Why does the iBook shut off after boot up and login?

I recently got an iBook from a friend, but it had no power adapter. I recently bought one and it booted up fine. When it logs me in, it opens up the desktop, but right after that, it completely shuts down. I tried taking out the battery and booting on AC power alone and it worked like a charm. Is it a battery issue or is it a literal power struggle? Let it be said that the battery was dead when I got it and I haven't been able to open it up due to the fact that the pull thingy under the battery was broken when I got it.