With the ac single field and no capacitor it does not rotate at all.
There is zero net torque.
However, if you spin it in one direction it stays there and starts
rotating and continues in that direction. My simple question was why?
With a capacitor you create two vectors of flux which add and give a
rotating magnetic field. That much is everywhere on the net and was
discovered well over 100 years ago. And that idiot also mentioned
shaded pole ac motors which also has a net rotating flux. It's my
first point that is not well explained. Actually even the book
explanation that I gave earlier isn't exactly compelling.
Ah, finally you ask a coherent question.
Imagine that the rotor was a bar magnet, perpendictular to the shaft.
Apply the AC current to the single stator coil, making a single-phase
field. The magnet will just sort of buzz, with no usable torque
applied to the shaft, unless you want a very weak vibrator. [1]
But spin the shaft up to 3600 RPM. The magnet now aligns with the
field at both the positive and the negative peak of the 60 Hz AC
cycle. If it gets a little ahead of or behind the time peaks, a torque
develops that tries to pull it back into alignment. That's a
synchronous motor. It won't start by itself, but it does fine once
you're up to synchronous speed. Think of the field as a stroboscope
that's illuminating the rotor only at its current peaks; that image
looks like a DC field forcing the bar magnet into one position.
Now replace the magnet with a cylinder of moderately conductive stuff.
At 3600 RPM, the field essentially magnetizes the rotor every time it
peaks, and the magnetization makes it act like the magnet in the
synchronous motor. The "magnetization" is actually eddy currents
induced into the rotor, and the best rotor isn't solid, it's shorted
turns of copper over a laminated steel core. Since the "magnetization"
isn't fixed like the PM rotor, it sort of slides around, so the
induction motor doesn't, and doesn't have to, run at synchronous
speed.
More detail tends to be mathematical.
Oh, don't thank me, I'm going to write a book on electronics for
beginners, and I need to practice explaining simple stuff to idiots.
John
[1] One interesting sort-of-countercase being the single-pole
induction motors that spin microwave oven turntables. They self-start
in a random direction.