I love Dave Smith's simple, elegant solutions. And yet if I can find an excuse to make something more complicated - I guess I've got the Rube Goldberg gene - I can't resist; after all, we're on the Hobby-machinist website, right? So I'm a fan of treadmill motors, cheap, reversible, easy speed control, use a footswitch (or two if you want to reverse without taking your hands from the tools). $50 used motor, $15 router speed control from HF (or heater control even better - slower minimum speed), $5 full-wave rectifier and $10 footswitch and $? lazy Susan from Amazon, and voila!
Two comments on other ideas: First, I've tried wiper motors and security camera motorized mounts set to scan back and forth to turn a wash pot full of solvent back and forth, and it was really too much for either to handle continuously, plus the wiper was a little jerky at the end of travel. I don't think the wiper sweep comes close to a sine function, more militaristic as in "hurry up and wait". Second, potentiometers alone often don't work out because you need pretty high current ones, $$ and hard to find; don't work with most AC motors and DC motors have low torque with them. (Motor speed controllers use very low power potentiometers in a circuit that pulses the current at a high enough value that it maintains torque even at low speeds.)
Two comments on other ideas: First, I've tried wiper motors and security camera motorized mounts set to scan back and forth to turn a wash pot full of solvent back and forth, and it was really too much for either to handle continuously, plus the wiper was a little jerky at the end of travel. I don't think the wiper sweep comes close to a sine function, more militaristic as in "hurry up and wait". Second, potentiometers alone often don't work out because you need pretty high current ones, $$ and hard to find; don't work with most AC motors and DC motors have low torque with them. (Motor speed controllers use very low power potentiometers in a circuit that pulses the current at a high enough value that it maintains torque even at low speeds.)