A disc is rotating with angular speed $$\omega $$. If a child sits on it, what is conserved

A solid sphere of radius R is placed on smooth horizontal surface. A horizontal force F is applied at height h from the lowest point. For the maximum acceleration of centre of mass, which is correct ?

As shown in the figure at point O a mass is performing vertical circular motion. The average velocity of the particle is increased, then at which point will the string break

For a hollow cylinder and a solid cylinder rolling without slipping on an inclined plane, then which of these reaches earlier