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If the furnace fan runs for a short time then stops, it sounds like the bearing might be seizing up.
Upon start up, a motor with a bad capacitor will hum but not start. If you manually start it, it will typically run until you turn off the electricity to it. Then on the next start up it will just hum again.
I would replace the capacitor first and see if that resolves the problem. Capacitors are much cheaper than a new motor and you will need to replace the capacitor anyway if you replace the motor (that is the motor manufacturers recommendation).
So, replace the capacitor first. If the same symptoms occur, replace the motor as well. Then all will be good.
Good luck. Let me know your results.
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