arcadiafilm, Motor does not start, but makes a buzzing noise then it may be faulty. To check motor, remove the drive/pump belts and try to start the motor. If direct drive model, disengage the pump/direct drive coupling at transmission, if the motor starts, problem is with the transmission or pump and which can be further diagnosed by reconnecting each one individually. If the motor does not start or continues to hum or buzz, then it will need to be replaced. Some times the pumps will accumulate debri or piece of small clothing wrapping around pumps propeller stopping motor from spinning. If you do not feel comfortable working on machine have a good repair Tech. have a look, diagnose and estimate repair. Check the attached links,instruction and guides, Good luck.
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Kenmore 110 Series Washing Machine Drive Belt Replacement
Kenmore 110 Series Washing Machine Water Pump Replacement