Question--computer just shutting off...

I need help--hoping maybe someone has some insight. My MacBook is about 2 years old and has started acting up. If I unplug it from the power source, it shuts itself off after about 5 minutes. The battery reads as "normal" and having a charge. It's fine as long as I have it plugged in, but if I unplug--forget it. Anyone have anything similar happen or have any suggestions?
Thanks.

Have you tried resetting the SMC? Here's a link explaining how to do so:
Intel-based Macs: Resetting the System Management Controller (SMC)
Good luck, and I hope it's a simple fix!