I have two wires that originate from the compressor (yellow and black). These wires connect to the contractor at the opposite poles. When I put my meter on them to check for resistance, I get a beep which indicates a short. This leaves me to believe that the compressor is bad.
I presume you meant to say: "These wires connect to the CAPACITOR at the opposite poles."
I would agree with you ONLY if the compressor motor was running at the time you checked the resistance across those wires. Otherwise, to my way of thinking, you should have gotten that beep because you're essentially measuring the resistance of a short circuit. Here, go to this page:
Single-Phase Hermetic Motors
and take a look at Figure 14-26.
That diagram shows a typical capacitor start motor wiring diagram. If the motor was not running at the time you checked the resistance between the yellow and black wires, then the start relay would have been closed, thereby including the start capacitor and the motor's start winding in the circuit.
(Once the motor gets up to speed, that start relay cuts the start capacitor and start winding out, and the motor continues operating on the run winding alone.)
Inspection of Figure 14-26 shows that measuring the resistance across the yellow and black wires will give you the resistance of the start relay coil, the run winding and the start winding all in series. You can see how a multimeter which operates on DC voltage and current could misinterpret this as a short circuit. It's just a long wire looped into three different coils, and it only makes sense and works properly with AC voltage and current. To a DC tester, it looks like a short circuit.
To properly check your compressor's motor for a short circuit, you need to access and remove the electrical connector that plugs into the side of the compressor. Once you remove that connector, you should see THREE electrical terminals of some type. One terminal goes to the start winding, and the end of the start winding wire goes to the "Common" terminal. Another terminal will go to the run winding and the other end of the run winding wire will go to that same "Common" terminal. And, the third terminal you see will be that "Common" terminal. Thus, inside the motor the start and run windings are connected in parallel to the common terminal. The electricity goes into the motor through the first two terminals, and the wire that connects to that common terminal takes the electricity back to the generating station (to use a hydraulic analogy).
The way you test your motor for short circuits is to measure the resistance between each of the three PAIR of terminals (S to C), (R to C) and (S to R). Two of those resistance measurements should exactly add up to the third. If they do, then there are no short circuits in your motor's windings. That's simply saying that measuring the resistance of the start winding individually and the resistance of the run winding individually should add up to the resistance of the start and run windings in series.
Also, check to ensure you don't have continuity between any of the terminals and ground. That's a problem for sure.
But, you'll need a multimeter that can measure small resistances without presuming it's a short circuit.
If the resistances measured this way don't add up, THEN you can legitimately conclude there's an electrical problem in the motor.
Also, hermetically sealed electric motors (that is, motors that are inside an airtight container) used in air conditioning and refrigeration will often use something called a "permanent split phase motor", which will use a run capacitor on the start winding, and that start winding will remain in the circuit all the time, not just while the motor is starting. Go to:
Split-Phase Hermetic Motor Windings and Terminals
and take a look at Figure 14-28.
Since the start winding remains in the circuit all the time the motor is running, permanent split phase motors don't have a start relay to cut the start capacitor and start winding out of the circuit once the motor gets up to speed. (This figure more clearly explains why measuring the resistance between the black and yellow wires may give you a low resistance, but it's certainly not a short circuit for AC voltage and current.)
So, phone up the company that made your air conditioner, or their local factory authorized service depot, and ask them if that's a "start" capacitor the yellow and black wires go to (in which case you have a capacitor start motor) or if it's a "run" capacitor the yellow and black wires go to (in which case you have a permanent split phase motor). And, while your talking to them, ask what you need to do to access the Start, Run and Common terminals to the motor on the compressor housing so you can check to see if the resistances through the motor windings add up, and so you can test for shorts to ground on it.
Also, I checked the capacitor and it appears to be fine. No swelling or shorting between the terminals. Interesting though, it is not registering any voltage.
You can't "check" a capacitor visually. I'd take it to any place that repairs electronics and ask them to check the capacitance on it to see if it's still good or not. A bad capacitor on a capacitor start motor WILL cause the circuit breaker tripping problem you're having, so it's critical that we have confirmation that the capacitor is good before proceeding. It's the most likely suspect here.
Also, be careful with capacitors, especially on television sets and CRT type computer monitors. These capacitors are very strong and can hold enough of a charge to give you a real good shock (and potentially kill you if Murphy's Law has any say in the matter).