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it is that displacement that we feel as tension and centrifugal force.
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That's centripetal force. It acts through the rope, and is different from gravity. It's simply a force.
Centrifugal force acts on every single bit of matter, exactly like gravity. Both, gravitational force and centrifugal force, are fictional forces. They vanish if you calculate things in a local inertial frame, where F=ma.