Tuesday, March 4, 2014

More on the Vernier clock

I've noticed on the statistics page that people are visiting the "Vernier clock" post.  All I have there is a simple animation of a clock with a Vernier scale but no explanation, so I thought I'd write a few more words about it.

Many years ago, before most of us were born in fact, watches with a second hand were expensive because they were difficult to produce.  Meanwhile, those of us who took physical science degrees before every device in sight became digital are very familiar with a type of scale called a Vernier scale.  With a Vernier scale, you really only need one hand on your clocks and watches, or at least, one dial, or, if you desire a second hand without any added machinery, you can turn the minute hand into a Vernier scale.  Or, perhaps you have second hand, but want it to read in hundredths of seconds.

The idea behind a Vernier scale, is that you make the measurement based not on which tick mark is being pointed at, but on which pair of tick marks--one on each side of the dial or sliding scale--line up.  In this way, it's not important how far apart the tick marks are, what's important is the difference in spacing between the first set of ticks and the second.

Lets go straight to the application to clocks, because it's actually simpler since the dials are cyclical.  Suppose we are dealing with a minute hand and we want it to read seconds as well.  On the outside, there are the usual number of sixty (60) stationary tick marks, one for each minute or second as the case may be.  Inside, there is a dial that turns around once every hour.  This dial has 59 tick marks.

Since the dial must move 1/(59*60)=1/3540 of an arc to go from lining up along one outside tick to lining up along an adjacent one, you'd think you're reading in 1/59 of a minute rather than seconds, which of course are 1/60 of a minute.  The seconds readout, however, is centred around the current reading for minutes, thus it moves forward every minute, producing an extra reading.  (Gaah!  this is wrong)

(Edit: I lied.  It's actually simpler for linear scales.  Take the number of divisions in the smaller units and either add or subtract 1 larger unit divided by the number of divisions to give you the spacing of your secondary scale.  E.g. if the units are millimetres and you want a reading in tenths of millimetres, your secondary scale (i.e. Vernier) will have a spacing of either 1.1 mm or 0.9 mm, depending upon which direction you want to take the reading.  It doesn't actually work for cyclical scales, that is you can't divide the dial up evenly and still have readings in minutes and seconds.  So my animation below is complete bollucks--the virtual dial is not circulating at a rate of one revolution per minute (this would be true regardless), and it's not reading in seconds either.)

It should also be noted that the ticks are lining up in a retrograde manner.  In the animation, the "virtual dial" is actually moving clockwise: thus there are actually 61 ticks on the moving dial--thus we aren't really reading in seconds, but in 1/61 of a minute.  I didn't really think that much about this aspect of it, however important.  I just made the animation as "proof-of-concept" and because it looks cool.

I find this concept of a "virtual dial" fascinating and think it might somehow be quite deep--sort of like a phase that travels faster than the group velocity in a wave because of how the different frequencies are interfering.  In other words, you can have electromagnetic waves that travel faster than the speed of light, although apparently you can't transmit any information with them.  I never seemed to be able to quite muster the mathematical chops to fully understand this stuff...

I thought about patenting the idea.  No dice: it was patented back in 1942 (U.S. patent US2293459)!  I also though about taking out a design patent on the overlapping slats idea, but even this is covered in the 1942 patent:

So is the idea actually workable?  Here's a guy who's actually built a couple of them: http://www.gizmology.net/watch.htm.

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