
A googol-to-one gear ratio [video] - klohto
https://www.kottke.org/20/03/heres-what-a-googol-to-one-gear-ratio-looks-like
======
gizmo686
I ran the numbers for going in the other direction.

Suppose that:

    
    
        * The 99 slower gears are massless.
        * the fastest gear weighs 10 grams
        * The fastest gear is a cylinder of radius of 2cm
        * The slowest gear completes 1 rotation every 3 days.
        * Relativity only applies when I want it to.
    

We have:

    
    
        * The fastest gear has an angular velocity of:
             w=10^95 radians/second = 10^95 s^-1 //Since radians are unitless
        * The fastest gear has a moment of intertia given by:
            I=mr^2 / 2 = 10^-2 * 4*10^-4 / 2 kg*m^2
        * The fastest gear has a rotational energy of:
            E=Iw^2 / 2= 10^-2 * 4*10^-4 / 2 kg*m^2 *  10^190 s^-2 / 2
            E =  10^184 kg*m^2*s^-2
         * Ignoring all of relativity except E=mc^2, we have
            m = E/c^2
            m ~ 10^184 kg*m^2*s^-2 / 10 ^ 17 m^2*s^2
            m ~ 10^167 kg
        * Acknowledging more of relativity now, we have:
          Schwarzschild radius = 2Gm / c^2
          Schwarzschild radius ~ 10^-12 m^3*kg^-1*s^-2 * 10^167 kg * 10^-17 m^-2*s^2
          Schwarzschild radius ~ 10^138 m
    

Giving us a black hole large enough to fit about 112 googol observable
universes.

I'm sure that I'm missing some relativistic effects here, but I don't even
know how to begin to approach that.

~~~
hnuser123456
>Giving us a black hole large enough to fit about 112 googol observable
universes.

*Giving us a black hole with a radius large enough to fit about 112 googol observable universes stacked end-to-end in a line.

~Triple the number of zeros for the number of universes to fit in its 3D
volume. 112 x (4/3) x pi followed by 336 zeroes, more or less.

~~~
hnuser123456
Doing some more order of magnitude approximations, in order to scale this
black hole down so it were the size of our observable universe, define "the
diameter of the universe in millimeters" (approx 10^30) as mmU. There are mmU
* mmU * mmU * 0.5 mmU universes in the diameter of this black hole. For each
millimeter in this scaled-down black hole, there are mmU groups of mmU
universes, with another half layer of nesting, but then you can divide that by
1000. To fit the number of universes in the whole volume in a line end to end,
repeat this process again, except without the extra half layer and division by
1000.

------
Benjammer
The original mentioned, Arthur Ganson's Machine with Concrete, is quite a bit
more artistic. The end is cast in concrete.

The idea is that there is a certain amount of energy lost as friction in each
gear, per turn. If you calculate the gear ratios and how many turns of the
earlier gears would be required to make the final gear move, even
infinitesimally, the amount of energy lost becomes more than all the energy in
the known universe. It's not really about RPMs.

~~~
noncoml
Impressive but imho he could take it one step further and instead of concrete
on the other end, he could rotate the gear in the opposite way.

~~~
Benjammer
It would take more energy than the total energy of the known universe to turn
the last gear.

~~~
numpad0
*the gear in the middle

~~~
Benjammer
Sure, if you are inverting the gear ratios half way through...

~~~
hnuser123456
Even without that, even 50 of these gears would take the mass-energy of
something between a star system and a galaxy cluster to turn at any human-
scale timeframe

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jxcole
Now if on the slow end you connected an exactly reversed series of gears you
could configure it such that the final gear moves at exactly the same speed as
the first gear, but the middle gear moves at 1/googol the speed. Of course
this is likely to break down at some point but it would be interesting to see
exactly what the maximum recoverable ratio is for gear movement and what the
limiting factors are.

~~~
rtkwe
Can't imagine how long it'll take to get the slack out of that gear train.

------
profmonocle
I would love for someone who knows more about physics to explain what effect a
single rotation of the front gear actually has on the last one.

Naively you would assume that every rotation of the first gear rotates the
last by one googolth. But (I assume) that can't be true in the real world,
since that distance is significantly smaller than a planck length.

