Can you use a rheostat to control the speed on a single phase(PSC), fractional horsepower motor? If so, how does it control speed without changing the frequency?
You can do it on motors with a high rotor resistance. You simply increase slip when you change voltage. A triac or a transformer with taps (or Variac) is more often used than a resistor. Suitable for fans and ventilators where power is speed^3
As Gunnar said, it is a pretty common method to control fans (like ceiling fans). The torque required to turn a fan increases very rapidly with speed so a rheostat (or capacitor) which limits torque will reduce speed. The high rotor resistance keeps the motor from overheating at high slip.
Stator side resistor works for NEMA Design D motor which has a curve of torque vs slip which is monotonically increasing.
Would not work nearly as well with typical NEMA Design B motor with a breakdown torque peak a few percent below sync speed.
Brad even though it's done on fans that doesn't mean it will work in any other application - it won't. However you haven't said what your application is.