There are several things that could be the problem. The motor not starting could be any one of several things.... bad capacitor, shorted or burned out coil, burned up relay, burned wire connection.
I have replaced a few motors and caps in my time but have also run into compressors with bad bearings that seized up when I put a booster cap on it to get it going. It made one heck of a racket for the 4 seconds it did run... then BANG!!! Silence...... OK... not good..... time to call my local HVAC guy.
While the HVAC crew isn't cheap, they do, for the most part, know how to troubleshoot and find the actual problem and get it repaired quickly, vs you trying part after part and since most stores will not take returns on electrical parts, you're out that money too.
My last case, with the compressor not starting.....I tried the booster cap.... $29, and that didn't solve the issue...... we ended up having to replace the HVAC gas pack due to it's age and the cost of replacing the compressor and still having a 20 year old gas pack setting there. I used that opportunity to have everything... including the ductwork under the house replaced with all new. They were in bad shape.
Most of the time, when my HVAC system fails, it's either raining, snowing, generally at about 10pm when it's dark..... it rarely happens in the morning of a nice spring day....... I had a blower fan go bad at 10pm on the coldest night of the year and there was an ice storm going on. My daughters had to use extra blankets in their upstairs bedrooms that night..... next day, I got a new motor and had the split unit blower running by noon.
It's always something..... oh the joys of home ownership.