My condenser fan went out a couple of days ago. I prelaced the motor and the fan. I did not replace the capacitor. The motor will not turn unless I push the fan, then it runs, until the unit cuts off. I believe this was the original problem that caused the motor to burn out. What would cause the motor not to start on it's own?

The capacitor is the thing that gives the motor the little extra boost of curent tto gt it started running, it is a one time boost each time you start the AC. So look on the motor and see what capacitor it requires... it will say something like 5uf 370 volts.