Saw an analog clock today at a school, and it surprised me. The minute hand moved every 15 seconds, and moved a quarter of the way to the next minute each of those times.
Now, the analog clock I have at home is continuously geared. The minute hand is always moving, just at 1/60th the speed of the second hand [illustrated below]
I couldn’t help but to wonder exactly how the internal gearing of that clock in the school worked. While we live in an age where a simple internet search and digging through some manuals and videos could explain this very quickly, it seemed remiss to pass on the chance to see what my own knowledge can afford me in an explanation.
My instinct in reverse-engineering something like this would be to think of the states that each hand can be. The second hand has 60 states, so give it a gear with 60 teeth. The minute hand moves every quarter minute and therefore must have 240 states, so give it a gear with 240 teeth. Now how to convert between the two gears? The seconds gear changes the state of the minutes gear only every 15 times it moves, so perhaps another gear on the same axle but with four teeth, corresponding to every 15 on the seconds gear [illustrated below]. The discontinuities between the four teeth would create the lack of continuous motion in between.
However, there was still one last puzzle. The minute hand and second hand moved exactly in sync during that one tick on the clock I saw in the school, which implied to me that the teeth must be the same radial size. Moreover, now that I knew the 1:15 relationship, I could imagine an easier gear pairing to make: Rather than a gear with 60 tiny teeth and another with only 4 teeth the same tiny size, how about a gear with 15 well-defined teeth and another gear with a single 1/15 tooth [illustrated below]. These can then be linked up to the appropriate 60-tooth and 240-tooth gears. Moreover, in this arrangement, instead of power traveling from the seconds hand then to the minutes hand and having to overcome torque of each in the chain, the power that drives the clock comes from a single axle that transfers power to both the seconds hand and the minutes hand at the right ratios.