I tried to repair a 2-speed table-top house fan that has a 120VAC motor that looks a bit like this.
(I no longer have the fan since it was returned to the owner.)

I have two questions that I genuinely cannot understand.
1) There are three connectors to the coil to select low speed and high speed. What is the physics behind the 2-speed operation?
2) When I try to power the fan (no changes made to the wiring from the switch to the motor) the motor just vibrates and does not rotate.
Trying to get it to spin by hand does not work.
(I no longer have the fan since it was returned to the owner.)

I have two questions that I genuinely cannot understand.
1) There are three connectors to the coil to select low speed and high speed. What is the physics behind the 2-speed operation?
2) When I try to power the fan (no changes made to the wiring from the switch to the motor) the motor just vibrates and does not rotate.
Trying to get it to spin by hand does not work.

