A single phase induction motor won't spin up without a push. This is
because until the motor is spinning there is no ratating magnetic
field. There are many ways to do this. One way is to use a starting
capacitor. This cap is connected to a different winding, called the
start winding, than the run winding and causes a phase shift, about
90 electrical degrees to the run winding. This gives the rotor a
direction to start spinning. There is a centrifugal switch that takes
the cap out of the circuit once the motor is about 85% of rated speed.
If left in circuit the start winding would overheat. Another cap
scheme is to have a cap that starts the motor and stays in circuit.
This scheme doesn't provide as much starting torque but does have the
advantage of simplicity and making the motor run smoother. There are
also motors with switched start caps and always in circuit run caps.
Eric