YOur load is too large and speed is too low for any reasonably sized direct drive motor. YOu need to "gear it down" in some way. You can use gears, chains, belts, or timing belts. Belts are probably the quiestest of the bunch, though a good gear drive is also quiet (and expensive, easily exceeding the cost of your motor by a lot). And yes, you probably need to use multiple stages of belt/pulleys or gears to do your job. That's why we use multiple stages after all...doing everything in one stage takes too much room.
Brushless motors are quiet...but expensive and also need expensive controllers. However, I think you might need an expensive controller anyways because changing direction of such a heavy mass smoothly and quickly isn't easy. Honestly, I think $50 is way too low. Little fist sized motors for radio controlled planes are largely in the range of $70-$100 and good ones easily go up to a couple hundred.
Typically required torque and power for a motor are:
Code:
Torque = Force[Newtons] * Distance from center of rotation (meters)
Power = torque(Newton-meters) * angular velocity(radians/sec)
= torque(Newton-meters) * RPM*2*Pi/60
HP = Watts/746
But your problem is a bit different for these calculations. YOur motor isn't actually lifting anything because you're spinning a balanced wheel. All the mass is balanced around the center counterweighting the mass everywhere else in all directions so technically it takes no torque to spin at a constant speed. But it does take force to accelerate the wheel. So the amount of torque you need depends on how fast you want to rev the wheel up and how quickly you want to be able to decelerate and change directions. The general equation for that is a angular/rotational form of F=ma which is:
Code:
Torque = (moment of inertia) * (angular acceleration)
The moment of inertia for a a solid disc of constant density rotating about it's center is:
Code:
Moment of inertia for solid disc spinning around center = 0.5*mass(kg)*radius(meters)^2
Angular acceleration is in radians per second squared. Assuming constant acceleration over the rev up period this is:
Code:
Angular acceleration = Change in RPM * 2 * Pi / 60/Time
ANd of course, after you figure out your torque and RPM, you can find power which you can use to find your motor. Then you can figure out what gear ratio is needed so that the power can be converted into the required balance between torque and speed.
Since you don't know the steady state torque reequired to actually spin the thing because you don't know the losses, you could estimate an upper bound for power by going
Code:
Upper bound for max power = (Torque required for reversing direction) * (maximum steady state RPM)
which is the worst case for both torque and RPM even though in practice the thing is probably never applying that amount of torque at that speed. I get about 20W. And maybe quadruple that number or even x10 or x20 it to take into account motor efficiency, friction, gearing losses, unforeseen factors, and the fact your wheels are probably not the perfect mathematical discs that we used.
Remember, that a motor will be smallest motor motor that will provide the power you need. Being the smallest it will also be the fastest with the least torque and will therefore require the most gearing. If you use a larger motor that provides excess power, it will also have more torque and be slower which means you don't have to gear as much (efficiency is a different story).