Small Engine Troubleshooting Problems? Ask a Mechanic Now.
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From what you are describing it sounds like the battery is too low to crank the engine. Can you manually turn the engine over by hand?
The high pitched humming noise is most likely coming from the starter solenoid which is normal when the battery is low. I would have the battery load tested to see what kind of voltage it holds under a load. Even with a new battery it is possible to get a bad one or one that is weak and not fully charged.
You can turn the engine manually by gripping the flywheel screen on top of the engine and turn it. You could also turn the engine manually by using a wrench on the bottom PTO of the engine and turn as if you were trying to tighten the bolt that holds the pulley on.