Friday, April 28, 2006

amazing stuff -1


Okay folks.. something different... a "puzzle " ....

Attached below is the email my cousin sent me, which I happened to see at 1 am in the nite !!
I did reply to him , but before I post my reply, I leave it you all to post your thots in the comments section !! Njoy.. its a friday folks

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I have a puzzle 4 u.. actually its no puzzle, just math.. and i have the answer too (i think)... what i need is a logical (not mathamatical if possible) explanation for it...

q. consider a ball as big as earth. a belt around it just touching the surface all around and having the highest possiable length (same circumfrence for belt and ball). now increase the length of belt by 1 meter. so now the belt is slightly bigger...! and if the gap between the surface of ball and the belt is equall all the way around....

wat will be the value ?

A. (as i solved)

let circum or earth be C1 and radius be R1
let new circum of belt be (C1+1) and its new radius R2
we have to find what is (R2 - R1)

C1 = 2 * 3.14 * R1
(C1+1) = 2 * 3.14 * R2

so (2 * 3.14 * R1) + 1 = 2 * 3.14 * R2

so R2 = (2 * 3.14 * R1) + 1
--------------------------
2 * 3.14

ie R2 = R1 + 0.159

so it should be around 16 cm...

and it will be 16 cm weather the size of ball is of the size of a football, or that of earth itself, if the increment is 1 meter....!!!!!!!!! (as per the equations)


BUT WHY ????
i cant believe this,..... we are just adding 1 meter to 36000000 meters and we get a increment of 15~16 cm !!!! and a cat can go under the belt easly ... i thought it will be only some micro-meters...!!!!!!

just explain me this physically..... i dont want to believe maths.....!(eventhough it is de easy way)

15 comments:

Surya S said...

The final conclusion says ///we are just adding 1 meter to 36000000 meters and we get a increment of 15~16 cm !!!! and a cat can go under the belt easly ... i thought it will be only some micro-meters...///

The statement intelligently tries to create a big and small opinions in the mind by using different parameters.

While talking about 1m increase, it is compares to the whole radius of earth, thus creating a feeling that 1m is very very small.

But when it comes to the final answer, it is now compared not with the same radius of earth, but with the size of a cat, thus creating the feel that it is indeed very big.

1m is very small in comparison to 3600000, but 16cm is still smaller compared to 360000. 16cm is good enough for a cat to sneak in, but then 1m is more than enough for 5 cats to go vertically.

The flaw is in the inconsistent comparisons.

Anonymous said...

Okay, few random comments

1. Has it even been surprising to you that no matter how big or small a circle is the ratio between its circumferance and diameter is always the same. As of now, it is not yet obvious to me, though a little bit of calculus makes it look plausible. In fact this is the defination of Pi which aryabhatta first calculated.
(http://www.answers.com/topic/aryabhata)

"Aryabhata also gave close approximation for Pi. In the Aryabhatiya, he wrote: "Add four to one hundred, multiply by eight and then add sixty-two thousand. The result is approximately the circumference of a circle of diameter twenty thousand. By this rule the relation of the circumference to diameter is given." In other words, π ≈ 62832/20000 = 3.1416, correct to four rounded-off decimal places."

2. (a bit of intellectual debate just for fun!)
what do you mean by you want a logical, but non mathematical argument. To me both are the same thing.

kaushik

Anonymous said...

Eat Math... Sleep Math.. and DRINK !

drisyadrisya said...

nice perspectives, surya and kaushik

Okay, attached below was my reply.. as soon as i saw the mail

excuse the bit of mallu in the beginning

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Raathri kedakkan pokunnente mumponnu mail checkiyamenu vechappo , manushyane swasthamayonnoranganum sammathikkilla !! iniyippum rathri muzhuvanum panthum kayarum swapnam kanendi varum !! :-)

Anyways, too sleepy to think much about this.. But I guess you are right, the gap will be
(increase in circumference)/2*pi always - in this case about 16 cm

I think we are facing difficulty imagining it because you aren't taking the relative distances into account.

Okay, start from small. Take a ball of radius 16 cm. Its circum is about 1m. Now add 1 m to it, Circumference becomes 2m and radius becomes 32 cm. gap is 16 cm . Circum is made double, so radius also doubled . Gap is 100% of radius

Now take a ball with 32 cm rad. Circum = 2m. Add 1m to it, so it becomes 3 m . The new radius is about 48 cm. "Gap" is still 16 cm. But this time the gap is only 50% of radius

Keep going up like this, you will see that the gap remains 16 cm, but the percentage compared to the radius keeps getting shorter as the ball gets bigger ..

Same case with earths radius, gap remains the same in quantty, but a very very small percentage of earth's radius .

