1 or 2 capacitors on this motor?
Sometimes there are two...
One is a "Start" capacitor and if a 2nd one, that one is a "Run" capacitor.
If only 1 capacitor and you have replaced it... if not a faulty capacitor, then your windings are shot?
When I test a motor with this symptom, I will use the "Clock Face" method of checking for "Dead Spots" on the winding/stator:
Once you have removed the shaft cap on the back of the motor, I presume you have done this since that is only way to turn the shaft manually to get it to run in the manner you described. Place a small mark (1/4" long?) on the outside perimeter of the shaft, say at the 12:00 position. See picture attached...
Then flip switch, does it start or hum?
Then rotate shaft until your mark is now at the 1:00 position. Flip power switch, running or humming?
Do this at all clock positions as you turn/rotate shaft until you reach 12:00 once more. If the pump hums at all clock positions completely dead motor or bad capacitor. If a dead spot is at only one clock position, then absolutely, bad motor - either rebuild or replace.
Coincidentally the motor may stop each time on a dead spot, where the motor won't start due to a fried portion of the stator/winding/armature, only a major repair fixes this. The purpose of the test is to place the position of the stator in a different position in relation to the windings. If humming all the way around, then possibly a bad capacitor, because it is not starting in any position. If it starts say at all positions except 4:00 and 9:00 then motor is bad.
Good luck
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