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If your dryer won't start, but the start switch is not the problem, pull the dryer away from the wall and check the power cord and plug. If the cord is frayed, burned, or otherwise worn out, you will need to replace it with the appropriate Kenmore dryer part, Kitchenaid replacement part, etc.
Some people get concerned when they see a burned power cord. The source of a burned cord can include a short circuit against the dryer housing. The burned cord is a fire hazard and should be replaced as soon as possible. Remember to get replacement dryer parts for both the power cord and the "terminal block" that provides the connection from the power cord to the dryer itself.