Furnace Blower Motor fan hums but wont start until I turn it by hand, then it runs fine. What should I check?
Yeah, could be the capacitor. A good quality multimeter with the ability to test capacitance can check it for you.
Discharge the cap, and remove it for testing (be sure your furnace is off). Then, set your multimeter to the F/uF setting and test it. The cap should show the cap rating in micro Farads (uF) and a +/- percentage. The reading should be within that range.
That said, I had the same issue. Cap was out of range (measured 9.6 on a 12.5 +/-6% cap). Replaced cap, motor seemed fine, but shortly thereafter would sometimes not start. I pulled the motor out, and dismantled it. The commutator was gummed up a bit, so I cleaned it. Worked ok for a bit, but the furnace would still sometimes limit out (probably because the motor wasn't starting). Ended up replacing the blower motor, and it works fine. Furnace was 23 years old, so I guess I can't complain.
Moral of the story is that it could be a combination of both. And, btw, if you replace the motor, always replace the cap, too.
Before I would check for any of the potential electrical issues I would try lubricating the motor shaft. Shut off the power to the furnace. Gain access to the fan and motor. Spray a small amount of lubricant (slick50,chain lube, etc) almost any light oil will work, on the shaft where it exits the motor. Some models may even have been equipped with a small oil port from the factory. You may have access to the opposite end depending on the design. Spin the fan by hand a few times. Apply a small amount of oil again. Wipe off any excess, reassemble and see if the fan starts.