I think where the mathematics ends, the issue of perception begins. A ball of 32 cm and a gap of 16 cm is someting you and I can visualise. And we feel that the gap is "BIG" because its 50% of the radius. But when it comes to earth's radius, its just too big for us to imagine a tape around it. And then when you increase it by 1 m. yes, the gap does increase by 15 cm, but we feel the gap is "SMALL" because its supposed to be a very very small percentage of earths radius.. the fact is, the gap remains the same, the reference object gets bigger and bigger , from a 32 cm ball to a huuuuuuuuuuuuuuuuuuuuuuuuge earth

So mentally we believe it should be a "small" gap.. the missing point is small compared to what ?
Call it relativity .. ha ha

******************************************************************

Believe me as I type it, I am facing the same problem, of mentally trying to picture it. And I am still not convinced what I types up is right.. am afraid, I could be blabbering in my sleep.. and may be you put a trap in it somewhere .. If you haven't it simply goes to show that this was an interesting thing to think abt and that mind is such a powerful thing .. I am reminded of Gita 6.5

atmaiva hy atmano bandhur
atmaiva ripur atmanah

good nite

Kalyan said...

hey arun, that was a good explanation. Surya's points are also very nice.

ranjith said...

Arun, Interesting puzzle!
Since R2-R1=(1/2*pi), probably this is a nice way of teaching kids (or anyone!)
the magic of pi !!

Since the puzzle is related to the perception of "size"
and "relative size", another interesting problem came to my
mind and that is related how "fluctuations" and
"relative fluctuations" scale in physics (this has
nothing to do with the puzzle here, but this is where i first
learned the idea that "size" and "relative size"
will scale very differently! And this idea is very important
in statistical mechanics!)

Take N random numbers between -1 and 1. Now find the
sum of these N numbers.

As N goes to infinity, one would think that for every positive
random number, there will be an almost equal negative random
number also, and hence the sum will be always very near to zero!

But this is not true!! It turns out that as N increases, the
probability that the sum is zero, decreases!!
If one calculates how much the sum will deviate from zero,
it turns out that the "deviation from zero" (also called
"fluctuation" in some physics context) increases directly
proportional to square-root of N (implies, deviation can be as big as
infinity!)

In physics, this sum problem is equivalent to a
"random walker" problem. It turns out that the distribution
of the sum is a Gaussian and has a width proportional to sqrt(N).
(So is the distribution of the end-to-end distance of a random
walker or the end-to-end distance of a polymer (with some assumptions!)).

But the relative fluctuation (defined as fluctuation/N), will go
to zero as N--->infinity. (or mean, sum/N, will tend to zero as N goes to infinty; but
not the sum itself!)

In this puzzle, though R2-R1 remains a constant,
the relative size, (R2-R1)/R1 goes to zero as R1 tends to infinity.

just some thoughts that came to my mind! may not be relevant
here!

ranjith

radha said...

you are talking about the mind and in the shlok you have atma. how r they same ?

drisyadrisya said...

Ranjith

Thanks for the post.. it does spin my head. I can imagine the part where sum/N goes to zero for N --> infinity . I can also imagine that there is a finite probability that the sum is non zero ..

But yeah, the part that the fluctutaion (or the sum) increases as sqrt N... beyond simple imagination I guess..

Interesting prob..

drisyadrisya said...

Radha

Interesting question... Methinks that the atma used here is not the Atma - the soul. (Neither is it Schopenhauer's dog, haha) ..

the atma is used to mean mind. I am not sure why manah is not used.. but looks like all translations mention atma=mind . See, even the next shloka

http://www.sulekha.com/groups/postdisplay.aspx?cid=1122&forumid=917899

bandhur atmatmanas tasya
yenatmaivatmana jitah
anatmanas tu satrutve
vartetatmaiva satru-vat

(Bhagavad Gita 6:6)


bandhuh--friend; atma--mind; atmanah--of the living entity; tasya--of him; yena--by whom; atma--mind; eva--certainly; atmana--by the living entity; jitah--conquered; anatmanah--of one who has failed to control the mind; tu--but; satrutve--because of enmity; varteta--remains; atma eva--the very mind; satru-vat--as an enemy.


Mind under control,
Is one’s best friend.
If not so,
It’s one’s greatest foe.

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Will post if I come across a specific reason !.. or anyone else comes across, will hopefully post here :-)

Rt said...

www.asitis.com/6/5.html

The word atma denotes body, mind and soul--depending upon different circumstances. In the yoga system, the mind and the conditioned soul are especially important. Since the mind is the central point of yoga practice, atma refers here to the mind.

Rt said...

You may find this exciting

www.gosai.com/chaitanya/saranagati/html/vishnu_mjs/math/

drisyadrisya said...

Thanks "rt" for both the posts. Took a quick look at the maths site and it looks interesting

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