Condenser motor won't start turning by itself, but if I turn it manually?

I have a split unit. A week ago, the fan motor stoped working. I went to buy another one and installed it. (I now know I should have gotten the capacitor too). I did not notice that the old motor was 1080 rpm and the new one was 825. (both 1/3 hp) . Yeserday, I noticed that the house ws not cooling again, compressor and fan would not come on. I removed the plate and I banged the dual capacitor with the screwdriver handle and the compressor came on, but not the fan. I turned the fan with a brittle piced of wood and the fan began turning. I bought a new capacitor and the fan motor will still not turn unless I turn it manually. I double checked most connections and breakers/fuses. all fine. My question is, do I need to add a start capacitor? if yes, why did it not need it for the first week? Or could the problem have been caused by the RPM difference, causing the capacitor to fail and therefore, the fan motor to go bad? why does the fan stay on if I turn it a bit? Plese help.