I agree with Tan on this one. I have had this same issue a couple of times. For me, it has been the CPU fan and the Video card fan that have failed. You are describing what would happen if the CPU fan failed. They will often make a progressively louder noise up until they actually fail. You can do as Tan said and open the case and simply look at the CPU fan while you push the power button to turn on the computer. It should start to spin Imediately.
You can easily replace the CPU fan itself, or buy a completely fan/heatsink kit to replace your own. The fan/heatsink combinations can be found at just about any store that deals with computers. If you just want to replace the fan itself, you can usually find those at places that tend to sell computer parts or repairs more esily than if you just look at the bigger stores. At least this has been my experience. The fan itself will probably sell for $5-10, and the fan/heatsink will probably run you $15-20. You can definately get more expensive items if you want
