iPod only turns on when in the wall outlet not in computer

PATTY PERRY, just because the battery indicates it is charged when plugged in, that does not mean it is holding a charge. If you have never replaced the battery, and it is still the original battery, it most certainly has come to the end of its lifespan. The reason why it comes on when plugged into an AC charger is the higher amperage supplied by it. The USB on your computer only supplies a max of 500mA and an AC charger 1000mA (1A) Replace the battery first, if nothing else, it will give you a known good starting point for further trouble shooting. Follow this guide to work on your iPod and the battery is available at multiple places. Just make sure that you get the one that fits your iPod. The 30GB has a different size battery. Hope this helps, good luck.