I envision the model being built to scale to around 6 feet tall so that the top of the lower block is at eye level. Then the top and lightest one-tenth of the structure is raised 12'' and dropped on the lower and more robust nine-tenths of the structure.l I invite the Readers to picture your ' crash all the way to the ground if you drop the top ' scenario
It's funny how you have, in 100% original truther fashion, completely dodged, ignored, missed the one important point of my previous post:
That of scale.
If you build your model 6ft (180cm) tall, that would be 6 spaghetti high, about 18 stories = levels of lateral bracing per spaghetto. Too bad I already ate my broken spaghetti, so I'll have to break another, make it 300mm/18 = about 16,5mm, measure the strength of that lenght of spaghetti...
but at least with 6 spaghetti, you are now up to a height-thickness-ratio of 900:1, which is reasonably close to scale.
*doing measurement on my kitchen scale*
Oh! Bad news: 16.5 mm are a lot more easily to handle for a clumsy person like me!
Turns out my first story-length piece (actually 17.5mm) of spaghetti supported at least 1100g before it broke.
My second piece (16mm) was even stronger: I gave up when I peaked briefly a little above 3000g because my thumbtip hurt from the tip of the piece poking into it.
Third piece (17mm) resisted 1500+g several times, where again I gave up as my thumb now seriously hurts. Dang, that spaghetti-stuff is strong!
So by experiment one column of story-height can carry a static load of at least 1500g.
Let's be conservative and say this is 5 times the actual load. Hence, let the top 10% (ohhh - can we make that 16.7%, so our top portion is as high as standard spaghetti are long?) weigh 300g per spaghetti column. As there are 522 columns, the total weight of the top block in our model would therefore have to have a mass of 522*300g = 156.6 kg.
522 spaghetti is coincidentally quite close to the number of spaghetti in a standard 1 pound (500g) pack that serves 4 to 5.
So would a pack of spaghetti withstand 156.6 kg (twice my body weight) if it falls onto it?
Hmmm.
Let's further improve our model and calculations:
I said that our spaghetti are laterally braced every 16.5mm - that is one story. Initially, our 156.6 kg would have to fall only those 16.5mm.
What velocity will the upper block have?
Let's see:
potential energy U = m*g*h
gets converted into
kinetic energy E
k = 1/2 m*v
2
m*g*h = 1/2 m*v
2<=>
v
2 = 2*g*h
<=>
v = sqrt(2*g*h) = sqrt(2 * 9.8m/s
2 * 0.0165m) =
0.56 m/s
Next step: The top block, falling at 0.56 m/s, falls onto our next set of 16.5mm long spaghetti, which excert a force up and will decelerate the block. At the same time, gravitation still pulls down (accelerate) at a rate of g.
The movement cannot be stopped instantaneously in this universe, as that would be equivalent to an infinite acceleration and hence an infinite force.
So as the falling block touches the tips of our spaghetti, the will get strained and bent. How much can you bend a spaghetto before it breaks?
Next experiment:
I bent a bavetta (a relative of the spaghetti family, see footnote in my previous port; mine are 257mm long) carefully in roughly a circular shape. Needless to say, the thing broke before it had formed a full circle, but it went round more than 180°. I am doing this in a very clumsy fashion - my best estimate is that I reached 240° of a circle, or 2/3 of a full circle. Full circle would have had a circumference of 257mm / (2/3) = 385mm. Radius therefore 385mm/2pi = 61mm.
Now comes the tricky math part - if I bend an upright column piece of 16.5 into a radius of 61mm, it represents an angle A of (16.5mm/385mm)*360° = 15°. What is the distance h
bent of the two ends of that column piece?
*consulting my Bronstein-Semendjajev Pocket book of Mathematics*
Aha!
h
bent = 2r * sin(A/2) = 2 * 61mm * sin(7.5°) = 15,924mm = 96.5% of column height.
In other words: Our spaghetti columns of 16.5mm can elastically bend until the floor resting on them has moved down by 0.576mm = 0.000576m
So if we want to stop the top floor's 0.56 m/s before the spaghetti break and the story collapses, deceleration to 0 has to occur within 0,576mm.
Now, the formula to derive distance from acceleration is
s(t) = 1/2 a*t
2 = 0,000576m
<=>
t = sqrt(2 * 0,000576m / a)
same for velocity vs. acceleration is
v(t) = a*t = 0.56 m/s
<=>
t = 0.56 m/s / a
so
sqrt(2 * 0,000576m / a) = 0.56 m/s / a
<=>
2 * 0,000576m / a = (0.56 m/s)
2 / a
2<=>
2 * 0,000576m = (0.56 m/s)
2 / a
<=>
a= (0.56 m/s)
2 / (2 * 0,000576m) =
272 m/s2 = 27.8g
Ok here is a problem: We said that our columns are designed to be able to carry 5 times the static load, that is, resist a force of 5g times the mass on top. However, we'd need columns that are nearly 6 times as strong! That means:
Our spaghetti will break before the top mass has come to rest. Collapse will continue!
Did we at least slow the fall somewhat? Sure, let's see by how much.
Initial velocity (down) v
i = 0.56 m/s
Max. upward acceleration that our columns can bear is a = -5g = 49 m/s
2But this -5g is diminished by the constant pull of gravity, so our spaghetti can effectively decelerate the falling mass by a = -4g = -39,2m/s
2
Velocity is
v(t) = v
i - 39.2m/s
2 * t
Distance (fallen) is
s(t) = v
i*t - 1/2 39.2m/s
2
I am too lazy now to figure out analytically at what t s(t) >= 0,000576m and what v(t) is then. I quickly ran the formulas through a spreadsheet:
t v s
0,0000 0,5600 0,000000
0,0001 0,5561 0,000056
0,0002 0,5522 0,000111
0,0003 0,5482 0,000166
0,0004 0,5443 0,000221
0,0005 0,5404 0,000275
0,0006 0,5365 0,000329
0,0007 0,5326 0,000382
0,0008 0,5286 0,000435
0,0009 0,5247 0,000488
0,0010 0,5208 0,000540
0,0011 0,5169
0,000592
So we see: After only 0.0011s, we have exceeded the maximum elastic deformation our spaghetti can bear, at which pint they break.
Velocity is then still 0.5169m/s, or 92% of the initial 0.56m/s
Our accumulated stories can now fall more or less freely for 15.96mm, during which velocity increases to about 0.7609m/s. Then we have another spaghetti crash which will again reduce velocity by only about 8% etc. etc.
Result: We can expect our 6' tower with 522 spaghetti columns and a total mass of 939.6kg to collapse within 10% of free fall speed