I agree much with Default, yet there is one general rule for any
induction motor: They are not variable speed. You can select
different speeds based upon the wound poles, but to cause the motor
to slip excessively is not an operable condition. The only way to
safely adjust the speed of any induction motor (aside from
shaded-pole) is to reduce frequency and voltage amplitude on a
measured scale. Shaded-pole motrs are horribly inefficient, but more
tolerant of moderate overloads.

Once the stator field loses grip on the squirrel-cage, the losses are
converted to heat in the stator as a result of decreased
inductive-reluctance in the stator, which results in increased
current. The heat will be turned on the resistive value of the
windings rather than the inductive properties that normally control
the current, and hence it will overheat, meaning the motor will lose
it's inductive properties and become closer to an unwound length of
wire to the line current....

If you are intending on varying the speed other than preset values for
an induction motor that are avaliable, you best either use a DC power
motor driven by SCR's such as a treadmill motor, or a brushless DC
motor, which is phenomenally-more expensive and rarely worth the
project.

My enthusiastic advice here is to either repair or replace the motor,
or convert yourself to DC. Then and only then will a
duty-cycle/amplitude modulation be useful for adjusting the speed.

In any case I advise anyone to get speed controllers from discarded
treadmills, they are the diamonds in the rough for anyone working
with anything DC-compatible, replacing the lab variac for DC
applications. Just be sure to salvage or at least know the pot
(varistor, not weed ya goofball) value for it, as guessing can be
hazardous to the controller. I now have one rated 28 amps DC
145+V-out, detuned to work on a standard 115VAC circuit at 15
amps...The motors couldn't be any more ideal for any application
anyway, so grab those too!

Gold I tells ya....gold....grab them like they are the last power
supply on Earth, because most of the cost is in these parts anyway...