Many of us like to see a smoothly-moving seconds hand, as opposed to that of a quartz watch which clunks it's way around the dial, only occasionally aligning with the marks thereon. And a good few of us think that the higher the beat rate the smoother it will be - which is true to an extent, but why? One could be forgiven for thinking that our so-called "persistence of vision" has something to do with it but, sorry to say, that is not the case. There might be a smooth running appearance if the beat was over 16 per second or even 24 per second like old movies but that would mean a beat rate of 86,400 bph!!
Yep, the hand does seem to run smoother on a 28,800 bph watch but only because it moves in smaller steps and takes more steps to go once round the dial. You can still see the steps, though. That got me to wondering how small a step would give apparently smooth motion. Well, apart from persistence of vision, another property of vision is that "visual acuity". Put simply, this is the ability of the eye and brain to distinguish between two objects that are very close together. Like the bars on the letter "E" when you visit your optician or optometrist.
Apparently most folks can distinguish objects quite easily if they are more that a certain angle apart. By "angle", I mean relative to the eyeball. That magic angle is 1/60 of one degree, also known as 1 minute of arc. At 20 feet distance, it is no coincidence that a person with 20/20 vision (6/6 in metric countries) should be able to read an "E" that is 0.35" tall at that distance because it is apparently 5 minutes of arc in size and each bar or gap is 1 minute of arc as seen by the eye.
However, when we glance at our watch, it's a little closer than 20 feet! Switching to metric, let's say 300mm, just less than a foot. This is the only information we need in order to calculate how far apart are two successive step positions of the tip of the hand that can be easily distinguished. Trigonometry tells us that the distance is 0.087mm. Less than a tenth of a millimeter!
Next, let's have a sweep seconds hand that is 13mm long from center to the tip. More trigonometry tells us that the tip of this hand sweeps a circumference of 81.681mm every minute. From here, we can go several ways but it might be instructive to calculate the beat rate in bph for the step distance of 0.087mm established in the preceding paragraph. It comes to 938 beats/min which is 56,267 bph. This tells us that, even at this very high beat rate, a person with normal vision will still see the seconds hand ticking!
Well now, what about those old-fashioned sub-seconds hands? They seem to run smoother too, huh? I have a watch with a sub-seconds hand 3mm long. Ergo, it sweeps a circumference of 18.85mm. That works out, for the same step distance, to only 13,000 bph. The watch actually ticks at 15,000 bph, so it is no surprise that the hand motion (to my less that perfect eyes) seems quite smooth with no apparent ticking discernible.
Best regards, xpatUSA