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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.