Love the last answer. I get it that this is a pool pump. The reason they won't start is 1 the pump is seized, or 2 the start capacitor is faulty. If you take the rear cover off the motor you will see the end of the armature in the center, it usually has a slot to use a flat blade screwdriver to see if you can turn the motor shaft. If you can turn it then the motor is not seized. Then I would be looking at the start capacitor as the problem. Its normally on the left side behind the rear cover. A black round cylinder about 3" long with 2 wires connected at the bottom. If it is faulty the pump will just hum and not turn.

The type of motor that uses a centrifugal switch, a brushless induction motor is considered too large and heavy for a portable vacuum cleaner though they are often found in permanent installations.

Portable vacuum cleaners use a series wound motor that is considerably smaller and lighter for a given power output and turns many times faster.

Both types of motor will hum if seized or locked and in that condition the series wound motor will soon destroy itself. It takes a little longer for the induction motor to overheat...

The centrifugal switch is not fitted to all induction motors and that is usually very reliable and rarely gives trouble except when unused for a long time in damp conditions.
Many single phase induction motors with or without the switch use a starting capacitor and sometimes a start/run capacitor. Often the capacitor is of inferior quality and will fail with the result the motor will just hum, though will start when the shaft is spun by hand.