I have some educated guesses as to how the rotation is actually being "stored"
if it's not physically moving the final gear, but I'm probably more ignorant
than I think - for example I'm not totally sure I understand what a "planck
length" is.

~~~
hnuser123456
There's slop in between the teeth of the gears, they aren't perfectly tightly
meshed. All of the gears can wiggle back and fourth that amount, but the
"wiggle room" of the last gear gets shifted one googlth of a rotation per one
rotation of the first gear.

If all of the gears were pre-loaded, such that the slop was "behind" the
contacting teeth, then the last gear would turn one googlth of a rotation,
with an incomprehensible capacity for torque if it were to encounter any
resistance in that very small distance. Being less than a planck length, you
might say that the probability of finding the gear's atoms one planck length
behind, and one forward, shifts continuously. Of course the atoms are
vibrating far greater distances due to ambient heat, but still bonded
together.

The frequencies are typically of the order of 10^13 Hz, and the amplitudes are
typically of the order of 10^−11 m. A Planck length is about 10^-35.

[https://en.wikipedia.org/wiki/Atom_vibrations](https://en.wikipedia.org/wiki/Atom_vibrations)

Maybe you could say that the gear that shifts approximately one atom vibration
(#11) or one planck length (#35) applies as much torque as needed to shift the
slower gears behind it, once the force applied becomes greater than the
friction resistance, which will happen quite often (about as fast as the atoms
are vibrating, if the gears are perfectly tightly meshed) because of the
tremendous capacity for torque. There will be higher pressure from the source
of the force than from the resistance of the further gears.

~~~
agumonkey
I wanted to make a geared divider for a "micrometric" slide but the slop/room
made me wonder if there was a way to know the actual precision.

~~~
Cerium
The slop is called backlash. On cnc machines they go to some effort to reduce
it through fancy gear and lead screw design. You could just go old school and
take it out by how you operate the machine. If you only ever measure while
traveling in one direction, then the slop will be taken out and won't matter.

------
lacker
If every electron in the universe came over and cranked that first gear around
a hundred times, it still wouldn't be enough to make the last gear visibly
move.

~~~
ehsankia
Stupid question, but realistically what would happen if you tried to manually
turn that last gear? Would nothing move? What if you put a lot of force, what
would happened / which part would break?

~~~
inimino
An interesting question!

As a guess, there should be some slack and elasticity in the last two gears or
so, which you could get out if you used enough force. Beyond that, the
effective torque is reduced by 100x or 1000x already, which means moving any
further gear even by the width of an atom would probably require more torque
than the final gear can bear, and either the teeth or some other weak point
would fail on the last gear. Of course, even what motion there is (up to
failure) would be hard to observe. Moreover, putting this much torque on the
end of the structure would also likely torque the entire machine end to end
(like twisting a rope) and generally mess things up in uninteresting ways.

Putting all that together, I think an observer would probably say nothing is
moving, right until the last gear and/or entire structure fails spectacularly.

------
_underfl0w_
Part of me likes to thing something crazy would happen if you turned it from
the other end. Turn one gear once and the gear at the far other end turns a
googol times but melts somewhere around a trillion rotations. Almost doesn't
feel too outlandish given a scope that already includes "more energy than the
entire known universe has" (per the article) for a measurement.

~~~
warent
All this time we've been fussing about clean energy and perpetual motion
machines when all we needed was this machine to apply essentially infinite
leverage and generate a universe of energy at the flick of the wrist

~~~
peteradio
Woops accidentally solved fermi's paradox and the mystery of what created the
big bang, in one shot.

~~~
tuco86
At first there was God. Then Adam and Eve. Then one of their descendants built
this machine and prayed for god to turn the last wheel. Bang! (a big one)

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robertfw
In the same vein, check out "Machine with Concrete"
[https://www.youtube.com/watch?v=5q-BH-
tvxEg](https://www.youtube.com/watch?v=5q-BH-tvxEg)

~~~
cortesoft
The post says this is what inspired it.

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xwdv
If you made this in a physics engine could you get the last gear to spin in a
reasonable time and not break the simulation?

~~~
autonoshitbox
Sure, just do something with Fourier transforms. Okay, it's not completely
accurate, but most simulations aren't anyway.

In fact, I can simulate this in text form with no floating-point or number
theory issues at all. Here's the simulation:

> For every 10^100 turns of the first gear, the last gear will move about one
> full turn.

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aj7
I bet the zero point fluctuation motion of the last gear dwarfs the
deterministic motion.

~~~
autonoshitbox
The word you're looking for is "thermal." The bulk of the gear behaves
completely classically at this temperature.

~~~
aj7
I think you are right in the sense that at room temperature, thermal
fluctuations dominate.

Better stated, my point is that if the entire assembly is at absolute zero,
the quantum fluctuations in the final gear are much greater than the
deterministic motion associated with one revolution of the first gear.

~~~
autonoshitbox
If this entire assembly is at absolute zero, something is very wrong in the
universe.

------
nkrisc
Are there any science museums that have something like this? When you look at
this, you think that surely you can make that last gear spin. After all we've
all spun gears, we know how they work. But it would be so counter intuitive to
spin one of those gears by hand and basically see nothing happen no matter how
much you spin it.

I know that would blow the minds of some kids (and adults).

~~~
jdstruck
Sciencenter in Ithaca has one of these. The last gear is attached to a small
platform with wine glass that you’re encouraged to try to break.

