My TV will not turn on. When I push the power button, I hear a click and what sounds like a hard drive spinning up and then slowing down. After spinning for about a minute, it turns off. In about a minute, it clicks and I hear the hard drive spinning up again. This happens again and again. But still, no picture. I'm assuming I need a new ballast, because I checked the bulb and its still good.

After you pushed the on button, did you hear any sound (like spinning sound)? pay attention on hard drive, did you see and flashing at all? Hard drive might just need to be reseated.

Can you hear the hard drive spin up for like a second and then turn off all of a sudden

so you've already taken out the bios battery and held the power button on for a couple of minutes before replacing te battery.
make sure the cpu heat sink isn't block with dust and fluff
next remove the ram boards and put them back in
if you still have the problem it's probably the power supply or motherboard (in that order)

maybe the power button is done thats what it sounds like not keeping the power on

Its in your logic board controller issue, usually the problem is like that but to see what is the another problem of your External Drive is to make or use any diagnostic freeware program to identify the device where it really connected in your PC. I have also encountered like your problem but I send my External to Electronic Tech to check the power connection and controller connection going to PC. Because you said that the hard Drive is still have a normal sound then it is working then but cannot see in your Computer. Try to Diagnose first before you bring to a Technician. Good luck!

Check the settings for power saving. Perhaps the system is trying unsuccessfully to shut the backup drive down. Putting the backup drive to sleep when it's not being addressed is actually a good idea and extends the life of the drive tremendously, it is afterall for backup so you want it to last. For example, I have a second drive in this Linux box, It comes to life only when accessed.

sounds like a fly back problem
have a qulifyed tech check this out its high voltage leaking you are hearing.
esp if its a sqweek hiss sound.

plz resit the ram

Sounds like a hard drive crash. Locate where the hard drive is on your machine and listen closely when you hit the power button. If you don't hear it start to spin, or if you hear any clicking or dull thunking sounds, then the hard drive is shot snd needs to be replaced.

Sounds like the hard drive crashed. Make sure the hard drive is seated back in properly, then hit the power button and listen closely. You should be able to hear it spin when the hard drive engages. If it does engage, you should hear it spin smoothly, if there are any clicking or dull thumking sounds, then it's shot and needs replacing.