Saturday, September 10, 2011

Stacking One Trillion Dollars

It is fun to watch Neil deGrasse Tyson. I think he does a very nice job even when speaking about politics. Ok, check out this video from Real Time with Bill Maher:

Not that I don’t trust him, but I guess I want to check. Would one trillion dollar (1 dollar bills I assume) stack to the moon and back four times?

I don’t usually carry cash in my wallet, but when I do I measure it. There were 5 bills. I measured the thickness of just one, then two and so forth. After all five were stacked, I started folding them over. Here is a picture.

Img 0512.jpg

Yes, this would be difficult to measure with a ruler. The device above is a micrometer. Ok, what about the data. Here is a plot of the measured thickness (in mm) vs the number of bills. Oh, I am assuming that a 5 dollar bill is the same thickness as a 1 dollar bill.

Here is a plot of the thickness vs. the number of bills.


I included a linear regression line with the data. It has a slope of 0.1 mm/bill. So, I will go with that value.

First, what is one trillion of anything? Sadly, not everyone agrees. In the USA, one trillion is 1,000 billion or 1012. In some other countries, one trillion means 1018. (see Wikipedia’s page on short scale vs. long scale)

So, if I stack 1012 bills, how high would it be? First, let me assume that the bills don’t get compressed. Why am I assuming that? I don’t know. The height of this stack would be:

La te xi t 1 2

The distance from the Earth to the Moon is about 4 x 108 meters. Okay, now there is a problem. According to my calculations, the stack of one trillion dollar bills would go one fourth of the way to the moon. Neil said it would go there and back four times (which would be 32 x 108 meters). His estimate of the height of the stack is 32 times too large (or mine is too small).

Let me try one other thing. If one trillion dollars goes to the moon and back four times, how thick would it have to be?

La te xi t 1 3

3 mm thick bills would be rather awkward. So, I think Neil messed up. Its ok. It happens to us all. Just don’t make a habit of it (although he did get the explanation for the tides wrong also). Anyway, the whole point would have been ruined. Could you imagine Neil saying this:

“Oh, and I would just like to point out something about a trillion. Did you know that if you stacked one trillion dollar bills, it would make it one fourth of the way to the moon?”

Oh well, what other things could we do with one trillion dollars?

Suppose you could stack the bills perfectly. As the stack gets higher and higher, it would be more likely to fall over from a slight nudge. Let me start with a block.

Untitled 1

For each stack, the red dot represents the center of mass. If the stack is tilted such that the center of mass goes over the edge of the base, the stack falls over. Yes, I am assuming the bills stick together. But you can see that the taller the stack gets, the smaller the “tilt” angle would be for it to fall over.

If the base of the bill has a width w and length t. For tipping towards the thinner side of the bill, we have a right triangle.

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Solving for ?:

La te xi t 1 4

Now, suppose the width of the dollar is 6.6 cm. Would would a plot of this “tipping angle” as a function of height look like for a stack from 1 meter tall to 10 meters tall.


So a 10 meter tall stack of bills would only need to be tilted 0.37° for it to be at the tipping point. Here is a plot for heights from 100 meters to 10,000 meter stacks. I had to make the vertical scale a log-plot.

Tipp 3.png

Ok, what if I take this up to 106 meters tall? That would be a tipping angle of 3.8 x 10-6°. And a trillion dollar stack (assuming it was all in a constant gravitational field – which it wouldn’t be) it would have a tipping angle of 3.8 x 10-8°.  Just for a comparison, Alpha Centauri A (a star) has an angular diameter of 1.9 x 10-6 °.

Suppose you could stack the bills and they wouldn’t fall over (and again assuming constant gravitational field). Would the bills at the bottom of the stack be able to maintain this weight? Ok, so I already set up something like this for the compressive strength of rock (when talking about the height of pyramids) Essentially, the paper can only take so much pressure before bad stuff happens. The point where bad stuff happens is called the compressive strength. I don’t know about paper, but wood has a a compressive strength from 3 to 27 MPa. For this case, I am just going to randomly choose 20 MPa as the compressive strength of a bill.

What is the pressure at the bottom of the stack? Well, that would be the weight of the stack over the area of a bill. Suppose a bill has area of 6.6 cm by 15.6 cm. This means the pressure at the bottom would be:

La te xi t 1 5

Where ? is the density of the paper bill and h is the height of the stack. So what is the density of a dollar bill? Well, I can get the volume (length 6.6 cm, width 15.6 cm, height 0.01 cm) and then I just need the volume. What about the mass? I put seven bills on a balance and found a mass of 6.910 grams. This would give a mass per bill of about 0.987 grams. So, the density of the paper bills is about 958 kg/m3.

Then what is the pressure at the bottom of my trillion dollar stack?

La te xi t 1 9

Really, the pressure would be smaller than this because the gravitational field gets weaker as the stack gets higher. I don’t think it matters. This pressure is way over the 20 MPa for the compressive strength.

If stacking won’t work, I am going to make a trillion dollar asteroid. I know the density of a dollar, so I know the mass of 1 trillion dollars. Maybe I should start with a picture.

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Why would you make a big ball of cash? Why wouldn’t you? You could call it a cashteroid. Ok, first the mass. If each bill is 6.91 x 10-3 kg, then 1012 of them would have mass of 6.91 x 109 kg. Assuming a constant density, this would give a volume of:

La te xi t 1 10

If this is a spherical cashteroid, I can then find the radius.

La te xi t 1 11

120 meters may seem small, but that is a ball 240 meters (780 feet) across. Here is a picture of the big ball of cash next to the International Space Station (approximately to scale):

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Maybe this is what Neil deGrasse Tyson should have said: “A trillion dollar bills would make a giant sphere 240 meters across that would orbit the earth and be brighter than the space station”.

Rhett Allain is an Associate Professor of Physics at Southeastern Louisiana University. He enjoys teaching and talking about physics. Sometimes he takes things apart and can't put them back together.
Follow @rjallain on Twitter.

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