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when rotating an object by a string a centripetal force from the string will act at the object towards the center and by Newton's 3rd law an opposite force will act at the string by the object . Then why it's said that the centrifugal force acts on the rotating object rather than saying it's acting on the string?

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You need to show where did you see this so we know the context. Without this, I could guess that the mention of "centrifugal force" you saw may be refering to the analysis of the motion in a noninertial frame. In this case a "fictitious force" is introduced, in the direction opposite to the acceleration (centripetal in this case). This type of "centrifugal force" is not the third law pair of anything as it is not due to the interaction between two objects. It acts on all the objects, when analysed in a rotating frame. The force on the string is a rel force, due to the interaction between the object and the string. And indeed it has a centifugal direction. Hovewer, it is not very common to see it described as the centrifugal force.

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The issue you have found here with Newton's 3rd law seemingly not being fulfilled happens because the centrifugal force is not a force. In reality no such force exists.

A force that does exist is a centripetal force pulling inwards in the rotational circle. And the string does feel pulled outwards while it exerts this force inwards. This is why weak strings will break when you swing something heavy from them.

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