Why single phase motor is not self starting?

In a single phase AC motor (at starting) there is only one pulsating field. For the sake of analysis only it could be represented by two rotating fields of equal magnitudes and rotating in opposite directions. If the rotor of this single phase motor is moved by any means, the stator field is still a pulsating field but the rotor induced currents will produce a rotating field and for the sake of analysis this rotating field is added to the component of the stator field rotating in the same direction. Therefore, the resultant air gap field rotating in the same direction as the rotor will be larger than the rotating component in the opposite direction. Consequently, the torque produced by the field rotating in the same direction as the rotor (which is has a larger magnitude) will be larger than the torque produced by the oppositely rotating component.

With the increase of the rotor speed, the positive torque will increase and the negative torque will decrease. The problem of initiation of motion at starting is solved by adding an auxiliary winding shifted in space 90 electrical degrees from main winding and this is connected to the same single phase supply but through a series capacitor to ensure the time phase shift between the fields produced by both windings. After starting the induction motor, the auxiliary could be switched off.

If you oversimplify the single phase induction motor so it is represented by a single electromagnet, when you apply AC voltage to it (the stator), you get a magnetic field that just flips back and forth from north to south 60 times a second. There is no rotation in the field in this case, so if the rotor isn't already turning, there is no force present to push it in either direction. When we place the start winding into the stator, it provides us with a slightly different set of north and south poles. The offset is determined by the difference in the reactances between the two coils. Now you have a primary north/South Pole with a secondary north/south that is slightly offset from the first (15-30 degrees) and gives you a weak rotating field which is able to kick the rotor into motion.