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.
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.
The Planck length is simply the characteristic length scale for processes where both general relativity and quantum mechanics play a role at the same time. Since these theories are incompatible, our current understanding of physics does not work for such processes. It is _not_ the smallest unit of distance or something like that. In both GR and QM, positions are continuous quantities with no discretization.
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.
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
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.