I have a motor whose specifications say that its torque is 190g*cm. If I have the formula for the mass of the wheel vs radius, is there any way I can find the radius to get the best linear speed?

The formula that relates mass and radius is

m=19.575πr^{2}+216.563

where the m is mass in grams and r is radius in cm

electricviolin:
I have a motor whose specifications say that its torque is 190g*cm. If I have the formula for the mass of the wheel vs radius, is there any way I can find the radius to get the best linear speed?

Is the wheel driving something or is it freely spinning?

What is the "best" in this context? Do you mean speed at the rim?

Normally if you want speed and there is no load you just choose the largest wheel possible,
but you'll have to wait for the thing to speed up.

The final speed of an unloaded wheel only depends on losses, not mass.

Perhaps you mean acceleration to a given set speed?

Based on the assumption you want to maximize the vehicle's acceleration, I'm pretty sure the you want the wheel to be as small as possible.

I think the gain you make by converting angular acceleration to linear acceleration with a larger wheel is lost, as the larger wheel simultaneously reduces the acceleration produce from the motor's torque (the longer lever arm transfers a smaller force).

If the wheel were weightless, then the radius of the wheel wouldn't affect the speed of the vehicle. (I'm not positive I'm right about this.) Since the wheel isn't weightless, you only lose by increasing the radius.

Of course there will be practical matters of how fast the motor can actually turn. It obviously won't be able to maintain a constant torque at all speeds. There are likely to be optimal speeds you want the motor to turn so the size of the wheel will be important but I don't think the optimal size can be computed with the information provided.

The reason the formula has a mass even when the wheel radius is zero is because I was accounting for the mass of the hub (which is used to attach the motor to the wheel). I suppose the minimum radius would be 3 cm using this idea. However, there is actually a small mistake in the formula; here is the revised formula:
m=1.14r-0.948

When I say "best liner speed", I mean the fastest speed at the rim. The wheel is free spinning, so ignore any sort of friction, air resistance (negligible), or applied force. I don't mind waiting for even a couple minutes for the motor to get up to speed - it just can't get up to speed if the wheel is too heavy for the motor's torque.

There are likely to be optimal speeds you want the motor to turn so the size of the wheel will be important but I don't think the optimal size can be computed with the information provided.

This is exactly what my question was (worded better). What other information would I need to find such a 'optimal size'?

electricviolin:
I don't mind waiting for even a couple minutes for the motor to get up to speed - it just can't get up to speed if the wheel is too heavy for the motor's torque.

Unfortunately the torque of the motor doesn't tell us anything about the motor's max rpm. I think it's fair to say low torque motors often spin faster than high torque motors (without a load).

The torque also does it tell us anything about how much weight it can spin.

Maybe if you posted a photo of your motor and wheels someone might be able to offer a suggestion on increasing the speed.

All the motors without gearbox examples I've seen use very small wheels. The wheels are often smaller than the motor's diameter so the motors are mounted at an angle to allow the wheels to contact the ground.

If I have a wheel, for example, that weighted 5 kg, a small dc motor is not going to spin it. Nothing will move and the motor will burn out. I know that once the wheel is spinning, it takes less of a force to keep it spinning granted that a is constant (and obviously m) in F=ma. Theoretically, the force would be zero, but clearly there is an opposite force acting on the wheel in the form of air resistance and friction between the motors shaft and its bearings. Therefore, the force needed would be more like F=ma+F_{air}+F_{f}. Remember that F_{f} is equal to F_{c} and that equal (mv^{2})/r. Clearly, the mass is significant in this problem.

Of course, I assumed that the wheel is already spinning. It it is not, the wheel first has first overcome the static friction, which requires even more force than the kinetic friction (μ_{s}>μ_{k}).

Your discussion doesn’t make much sense because you are confused by the differences between force and torque. For example, wheel radius is extremely important. Here is a good introduction to the differences.

I gather that you want to spin up some sort of wheel that will launch ping pong balls.

If so, you need to describe the entire mechanism that will be spun up, which will include the wheel, the launcher and the balls. That description must include all dimensions of each component as well as the mass or mass density of each, so that the total moment of inertia can be calculated.

Knowing the moment of inertia, the final RPM and the time to spin up, it is easy to calculate the required torque.

The only reason the small motor won’t move the heavy wheel is because the gearing is wrong. Of course if you gear down for more torque the top speed will be lower. But small motors run at 10,000 to 20,000 rpm so there is plenty of scope for a gear reduction.

If you want a device th fling balls you need to figure out the speed required to get the balls to go where you want. And then work onwards from that.

The brushless motors that are used in model aircraft may be of interest. Some of them are designed for high torque at moderate speed and the model aircraft speed controllers are usually easy to control with an Arduino using the Servo library.

I gather that you want to spin up some sort of wheel that will launch ping pong balls.

Yes, that is correct.

The ball's weight about 5 grams. The whole mechanism is just a ball going into two opposite spinning wheels similar to this

I guess to calculate the moment of inertia, you need the mass of the wheel and the radius? I haven't learned that in physics yet, so I am not too sure about it. Lets start with a mass about 22.6 g at a radius of 7 cm (I made an error in the formula again). Lets say the time is 1 minute (time is not crucial here, so this number can be played with if needed) and the RPM is 14000 rpm (the motors max is 15200, so I'll go with 14000 for now). Please also explain how you are calculating this so I can recalculate if numbers change.

I can't help feeling that a lot of this theoretical complexity is unnecessary for this project. Presumably, once up to speed, the wheels keep spinning so the only extra energy required is to overcome friction and to accelerate the lightweight balls.

What surface speed do you require to fling the balls out?

You need enough power to overcome losses and the load.

You need enough speed to go fast enough.

You choose the speed to be right, then enough power to do the job I think.

Small motor and large flywheel puts more requirements on the balance of the
wheel, note. The bearings need to be strong enough for the off-axis forces.

Larger wheel will have more bearing losses, note.

You also need to figure out the momentum / angular momentum needed for
propelling the ping pong balls - what's their mass and what velocity are you
hoping the balls to go?