Electrical – Using an AC drill motor with DC

In general, it should be OK to run your motor from a 9V 1A power supply. The 300mA rating ought to be the current it draws while running. Stall current (how much current it draws if you block it from turning) may be higher, in which case it might get hotter than it should - possibly burning out if you leave it blocked too long.

If the motor has a thermistor, then the stall current will be limited and you won't have to worry about burning out the motor.

The site @codo referenced (Lego motor tests) gives more detailed information about the various motors, including which ones have thermistors.

Don't block the motor from rotating and you should be fine with the power supply you have. If you expect blockages to happen, use a motor with a thermistor or use a power supply that limits the current.

You probably have a single-phase induction motor. Variable speed is done all the time with 3-phase induction motors for industrial applications. The device that controls the speed is called a Variable Frequency Drive or VFD. Like it sounds, a VFD varies the frequency of the AC that drives the motor, and the motor changes speed accordingly.

Look at some industrial supply places to see if you can find a VFD with single-phase in and out. Once you get the hang of what's there, buy one from ebay so that you don't get the 100x markup of a new one.

Also note that a motor has to be over-built to some extent to handle a VFD running slower than its native line frequency. Those that aren't over-built will still work, but they may overheat more easily because the internal fan doesn't work so well at lower speeds.