The motor you mentioned in your notes that sells for $115 has a no load RPM of 550 RPM. I have NO EXPERIENCE with this motor. I have NOT had very good experience with planetary geared motors and tend to shy away from them at least for the non standard systems I have been working with. Both of the geared motors have a fan in them. TNC Scooters makes more profit on the planetary geared motor and tend to make it look more attractive. I wouldn't use that motor unless I really needed the higher 550 RPM shaft speed. In your case, since you are trying to drive the rear wheel directly from the motor, the higher shaft speed may be more appropriate than the slower speed geared motor.
And forget the 56 tooth gear on the wheel.
That will NEVER WORK unless you intend to go VERY slowly. I think you missed something. Go back and re-read my previous post. The chain on my system runs from the motor to the CRANKS
not the rear wheel !!!
REPEAT: I have the motor driving the CRANKS NOT the rear wheel. The cranks (in my case) turn at 100+ RPM as that is my typical cadence. I've been told that I am a semi-pro bike rider (whatever that means) and tend to have a bit higher cadence than a non pro rider. My typical power output at that cadence is the 600 to 750 watt area. It is not uncommon for me to stretch or break chains or bend bicycle frames using human power only (NO MOTOR) during a sprint.
On the system I was talking about in the previous post I mounted a second chain wheel up front on the opposite side of where the normal bicycle chain is and drove that gear with the motor. The motor had a 16 tooth gear on it and the secondary crank gear is 48 teeth for a gear reduction of 3 to 1. When the motor is running wide open throttle at 325 RPM that turns the cranks at ~108 RPM. Since the motor is now turning the cranks that means the motor can take advantage of the regular bicycle gearing and is not locked into the single gear system like you are trying to do.
Although I have NO EXPERIENCE with the 550 RPM motor you have chosen to use I offer the following assuming you intend to drive the rear wheel directly and NOT THROUGH THE CRANKS as I was doing:
I have no idea what top speed you are shooting for. I'll assume 25 MPH. A 700c wheel will be doing ~320 RPM at this speed.
Since the motor has a no load speed of 550 RPM I will assume 450 RPM under load although something tells me this motor may run closer to 400 RPM under full load....
The motor comes with a 10 tooth gear. Since I have NO EXPERIENCE with the motor I'm not sure, but it looks to me that this motor also takes the wider chain (NOT STANDARD BICYCLE CHAIN). Assuming you are prepared to deal with the wider chain you will need a gear reduction of (450 rpm motor speed /320 rpm wheel speed) or 1.406. This suggests a 14 tooth gear on the rear wheel (standard small rear wheel gear !!!) so we end up with the following:
10 tooth gear on motor turning at 450 RPM
14 tooth gear on rear wheel for gear reduction of 1.4. (10 / 14 = 1.4)
450 RPM / 1.4 = 321 RPM or 25 MPH using a 700C wheel.
Although math is not my forte, I presented the above to give you some idea of how to calculate gear ratios and figure out what size gear you need to get the top speed that you want while permitting the motor to spin at a speed that will NOT burn it up....