Another way to look at it, which is mathematically simpler (I can easily do it in my head in the field) and more intuitive for me is:
rotation per second * shutter speed * 360 = blur in degrees
revs per second = 10 shutter speed: 1/100s
10 * 1/100 * 360 = 36 degrees
I can relate better to 36 degrees of blur than the tips traveling a certain distance around a circle of a certain circumference. But others may relate better to the distance method.
For a known rotation speed, we can solve shutter speeds for a desired angle of blur:
shutter = blur angle / (revs per second * 360)
I want 45 degrees at 10 revs per second...
45/(10*360)= .0125s = 1/80s
Truth be told, even on a good day I can't divide 45 by 3600 in my head while shooting an air show.
However, a good rule of thumb is that 45 degrees of blur is a nice blur and a good target. Forty five degrees is 1/8 of a turn (45/360).
If you want 45 degrees, multiply the rotational rate (per second) by 8 to get the reciprocal of the shutter speed: 10*8 = 80 = 1/80s
Even a cave man can do that one in his head while tracking oncoming aircraft in the viewfinder while the air show fanatic next to him recites the rpms of all the performers. He has to be good at dividing by 60, though.
(this is real life for me- if you don't know the numbers then stand next to a guy that knows all the RPM numbers in his head)
If something other than 45d is desired than that optimum angle can be converted into a number to multiply by the revs/second to get the optimum shutter speed. If 90d is desired, the number is 4. Using 10 gets you 36d, and so on.
Engine speeds are usually expressed in RPM's, requiring division by 60 for all of the above.
If you accept 1/6 turn as an optimum or base rate, then simply divide the RPMs by 10 to get the shutter speed:
600 RPM Shutter speed = 1/60s
That is 60 degrees. For 30 degrees, double the speed to 1/125s. "Neil's divide by 10 rule" gives you lots of blur, and if you think that is too slow to deal with or too much blur, you can double it and still get nice (30 degree) blur. If you double it again to 1/250s you only get about 15 degrees of blur and that may be marginal.
That's about as much thinking as I want to do while shooting prop planes at an air show . I usually forget to prepare all this in advance so I just try to remember the divide by 10 rule.
It's better, of course, to prepare in advance as Hal is doing. Trust me on that one . But the rule of thumb works in a pinch.
(Air show prop planes tend to run around 2500-3000 RPM. Helis are slower, more like 200-250, since their props are so much bigger. It may be tougher to shoot a heli, especially if it is moving fast and is close by)
One more idea. I think the tail rotor turns at a ratio of between 3:1 and 6:1 relative to the main rotor. So if the chopper has a main rotor speed of 300 RPM the tail rotor will be turning somewhere between 900 and 1800 RPM.
If you set up a shot where the main rotor turns 30 degrees, the tail will turn somewhere between 90 and 180. If you know the ratio (which varies by model and design) then you may be able to figure out a way to get both well blurred. It can be a tough needle to thread.