What Should You Use to Build a Cashteroid?

This is a trillion dollar cahsteroid next to the ISS.

I will go ahead and define it:

cashteroid: A giant pile of money that is so large it could be treated like an asteroid. The best cashteroids have a gravitational field that is large enough so that you could walk on the surface.

I originally talked about cashteroids when I looked at the amount of dollar bills you would need to stack to the moon. In our recent Google Hangout (Uncertain Dots Episode 23), Chad Orzel asked the question – what would be the best coin to make a cashteroid? I can’t remember why we were talking about cashteroids, but I’m sure it had something to do with a big pile of money. Chad also guesses that dime would be the worst thing to make a cashteroid out of since they are pretty small for their value.

Currency Data

Ok, let’s get right to it. I can look up the dimensions of different coins on Wikipedia and I get the following data. I’ve included both the older (almost pure copper) pennies as well as the new zinc pennies since you can easily find both. I did not include any silver quarters since those aren’t so easy to find (I know from trying).

• New penny: m = 2.5 g, d = 19.05 mm, t = 1.52 mm


• Old penny: m = 3.11 g, d = 19.05 mm, t = 1.52 mm


• Nickel: m = 5 g, d = 21.2 mm, t = 1.95 mm


• Dime: m = 2.268 g, d = 17.9 mm, t = 1.35 mm


• Quarter: m = 5.67 g, d = 24.67 mm, t = 1.75 mm


• Half dollar: m = 11.34 g, d = 30.61 mm, t = 2.15 mm


• Dollar coin: m = 8.1 g, d = 26.5 mm, t = 2


• Dollar bill: m = 1 g, length = 155.96 mm, t = 0.11 mm

From this, I can calculate the volume of each coin (and the dollar bill) as well as the price per mass and price per volume. Oh, it seems the thickness of the dollar bill isn’t so obvious. For some reason it’s not stated on the US One Dollar Bill Wikipedia page. However, in a previous post I experimentally determined a value of .11 mm.

There’s something else that is quite interesting. When calculating the volumes, I found that the volume of a dollar coin and a dollar bill are almost exactly the same at around 1.1 x 10^-6 m³. I just thought that was weird. Or maybe it’s not weird and the US Department of Treasury planned it this way so that it wouldn’t matter if you built a cashteroid out of dollar coins or dollar bills (at least in terms of volume).

Now for a comparison. This is what you have been waiting for (maybe). Here is a graph showing the price per kilogram for the different monetary options.

From this you can see the old penny is the cheapest currency per mass with a cost-mass density of 3.2 x 10^-6 USD/kg. The newer penny is just slightly more expensive and probably easier to find. Also, it looks like the dollar coin is the most expensive option but that’s wrong. I left off the dollar bill because it was too high. The dollar coin is 1.23 x 10^-4 USD/kg and the bill is 10^-3 USD/kg.

What about price per volume? I’ll call this this cost-volume density.

Again, the penny wins as the most affordable option here. Both the old and new penny have the same volume so they are equivalent in cost-volume density. Since the dollar coin and bill have nearly the same volume, they are both terrible options for a volume-based cashteroid.

Ok. One more thing. What if you want to line money up? You know, end to end? What would be the cheapest option?

The penny wins again. Of course the dollar bill is better than the dollar coin since it is much longer. Still, it doesn’t beat the almighty penny.

How Much Would a Cashteroid Cost?

I gave a terrible definition for my cashteroid. I said it should have a large enough gravitational field so that you could walk on it. But what is the smallest gravitational field for which you could walk? I’m not exactly sure.

In a previous calculation, I looked at the possibility of walking on the surface of comet 67P. I found that it has a surface gravitational field of about 5.24 x 10^-5 N/kg (compared to 9.8 N/kg on Earth). This was not a large enough field for a human to reasonably walk. In fact if an astronaut jumped with a speed of just 0.46 m/s, that human would be going fast enough to leave the surface and never come back.

For a walkable asteroid, let’s just guess (for now) that it would have a minimum gravitational field of 5 x 10^-4 N/kg (10 times larger than 67P). Now I can calculate the cost of my penny cashteroid by looking at the surface gravitational field (assuming that the pennies are tightly packed and the penny density remains constant). If I have a spherical object, the surface gravitational field can be calculated as:

Here M is the mass of the casteroid and R is the radius. These two things are not independent. If the density of the cashteroid is ρ, then I can write the mass in terms of the radius.

This gives the following expression for the gravitational field on the surface of the cashteroid.

Now I just have to put in my value for g and solve for R assuming a penny density of 5770 kg/m³. Note that I am just using the new pennies – however, I should probably find the ratio of new to old pennies in actual coin change to get an average penny mass. This would be a fun future blog post. It would be very similar to the same issue with isotopes.

Using the penny density, I get a cashteroid with a radius of 310 meters. Wow, that is smaller than I would have guessed. But I guess it’s not too crazy. Comet 67P has an estimated radius of 2,000 meters but a much lower density.

I can also find the mass of my penny cashteroid. It would be 7.2 x 10¹¹ kg. Again, notice that this is a lower mass than comet 67P (with a value of 3.14 x 10¹² kg), but it is also much smaller in radius. The gravitational field depends on both the size and mass.

Ok, so how much would this cashteroid cost? Using a cost-mass density, I get cost of 2.8 million dollars. That’s not too bad.

Homework

I am going to write your homework assignment here on the chalk board. Copy it down.

• Estimate how long it will take for the word “cashteroid” to appear in the official dictionary.


• How much would it cost to make a cashteroid out of dollar bills?


• What if you wanted to put your cashteroid in orbit around the Earth (at the same altitude as the ISS). How big would it have to be so that you could see it with the naked eye? What should you make it out of?


• Calculate the escape velocity for both penny cashteroid and a dollar bill cashteroid. Why are they different?


• Collect 1,000 pennies in the wild. Find the ratio of old to new pennies and determine the average penny mass.