i know that a capacitor with a motor increases the speed and torque, thats because when the coil in the motor is opposing the magnet the capacitor discharges to repel or attract the magnet and cause rotation, and when it is discharges, while the motor is rotating it charger again, this can happen thousands of times in a second.
°I can never remember which direction this rule of thumb leans.....insofar as ac phase goes, they say " resistance lags, capacitance leads"
The start coils on an AC capacitor-start motor - controlled by the centripetal switch, offer an additional field 90° apart from the run coils, to give the added oomph to get a static load moving. When the motor gets up to approx. 75% of rated rpm, the switch opens and kicks the capacitor / start coils out of the game.
A permanent split-capacitor motor, as in ones employed in an air-conditioner condenser fan, the capacitor affects phase, to give more start torque.
This from 8 years in HVAC / motor repair [ though not rewinding ] tho' give an old retired fossil slack for fading memory
Hmmm... Memory..... use it or lose it