single phase motor

hi every body

i understand that the three phase motor turn because the rotor is following the magnetic flux of phase A,B,then C and so on.

but i do not understand about single phase motor, normaly there is run capacitor connected on series with the start winding.

is that capacitor used to delay the flux of the start winding? so there is flux on run winding then on start winding?

also i noticed that the motors which drive light loads '' fan motors'' and it has only one speed ''high'' has no run capacitor, only centrifugal switch and some times start capacitor.

is that because motor with run capacitor stronger than the one without, when it is running at low speed or because the motor without will stall when it is loaded and running ot low speed or it is the design?

please advice