im a hvac guy when im not wheeling. this should help. if the compressor was running for up to a minuite before shutting down, i would lean toward being low on refrigerant. many a/c units have low pressure switches that cut out when low. at rest, your pressures are equal, but when compressor starts pumping, the low side gets low, while the high side raises. if low side gets too low, switch will cut off the unit. then as the pressures equalize, it will let it come on again. this is a saftey to protect your equipment.
howeavor, since i think you said the compressor wont even come on, just trys, hums, and quits, let me point you in a differant direction. since you know its the compressor, not the fan, double check some stuff and verify. you said your reading 120 volts- most a/c units are 240 volts. make sure youve got a full 240( plus or minus 20%) to the contactor. then make sure you get the same voltage on the other side of the contactor when its pulled in. somtimes ants will get in there and cover the contacts. you should be able to physically see the contacts. if this is the case, or if you dont have but one leg(120 v) sometimes the fan will run but the compressor will try and fail. make sure you have both legs(240v) make sure your getting 240 from the house. thats the easy fix.
if that dosent help, heres the bad news. one of the purposes of the capacitor is to
"jump start " the compressor, because ,like any motor, starting it takes the most effort, and draws the most amps. which is why your lights in the house dim upon startup. if youve got the correct voltage, and a good capacitor, that tells me that the copmpressor is hving trouble getting started. when hvac compressors try and fail, their internal overload- built inside the shell- will trip. these will auto reset when it cools down. thats why you hear it try to start every so often. try a super boost capacitor- a special one to give it a kick in the pants to start.parts houe, ask for a malco superboost #spp6. if it still wont, your compressor is shot, locked up. not repairable. if its under warranty, dont tell the company that you worked on it. even though youve done nothing wrong, some will say youve voided their warranty. heck, if under warranty, let them work on it 4 you.
ps: you have a double capacitor- 40 mfd for the compressor, 3 for the fan. you should be ok there. they just share, instead of having 2 seperate.