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Valgrim
What happens to the energy used to rotate the device? Is it dissipated as heat
before it reaches too far?

~~~
Khoth
Friction basically. In a perfectly idealised version of the system with no
friction/slack/materials being deformed/etc, if you started it off then let go
then everything would keep spinning forever. You don't need to pump a whole
lot of energy into the device to make it go.

~~~
kangnkodos
"...if you started..."

That's a big if. I haven't done the math yet, but I'll bet that the outside
rim of at least one of the gears would have to be started moving faster than
the speed of light. I'm not sure how you would do that.

The gear just before the first gear at light speed would have the outside rim
moving at about 100th of the speed of light. It would take quite a bit of
energy to do that.

(Others have pointed out that at speeds well before the speed of light, some
of the gears would melt. I'm ignoring that.)

------
pbhjpbhj
Flip it around .. what's the fastest rotational speed achieved via a hand
powered machine?

~~~
dTal
I imagine you could get several thousand RPM at least with a hand pump and a
dentist's drill, if mechanical gearing is not a requirement.

~~~
praptak
I think ~100k RPM is achievable with a shirt button on a bunch of twisted
strings, pulled on both ends.

This was actually used to construct a cheap mechanism that needed the high g
to separate some chemicals to analyse samples. I can't remember the details
though.

~~~
astatine
125k RPM. For malaria detection using a whirling paper centrifuge, powered
entirely by hand. [https://news.stanford.edu/2017/01/10/whirligig-toy-
bioengine...](https://news.stanford.edu/2017/01/10/whirligig-toy-bioengineers-
develop-20-cent-hand-powered-blood-centrifuge/)

~~~
praptak
Thank you! I was not able to dig this up.

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forkexec
Nice!

I'm surprised Oskar van Deventer (YT: OskarPuzzle,
[https://oskarvandeventer.nl/](https://oskarvandeventer.nl/) ) hasn't done
more than 10^9:1 reduction.

[https://youtu.be/b_pbOCIg_nI](https://youtu.be/b_pbOCIg_nI)

I think this calls for 10^10^2 reduction using a series of cycloidal drives
(34x 1000:1 should do it just fine)

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bilekas
This is insane. My OCD is tempered and my love for all things mechanical has
been re-ignited.

I am curious how this ratio is calculated though, I didn't drill into the
relative ratios.. If anyone has a link I would love to see!

~~~
elihu
It's pretty simple: for each pair of gears that mesh with each other, one has
ten times as many teeth as the other, which gives a ten to one reduction.

If you chain 100 such reductions, you get 10^100 to one reduction.

~~~
bilekas
But the construction of it ??! It's one thing to put the numbers down, we know
they round out nicely, but in my head this needs a very minute level of
construct engineering..

For example: If 1 gear(cog?) is 1/1000 of a cm out, would that not effect the
ratio over this 'distance' ?

Edit: I might just sound like an idiot right now, but when we get down (or up)
to these numbers, i can't help but feel manufacturing numbers play a bigger
role

~~~
mkagenius
As in the video, they just reuse the same ratio gears in cascade?

~~~
bilekas
You're right, took me a bit to catch up.

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raverbashing
Ok what if instead of gears we use belts to connect? I think some of the slack
will be taken out (as long as you tension the belt accordingly) and
transmission of power should be more uniform.

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Thorentis
Why can't it spin faster?

~~~
chrisseaton
I'm sure it can spin a bit faster. Why can't it spin so fast that the final
wheel visibly turns? Well the first wheels would melt from friction or be torn
apart before then, and where are you going to get that much energy to turn
them so fast? At some point there's the speed of light as a limit as well.

~~~
owenmarshall
There are 10^80 electrons in the universe, Eddington thinks.

Even if you managed to make a contraption that turned one electron into one
rotation of the first wheel, and you fed the entire universe to your
contraption... you’re coming up a few dozens order of magnitude short to make
that full turn :^)

~~~
chrisseaton
That's what I already said

> where are you going to get that much energy to turn them so fast?

~~~
owenmarshall
I guess you just have to make a machine that manages _two_ turns per electron.

(I know you did - but any chance to introduce Eddington’s number is fun)

~~~
chrisseaton
> I guess you just have to make a machine that manages two turns per electron.

But that just isn't going to be tractable.

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SamReidHughes
Now put a shaft connecting the last gear to the first.

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high_pathetic
Just for fun, I would connect the first gear to an electric motor and let it
run for an hour or so.

~~~
tempestn
Here you go:
[https://www.youtube.com/watch?v=ApqfqiFTO4E](https://www.youtube.com/watch?v=ApqfqiFTO4E)

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kjaku
So after rotating enough to reduce clearance between teeth, you can move last
gear lets say tooth to the distance 10^65 smaller then Planck Length --
smallest known length, wchich is 1.6 x10^35 m and where all strange things
like quantum foam, or micro wormholes are supposed to happen ...?

