This noise started to show up only when I was running Sims 3 World Creator which is pretty graphically intense. The computer really warm when it happened, so I downloaded some monitoring software and the numbers weren't great but not horrible either. So I just started shutting the computer off at night to let it cool down.
So after some time passed, it started making the same noise, shutting itself down more often and not just during gaming. I tried to find answers online and it seemed like, since my Radeon card was ~1 1/2 years old, it might need the thermal paste replaced on the heatsink, since it was always so hot. And the whirring noise did seem to be coming from the graphics card fan, which would slow down and kind of stutter. So I finally took the dive and tried to replace the thermal paste this last Saturday. I got the computer back together and it was working that night. Sunday, after working all day, I got home and, though the computer starts, the whirring begins immediately and not even the glimmer of an image shows up on the monitor. Is it very likely that it's just the video card doing all this? If I get a new one, will it take care of the problem? I would post more details if I can, but I'm on a work computer since mine is currently dead.
WAIT! If you are sure that the sound is from the Graphics Card (100% sure) then get a new GPU. There is a possibility that your CPU is overheating then shutting down. Check to see which is the problem.
I did have another GPU I could test on it. And it worked great! I thank you both for suggesting it! It just never occurred to me to try an old one, probably too busy fretting that I had killed the entire computer. With the older, but still working, card, the noise is gone completely and the computer is NOT hot! And its got a really decent picture too. Still plan on buying a new, higher end card but now I can wait a bit for it at least.