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Can a pinned rod (not just its particles) rotating around that pinned spot be considered undergoing circular motion? That is, is it rotating because after it has received a torque, it starts moving away from the pinned spot and the pin exerts a force on the object, accelerating it inwards and acting as a centripetal force?

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Yes, that's a valid interpretation. If we view the rod as a system, then the center of mass of the system is accelerating towards the pin at all times, and so there must be a force exerted on the system towards the pin. This force is, as you surmise, exerted by the pin.

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