please show me how to solve symbolically while solving the problem

10. A playground carousel is rotating counterclockwise about its center on frictionless bearings . A person standing still on the ground grabs onto one of the bars on the carousel very close to its outer edge and climbs aboard . Thus, this person begins with an angular speed of zero and ends up with a nonzero angular speed, which means that he underwent a counterclockwise angular acceleration. The carousel has a radius of 1.68 m, an initial angular speed of 3.12 rad / s and a moment of inertia of 123kg * m ^ 2 The mass of the person is 48.7 kg. Find the final angular speed of the carousel after the person climbs aboard. 11. A small 0.330-kg object moves on a frictionless horizontal table in a circular path of radius 1.90 m. The angular speed is 8. 13 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 114 N, what is the radius of the smallest possible circle on which the object can move? 12. A solid disk rotates in the horizontal plane at an angular velocity of 0.046 rad/s with respect to an axis perpendicular to the disk at its center. The moment of inertia of the disk is 0.077 kgFrom above, sand is dropped straight down onto this rotating disk so that a thin uniform ring of sand is formed at a distance 0.35 m from the axis. The sand in the ring has a mass of 0.36 kg. After all the sand is in place, what is the angular velocity of the disk