Had to attached my pool pump to an extension cord due to faulty outdoor outlet. Pump hums for a short while and then stops. I read another post that warned against using extension cord (even though the one I am using is heavy gauge ... it is at least 50' long). Have I burned the motor up?

Maybe not, I don't think it would hum. It usually hums when there is debris in the intake. You need to remove the pump and with a pressure hose flush out the intake on the bottom. See if that does the job.

Capacitor-start motors will hum when the capacitor blows. Is it a capacitor-start motor? They hum because they don't have a direction to turn. You can usually get them to start by carefully spinning the motor by hand.

that is how i start my grinding wheel. what can i do to fix it? i don't have to fix it 'cause i just jump start by spinning a grinding wheel and it and it goes.

"Faulty outdoor outlet" ??
Motors can make wreched humming sounds if there is a electrical issue in the windings, or wiring. Are you able to test the feed circuits? Is it tripping the breaker? I agree with other posts, but make sure there is not a major electrical problem.
It could be siezed too, which would also explain your problem but may not trip a breaker until it starts burning wires. If your extension cord is getting hot, unplug it!!! That is the main reason not to use cords, pumps do alot of work, a long 14ga ext cord may not be rated for your pump.
Safety first, then swimming.