Notebook shutting off when idle

Here's how it all played out:

About a week ago my cord stopped charging the notebook, but it would still run when plugged it, it just wouldn't charge it, then on Thursday it suddenly decided to charge again. I left the house came back and it as completely dead. I couldn't get it to come back on, the plug very very faintly glowed light green but not really. I went out and bought a new cord. Came home plugged it in and it worked fine, it looked as though the battery had just run out as it restarted with all the applications already running including the note about the battery. Then about an hour later I got the do you really want to shut down your computer window without my having prompted it, I said no. Then it came up another time or two, I said no. Then I set it aside to do something else, and when I came back it was shut down entirely. Not just asleep. I experimented and found it's shutting down after about a minute of being idle. If I'm using it there's no problem just as soon as it's idle. My energy saving preferences aren't the problem, I tried that already. Anyone have and fix this problem?

I have the early 2008 macbook