I have the same problem with our G5. I check the cord and it's good. Changed the switch, but that actually wasn't the problem. I've tracked my issue down to the motor brushes. When is does run, sometimes blue sparks are seen on the right-side of the motor near the right set of brushes.
I was able to rotate the motor spindle simply by turning the front attachment belt lever (the thing on the front to take off the vacuum head). This rotates the motor just enough to get better contact with the motor brushes.
I'm next going to replace the brushes and see if that fixes my g5. Good luck.