Any constant-speed motor will work fine for spinning the platters.
The sculptures seen in this first image were made by 3dscanning and 3dprinting my head.
Step 1: Phonograph Turntable and Strobe
It has a knob that adjusts the rate at which the strobe flashes.
The turntable you want is one that spins at 78rpm.
That's the old victrola speed which is 1.3 revolutions per second.
If you draw 8 evenly spaced cels of animation and adjust your strobe til it flashes about 10 times/second you'll see 8 copies of your animation all going at once, each at a different phase of the sequence.
Step 2: Turntable Alternative
Use a variable dc powersupply to adjust the voltage that drives it to change how fast it spins.
The speed won't be perfectly steady but it will be good enough.
Most dc powersupplies have a voltage adjustment somewhere in their guts if you look around. Don't get electrocuted.
This motor is a 24 volt Pittman GM8713G883 with a 31:1 gear reduction. I'm happy with it.
Step 3: 3Dprinted Zootrope
Then 3dprint it out and point a strobe at it. I used a Zcorp 3dprinter to print this one way back in the day with experimental materials. Zcorp parts look a whole lot better now.
The zcorp machine is fast and inexpensive compared to other systems. I'm a founder of the company so of course it's awesome. http://www.zcorp.com
Pixar has been making tons of 3d zootropes on Zcorp printers lately.
Step 4: Strobo Claymation Playground at Sea of Dreams 2010
We were a huge hit. All night long people were clustered around making claymation platters.
I made this one of a swimmer diving in and out of the water.