IF it's an induction motor as opposed to a universal motor, the problem is probably either the starting capacitor or the contacts on the starting switch (which cuts in the starting capacitor) is fouled by dust.
If it's an induction motor, try blowing it out with compressed air.
A test. Disconnect the belt (if it's belt driven) and give the motor a good spin. While it's spinning, turn the power on and see if it starts. It doesn't matter which way you spin it.
If there's no belt, it's probably a universal motor.
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