I would suggest using an old furnace blower motor

Update:30 Sep 2018
Summary:

I would suggest using an old furnace blower motor. I li […]

I would suggest using an old furnace blower motor. I like these because the basic version might be 1/6 horsepower. This seems low, but that’s on purpose. You will run it continuously and you don’t want to run up a big power bill. They are pretty energetic, similar to other fractional horsepower motors. Some models offer three speeds. But those usually require forced air (those motors are meant to mount inside the squirrel-cage blower). I prefer an older style motor, one that mounted beside the squirrel cage blower. These used a V-belt for coupling. One friend wanted to use bald tires as tumblers. He envisioned a few tires rolling side by side with different grits in each, with a final walnut-shell finish tumbler included in the lineup.

What kind of motor should I use to spin a large homemade tumbler (see comment)?

This is going to seem really odd, but I would use a dynamo from an old car. Needs to be an old car - before alternators, say 1965ish. From a scrap yard.

These can be run as DC motors, and are really easy to controlspin motor  with a (high wattage) resistor in the field. Massive torque for starting, and can run at say 200watts for ages. A 12 volt supply from a truck battery.

Reason i say this is that buying an induction motor, and an inverter drive to give speed control and reversing is going to be expensive, which might cost more than you saved by building the machine yourself.

Also, by using 12 volts, you get rid of the voltage related safety concerns with using mains power.

Maybe, if your project is a success, then you could think about spending money on a mains power set up, but certainly at the experimental stage, keep the costs down.

You could use the pulley wheels and belt from the car, maybe experiment with the different sizes of pulley found on the car to fit to your machine to get speeds you like.