) = 1
    R - r = 1/(2*pi)

    Since H = R - r
    then H = 1/(2*pi)

    ReplyDelete
  7. hey "anonymous," your a idiot. 2*pi*r is the CIRCUMFERENCE not the diameter.

    ReplyDelete
  8. This is interesting, and correct -- though not immediately intuitive. For the calculus-inclined, it is the same thing as saying that a change in the circumference of a circle has a linear relationship with a change in its radius. Or:

    R = C / (2*pi)
    dR/dC = d/dC ( C / 2*pi )
    = (1 / (2*pi)) * d/dC(C)
    = 1 / 2*pi

    So the derivative of the radius with respect to the circumference is always 1 / 2*pi. :)

    ReplyDelete
  9. This seems to have no purpose.

    ReplyDelete
  10. OK senator - a fair comment - blame a seniors moment. Make it

    let c be the circumference
    let r be the initial radius
    let R be the final radius

    So
    2*pi*r = c
    2*pi*R = c + 1

    Therefore

    2*pi*R = 2*pi*r + 1
    2*pi*R - 2*pi*r = 1
    2*pi(R - r) = 1
    R - r = 1/(2*pi)

    Since H = R - r
    then H = 1/(2*pi)

    Happy ?

    ReplyDelete
  11. "your a idiot"

    Ohhh the irony in that statement...not just once but twice...

    ReplyDelete
  12. Whichever way you do it it always ends in 1/2pi, provided you do it correctly. :)

    ReplyDelete
  13. Sort of counterintuitive at first but it makes sense after analyzing the properties of a circle.

    ReplyDelete
  14. your - possessive, as in this is your life

    You're - contraction of you are, as in you are an idiot

    ReplyDelete
  15. OMG!
    The math is simple and it works, but I trust the voices in my head more, and they say this is crazy talk! How is this possible?! No matter how many times I do this in my head visually, it doesn't work.

    ReplyDelete
  16. possibly a simpler way of looking at this.

    If your circumference changes by 1 unit then since the radius is related to the circumference in the following manner

    C=2pi*r => r=C/2pi

    then the change in radius shall be r=1/2pi (for C= 1)

    ReplyDelete
  17. there is no way this is right. i can't prove it yet but stay tuned, it will be disproven

    ReplyDelete
  18. Yep this is right. And I agree that it is counter-intuitive. But radius increase linearly with circumference, so there. hey ho.

    ReplyDelete
  19. Easy thought experiment:

    Convert the "globe" to a 2 dimensional circle. Now squish that circle into a perfect square. Now chop the 1 meter extension ribbon up into 4 equal pieces. Add each of the 4 pieces to each of the sides of the ribbon wrapped around the square. (enlightenment happens here)
    The effect would be even more pronounced for a circle. Think of "pi" for a square as "4" whereas pi for a circle is 3.1415.......

    You're welcome.

    ReplyDelete
  20. Hey "senator_mendoza," go look up anonymous in the dictionary, it isn't his name. He just said "d" instead of "c". You said "your a idiot" when you should have said "you're an idiot". Before you criticize somebody else's comments, look up third grade level words and grammar.

    ReplyDelete
  21. Should I bother?????? I might aswell. He knows it's not his name; hence the " which you used yourself. Duh.
    I wish the calcs were done properly the first time round. BTW Aaron not aaron lol
    Yes, I know I havn't used perfect grammar. I'm not being graded.

    ReplyDelete
  22. Thanks to hatOFtin for that very simple explanation of how this is possible. It makes the math seem rather irrelevant.

    ReplyDelete

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