Is it a louder hum? There are basically two main parts in the AC unit, a compressor and a fan motor. If the compressor is running without the fan this noise will get pretty loud. If the compressor is not running then the hum can be pretty low and steady and just be the sound of the contactor. Can you tell me which scenario fits better?
Can you feel the smaller copper line coming out of the outdoor unit and tell me what it feels like ? It may be hot so be careful.
Yes , if you could, about one minute after start up that line should be fairly hot if the compressor is running without a fan, that will help me tell whats going on there.
OK it sounds like you have either lost high voltage power to the unit, start by checking the main breaker and/or fuses , do you have a volt meter ? The other scenario is that the contactor ( what you said was a relay, is not functioning, this is very possible if it is a loud hum ) A volt meter will make short work of it.
OK, if you can open the electrical cover on the unit and you will see a contactor, you just want to measure voltage on the larger wires of it, if you get 220 anywhere you have power and the contactor has failed. If you don;t have 220 then you have a fuse or breaker blown somewhere. Be careful , you can turn the power off while removing the cover.
OK does it sound like the noise itself is coming from that relay ?
OK, shut the power down to it and look for a capacitor, its an aluminum part with wire connections on it, tell me how many terminals are on it it will be either two or three. Also try to note the condition of it physically if it is bulging out at the top, it should be in the same cabinet as the contactor. Don't touch those terminals even with the power off as it holds a charge.
Three wires & it looks to be in good condition. I have looked at the capacitor again. It not flat on top, it is slightly arched or bulged.
A uniform convex arch is normal , it does not sound like the compressor is running from what I can understand and that the fan motor is stuck and humming ? This would usually be due to a faulty capacitor , with the unit unpowered can you spin the fan motor ?
I would try replacing that capacitor, you must replace it with the same microfarad rated one as the existing , you will have two ratings on it because it is a dual capacitor. One side drives the compressor and the other the fan motor, it should say something like 35 / 5 on the microfarad rating, can you see that ?
Now what you want to do before touching that part is take a screwdriver and short out the terminals to each other after power is turned off. Once you do that you can handle it without any chance of getting a zap from the residual charge in it. Obviously , hold the screwdriver by the insulated portion and touch the terminals with the metal part.
Once you get a closer look at it you will see the terminals marked H and F and C , you have to mark all your wires and know for sure that you get them back on the correct terminal.
This is the best I can do from here without replacing it but capacitors are a typical problem.