Ping Pong balls

[Grok]

A stack of ping pong balls are arranged such that each level rests compactly on top of the level below it.  The top level has 1 ball and sits on top of the level with 3 balls, which sits on top of the level with 6 balls, which sits nicely on top of the level with 10 balls, etc.  You'll have to come up with your own visual.

Assuming incompressible 1" diameter ping-pong balls affixed in a stack of 5000 levels, at sea level, how much pure water is displaced by this stack?  Assume a ping-pong ball weighs 9g.

[Adron]

Archimedes tells us that the amount of water displaced by something floating in the water will weigh as much as the floating item.

Now for a formula for number of balls in stack...

Level n of the stack contains n balls more than the level above it. (Get that from visualizing how you add another line of balls, 1 ball longer than the previous for each new level, to be able to put the previous stack on top of the new level)

Balls at level n :

Sum for i = 1 to n of i  = 1/2*n^2 + 1/2*n

Total number of balls in stack of height m:

Sum for n = 1 to m of 1/2*n^2 + 1/2*n = 1/6*n^3+1/2*n^2+1/3*n

In a stack of 5000 levels, there are thus 20845835000 balls, total weight 187612515 kg, and that much water will be displaced by the balls.

Correct?

[Grok]

I first ran across that problem at age 12, in issue #1 of Games magazine.  It was several years before I had the math to solve it.

[Yoni]

Adron's calculations look correct, and since I have nothing to add to that, I'll provide a link with unsurprisingly much information about these numbers.

http://mathworld.wolfram.com/TriangularNumber.html

[register]

Calculations are confirmed correct, with balls and weight.

But the answer to the question:

at sea level, how much pure water is displaced by this stack?

Can't be answered.

1. At Sea Level
2. Pure Water

We need a third piece of information.

3. Temperature

Then we can determine how much pure water at sea level is displaced.

[Maddox]

Damn, I could have used this for 2C I took on saturday.

[Adron]

I'm not sure why we need temperature.. If we have the total mass of all the balls, that should be the amount of water, oil, or whatever you submerge it in displaced. If I wanted to give the amount as volume instead of mass, I'd need to mess with densities, but I'm used to measuring amounts as mass.

Well, if you want to be really really accurate, you might want to subtract the mass of the amount of air they displace too. And perhaps take into account the differing densities between the bottom of the stack and the top of the stack. And ...

[register]

Assuming incompressible 1" diameter ping-pong balls affixed in a stack of 5000 levels, at sea level, how much pure water is displaced by this stack?  Assume a ping-pong ball weighs 9g.

To Adron's last post: Yes. I'll allow you the attempt to provide the correct answer, assumning Grok provides complete information to answer the question asked.

[Grok]

OK Smarty pants, the temperature is 28 C on the surface, at the center of where the balls enter the water, and decreases at an even gradient of rate 0.04 C per meter in depth below the surface.  Now you have your temperature information, go for it.

There were enough "clues" in the question to tell a reasonably intelligent person to ignore microvariations due to physics.  Words like "incompressible", "sea level", "pure" are good examples.  Adron and Yoni seemed to have no trouble grasping the question's focus.

But by all means, solve the much harder problem and share the answer.  I do not know how to do it, so I will learn from you!

[Grok]

To be really accurate (and annoying), you would also want to know the air temperature, humidify, barometric pressure, wind velocity (stack might be tilting), lattitude and longitude, time of day ... all could effect the final answer.

[iago]

What about speed that the earth is moving at the time, gotta figure out for relativity

[muert0]

Is it saltwater, springwater, distilled, potable, or maybe brackish even?

[Adron]

Assuming incompressible 1" diameter ping-pong balls affixed in a stack of 5000 levels, at sea level, how much pure water is displaced by this stack?  Assume a ping-pong ball weighs 9g.

To Adron's last post: Yes. I'll allow you the attempt to provide the correct answer, assumning Grok provides complete information to answer the question asked.

That's OK, I already got to do half of it, so I'll let you do the other part. Can't do all the fun myself