Check the start capacitor. It's job is to give the motor a little extra kick to get it turning. When it quits it gives the exact symptom you are describing. It is about 6 inches long and a couple inches in diameter. It is located right beside the motor.

Thanks for the response Don. The start capacitor looked okay. I wasn't really sure what to look for, but all four wires were connected tightly. After further tinkering I think the problem is that the shaft coming out of motor with the worm drive on it does not turn freely. I have a similar unit on my other garage door and it turns very easily. Any thoughts? Possible that the shaft is bent??

I can't imagine that shaft bending. Maybe the motor bearings are worn out. The motor is expensive - around $100.00. Might be time to get a new opener.