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Let us suppose an object is performing vertical circular motion in the first quarter(where the possibility of the object being detached is $0$). At an instant,let the tension of the string be $T$ and component of $mg$ be $mg\cos \theta$. Now,we know that $T-mg\cos \theta$ must be equal to the centripetal force. Is there any possibility for $mg\cos \theta$ to be greater than $T$ which would imply that the centripetal force becomes negative?Thanks in advance.

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If the rope breaks, then $T$ will quickly fall to zero, and the object will simply fall like a projectile under the influence of gravity.

When you talk about a negative centripetal force, be careful. The centripetal force just changes a particle's direction but not its speed. The centripetal force always points towards the instantaneous center of curvature (in the case of circular motion, the instantaneous center of curvature is the center of the circle). If the tension $T$ falls to zero, the centripetal force determining the object's path will simply point to a different center, but calling it negative doesn't make much sense.

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