Angular momentum and conservation of angular momentum problem

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A small 0.531-kg object moves on a frictionless horizontal table in a circular path of radius 0.85 m. The angular speed is 6.30 rad/s. The object is attached to a string of negligible mass that passes through a small hole in the table at the center of the circle. Someone under the table begins to pull the string downward to make the circle smaller. If the string will tolerate a tension of no more than 105 N, what is the radius of the smallest possible circle on which the object can move?



conservation of angular momentum
angular momentum = moment of inertia x angular velocity
moment of inertia = mass x radius squared
torque = force x lever arm
torque = moment of inertia x angular acceleration
maybe more equations I'm not sure.



I tried to use the conservation of linear momentum equation since you can find the momentum of the object when it is going around in the outer circle. I don't know what to do with the force or how to work it into the equation to solve for r of the little circle.
 

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