It's probably the brushes on the motor. When they wear down they stop making contact with the commutator and no electric current flows into the motor. The symptom is usually the motor does not turn at all, or only turns if there is no load (eg it turns when there are no clothes in, but won't turn with clothes in or when spinning).
Have a look at the brushes - are they very short? If so, its time for a new set!
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