Caveat: Powers of Ten

Here’s an interesting thing. This guy made a gear reduction gadget – like a car’s transmission. It seems small and reasonable enough on first glance. But it’s not. Here’s his gadget.

Each gear reduces by a power of 10 from the previous gear. Since there are 100 gears, that means that the slowest gear spins 10¹⁰⁰ slower than the first gear. (10¹⁰⁰ is the number called googol). Just watching the gadget, it appears the fastest gear is spinning about once every 4 seconds. The slowest gear, therefore, would spin once every 4×10^-100 seconds. If there are about 3.15×10⁷ seconds per year, and approximately 1.3×10¹⁰ (13 billion) years since the beginning of the universe, the age of the universe is about 4×10¹⁷ seconds. So… there would be about 10⁸³ “age of universe” units of time before the slowest gear makes a full turn. That’s 10⁹³ years, i.e. 1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 years. Actually, I just googled “how long until heat death of universe” I got “10¹⁰⁰ years if protons all decay”. That works out nicely: the slowest gear would almost complete one turn before the universe dies.
I had an interesting thought, though. What if we attached the motor to the other end. It would need a lot of torque: more torque than there is energy in the universe. I was thinking about how many gears would get the spin speed up to relativistic speeds. It’s very few gears. At one turn every four seconds, the rim speed is about 0.0785 m/s (7.85×10^-2 m/s). Since the speed of light is about 3×10⁹ m/s, by the 12th gear the rim speed would exceed the speed of light under classical rules. So if one could actually get a motor with a enough torque to spin the gadget from the slow end, the 12th gear would become a supermassive black hole (in the sense that relativistic speeds increase mass toward infinity) and who knows about the gears beyond that?
This gadget is interesting, because if fits the potential for all these “bigger than the universe” numbers into an analogue device that fits on a table. I like thinking about it. I’m just speculating a bit – so forgive any errors in calculation – at such astronomical scales a few orders of magnitude one way or the other don’t impact the tone of the speculation much anyway.