## Tuesday, January 25, 2011

### Fun With Numbers: A Demon Gearbox

Not everything has to be strictly useful, right? I admit (and this is a sort of a Disclaimer) that this is something without even remotely imaginable practical purpose, but it may provide some fun with numbers. Namely, it's a "Super-Gearbox" that gears the initial axle's rotation down through sixteen stages.

The input axle (a small black rubber tyre at the far left hand side) passes through nine worm gears that drive 24-tooth gears in sequence, and proceeds through another seven stages of various gears, to end on the right hand side, on the linear rack (near the beige gear).

So, what is the input-output ratio? To have the output axle (8-tooth gear on the rack) make one full turn, the input axle needs to be rotated 5349660268953600 times, or about five point something million billion revolutions. Of course, the gearing could be reduced even further.

Let's imagine this gearbox to be constructed with hypothetical infinitely strong and hard parts, and that the axle friction is negligible (yes, I told you there would be no practical usage). What would that mean?

Suppose we put an NXT motor on the input axle. It can provide about 16 Ncm of torque (thanks to Philo for this information) on its output. Through the gearbox, the torque would increase to 856 TNm (terranewton-meters), and the rack would exert a force of 171 EN (exanewton). In other words, it could lift (and still have some force to spare) all the water in Lake Superior. However, with the NXT motor's output speed of 117 rounds per minute, it would take about 2.8 million years to lift it one millimeter. Though, I suspect the batteries would get flat somewhat earlier.

On the other hand, if we rotate the output shaft with a speed of one revolution per second, a little rubber wheel on the input would have a tangential speed of almost a million times the speed of light (if there was no relativity to prevent that).

So what have we learned from that? Nothing. Nothing, except that in reality we don't have the theoretical infinitely hard Lego parts, but ABS plastic. Gear the motor up more than ten, fifteen times, and the combination of friction and low torque will make it difficult even to rev up, and forget the work done; gear it down more than ten, fifteen times, and if something gets stuck on the output terminal, prepare to hear a thin crack and say goodbye to some of your parts.

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