Answer to "Wont boot after running out of power"It's the easiest thing to solve, first remove the battery and the AC adaptator , and press power button for 10 second . plug in the battery and the AC adaptater and start it up.
Answer to "iBook g4 stops on the apple logo when I power it up"First, it's dangerous to use a computer without battery, cause when the AC adaptater is unplugged , the machine suddently shuts down, and this is not recommanded for computer health ( It can damage files, hard driver, or something else) . So don't do it again. According to what you tell us , I can say that there is 2 possibles causes: 1-Your files has been damaged, so that Mac won't boot 2-Your RAM Memories can be damaged too. Try this , and tell us more about after doing them: 1-remove the RAM memories and try each one of them separately (plug one and try , then plug the other and remove the first one and try again. 2-At Start up , press "Commande+S" to boot in single user mode and type "/sbin/fsck -fy" exactly like this ( there is a space between "k" and the dash. Do it until you get this message : The volume _name_appears to be OK! and type "reboot" to restart the mac. 3-if it won't boot, you will have to reinstall it . 4-Force mac start up holding "Commande+control+powerbutton" If all this have help...