The problem isn’t finding “a way to relate the freezing of the gear teeth to the actual VEX motor RPM at the output shaft”, it’s determining which of the infinite number of ways of relating them that’s occurring.
To determine the rotational period by using a strobe, one varies the period of the strobe until it corresponds to the period of rotation. That way, every time the light flashes, the rotating object is in the same position (the phenomenon you call “freezing”.)
By definition, as the motor is what you are trying to measure, the motor is not “too slow”. The DIY strobe is too fast. Why not just slow it down until it is flashing at the motor’s rotational frequency:
100 rev/min = 100/60 rev/sec = 1.67 rev/sec
If you take the reciprocal of that:
1/(1.67 rev/sec) = 0.6 sec/rev
To be safe, I’d build a strobe that could be adjusted to a period of at least 1.2 sec; 2 sec would be better. Here’s why:
The problem with using a 36-tooth gear is this: You don’t know how many teeth went by between flashes of the strobe. If the gear had advanced by one tooth in t seconds, then the whole rotation would take (36 * t). However, if the gear had advanced 2 teeth, the whole rotation would take ((36/2) * t) seconds. This goes on ad infinitum.
In fairness, this problem isn’t limited to having multiple, indistiguishable marks per rotation. If you have a single mark and it appears not to have moved between flashes, you can’t tell if the wheel has turned once, twice, or some other integral number of times.
The way to deal with this is to adjust the strobe until the rotating object appears to stop, then increase the frequency until it is no less than twice the frequency at which the object appeared to stop. If you find no more points at which the object appears to stop, you’ve found the right period. If you find another frequency at which the object appears to stop, keep increasing to twice that frequency. Keep looping until you find a frequency two times which does not also “freeze” the object.
If I start at 0.9 Hz (0.9 flashes per second) and the object is “frozen” when I get to 1 Hz, I know the object completes some integral number of rotations in 1 second. What I don’t know is how many.
So, I increase the strobe rate, looking for other “freezing” points. If the image “freezes” again at 2 Hz, then I know the rotational frequency is some integral multiple of 2 Hz, so I keep adjusting to at least 4 Hz. If the object doesn’t “freeze” at 4 Hz, then the rotational frequency of the object must be 2 Hz. If it does “freeze” at 4 Hz, then I need to keep going.
The question may well have occurred to you: But what if the rotational frequency is between 2 Hz and 4 Hz, for example 3 Hz? Wouldn’t that also “freeze” at 1 Hz? The answer is “yes”. The way you’d find the correct frequency, though, is in the scanning between 1 Hz and 2 Hz. An object rotating at 3 Hz will “freeze” at a strobe rate of 1.5 Hz. (At a strobe rate of 1 Hz, you’re seeing every third rotation; at a strobe rate of 1.5 Hz, you’re seeing every second rotation.) Once you see the “freeze” at 1.5 Hz, though, you must keep increasing the strobe rate to at least 3 Hz. The image will “freeze” again at 3 Hz, but nowhere above that.
You can simplify the procedure by starting the strobe at its maximum frequency and adjusting it downward until the image “freezes”. What you need to keep in mind, though, is that if the first “freeze” is at more than half the strobe rate at which started, you may not be getting the right answer.
So, back to the question of the 36-tooth gear: If the motor is turning at 100 rpm, the gear is advancing at:
(36 teeth/rev) * (100 rev/min) = 3600 teeth/min = 60 teeth/second.
If you start with a strobe rate of 10 Hz, you’ll find the image has “frozen”, as the gear has advance exactly 6 teeth between flashes. It will also “freeze” at:
12 Hz (5 teeth between flashes)
15 Hz (4 teeth between flashes)
20 Hz (3 teeth between flashes)
30 Hz (2 teeth between flashes)
60 Hz (1 tooth between flashes)
However, at no strobe rate above 60 Hz will it “freeze”. Remember that you need to increase the strobe rate to at least 120 Hz to know you’re not missing anything.
If you take 60 Hz, convert it to teeth per minute and divide it by the number of teeth, you get the number of revolutions per minute:
(60 teeth/sec) * (60 sec/min) / (36 teeth/rev) = 100 rev/minute.
PS: I wrote this a couple of hours ago, before anyone else had responded to original posting. Despite that, it appears to contain some things that may be of use, so here it is.