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From: mina_world on 22 Feb 2007 11:13
Hello sir~

Like many other performances, this one begins with a deck of cards.

Take an ordinary deck of 52 cards, lying on a table, all four sides of the
deck squared away.

Now, with a finger slide the topmost card forward without moving any of the
others.
How far can you slide it before it tips and falls?
Or, to put it another way, how far can you make it overhang the rest of the
deck?

The answer, of course, is half a card length.
If you push it so that more than half the card overhangs, it falls.
The tipping point is at the center of gravity of the card, which is halfway
along it.

Now let's go a little further. With that top card pushed out half its
length-that is, to maximum overhang-over the second one, push that second
card with your finger.
How much combined overhang can you get from these top two cards?

The trick is to think of these top two cards as a single unit. Where is the
center of gravity of this unit? Well, it's halfway along the unit, which is
altogether one and a half cards long;
so it's three-quarters of a card length from the leading edge of the top
card.

The combined overhang is, therefore,three-quarters of a card length.
Notice that the top card still overhangs the second one by half a card
length.
You moved the top two cards as a unit.

If you now start pushing the third card to see how much you can increase the
overhang,
you find you can push it just one-sixth of a card length. (***)
Again, the trick is to see the top three cards as a single unit.
The center of gravity is one-sixth of a card length back from the leading
edge
of the third card.
----------------------------------------------------------------------
My question is....
I can't understand (***) part.
Why can I push it just 1/6 of a card length ?
Can you explain it ?


From: Robert Israel on 22 Feb 2007 13:55
"mina_world" <mina_world(a)hanmail.net> writes:

> Hello sir~
>
> Like many other performances, this one begins with a deck of cards.
>
> Take an ordinary deck of 52 cards, lying on a table, all four sides of the
> deck squared away.
>
> Now, with a finger slide the topmost card forward without moving any of the
>
> others.
> How far can you slide it before it tips and falls?
> Or, to put it another way, how far can you make it overhang the rest of the
>
> deck?
>
> The answer, of course, is half a card length.
> If you push it so that more than half the card overhangs, it falls.
> The tipping point is at the center of gravity of the card, which is halfway
>
> along it.
>
> Now let's go a little further. With that top card pushed out half its
> length-that is, to maximum overhang-over the second one, push that second
> card with your finger.
> How much combined overhang can you get from these top two cards?
>
> The trick is to think of these top two cards as a single unit. Where is the
>
> center of gravity of this unit? Well, it's halfway along the unit, which is
>
> altogether one and a half cards long;
> so it's three-quarters of a card length from the leading edge of the top
> card.
>
> The combined overhang is, therefore,three-quarters of a card length.
> Notice that the top card still overhangs the second one by half a card
> length.
> You moved the top two cards as a unit.
>
> If you now start pushing the third card to see how much you can increase
> the
> overhang,
> you find you can push it just one-sixth of a card length. (***)
> Again, the trick is to see the top three cards as a single unit.
> The center of gravity is one-sixth of a card length back from the leading
> edge
> of the third card.
> ----------------------------------------------------------------------
> My question is....
> I can't understand (***) part.
> Why can I push it just 1/6 of a card length ?
> Can you explain it ?

If you take the card length as 1 and the leading edge of the third card
as x = 0, the centres of gravity of the three cards are at x = -1/2,
x = -1/4 and x = +1/4. The centre of gravity of the three together
is at the average of these, namely x = (-1/2 - 1/4 + 1/4)/3 = -1/6.
You can push the three-card unit until that centre of gravity is over
the leading edge of the fourth card.
--
Robert Israel israel(a)math.MyUniversitysInitials.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
From: mina_world on 22 Feb 2007 19:56

"Robert Israel" <israel(a)math.MyUniversitysInitials.ca> wrote in message
news:rbisrael.20070222185013$7ab0(a)news.ks.uiuc.edu...
> "mina_world" <mina_world(a)hanmail.net> writes:
>
>> Hello sir~
>>
>> Like many other performances, this one begins with a deck of cards.
>>
>> Take an ordinary deck of 52 cards, lying on a table, all four sides of
>> the
>> deck squared away.
>>
>> Now, with a finger slide the topmost card forward without moving any of
>> the
>>
>> others.
>> How far can you slide it before it tips and falls?
>> Or, to put it another way, how far can you make it overhang the rest of
>> the
>>
>> deck?
>>
>> The answer, of course, is half a card length.
>> If you push it so that more than half the card overhangs, it falls.
>> The tipping point is at the center of gravity of the card, which is
>> halfway
>>
>> along it.
>>
>> Now let's go a little further. With that top card pushed out half its
>> length-that is, to maximum overhang-over the second one, push that second
>> card with your finger.
>> How much combined overhang can you get from these top two cards?
>>
>> The trick is to think of these top two cards as a single unit. Where is
>> the
>>
>> center of gravity of this unit? Well, it's halfway along the unit, which
>> is
>>
>> altogether one and a half cards long;
>> so it's three-quarters of a card length from the leading edge of the top
>> card.
>>
>> The combined overhang is, therefore,three-quarters of a card length.
>> Notice that the top card still overhangs the second one by half a card
>> length.
>> You moved the top two cards as a unit.
>>
>> If you now start pushing the third card to see how much you can increase
>> the
>> overhang,
>> you find you can push it just one-sixth of a card length. (***)
>> Again, the trick is to see the top three cards as a single unit.
>> The center of gravity is one-sixth of a card length back from the leading
>> edge
>> of the third card.
>> ----------------------------------------------------------------------
>> My question is....
>> I can't understand (***) part.
>> Why can I push it just 1/6 of a card length ?
>> Can you explain it ?
>
> If you take the card length as 1 and the leading edge of the third card
> as x = 0, the centres of gravity of the three cards are at x = -1/2,
> x = -1/4 and x = +1/4. The centre of gravity of the three together
> is at the average of these, namely x = (-1/2 - 1/4 + 1/4)/3 = -1/6.
> You can push the three-card unit until that centre of gravity is over
> the leading edge of the fourth card.

Yes, I see.
Wise explanation.
Thank you very much.


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